Monday, December 17, 2018

[zomozzyk] Depletion of a social class

Some social class decreasing in population within a stratified society seems a somewhat common occurrence, perhaps with enough instances to draw general conclusions about how society responds.  How does society respond?

Examples of depletion of an upper class:

  1. Eastern European brain drain during the Cold War, especially prior to the construction of the Berlin Wall
  2. Katyn massacre
  3. Civil rights reform in America allowing the black middle class to leave segregated black neighborhoods.  The black middle class were an upper class relative to the black lower class, the latter who then suffered in urban decay due to the departure of the black middle class from their neighborhoods.

Examples of depletion of a lower class:

  1. Some barrier to upward social mobility becoming removed or decreased.  (Examples?  Maybe public education.)
  2. High-casualty war fought by soldiers conscripted from a lower class.
  3. Availability of abortion decreasing the incidence of women (and families) becoming demoted in class due to the financial burden of raising a child (that they don't want).  Most abortions are for economic reasons.  Children raised by a parent or parents who don't want them or are unable to support them are less likely to do well in society, so would have gone on to populate a lower class.

This last one was the inspiration for this post.  Fear, uncertainty, and doubt about the large-scale social effects of abortion could explain poltical support of making or keeping it illegal or hard to obtain.  Does depletion of a lower social class cause to the classes above increased pressures and incidences of class demotion?  It seems like it would have to.  This would yield a predictive model of who supports or opposes abortion: those who are most insulated from class demotion will support it.

Previously: (1), (2)

[ucpfnmju] The villain travels backward in time

The hero and villain are (somehow) the same, except traveling in opposite directions in time.  Making the world a better place from one point of view appears as making the world a worse place from the other's point of view.

Inspired by particles and anti-particles in particle physics.

There is of course an explosive confrontation when the hero and villain meet.

Saturday, December 15, 2018

[rtnsyqie] Inline shower temperature

Create a device that measures the temperature of the water being emitted by your shower.  Shower heads have standardized connectors, so it should be easy to produce a device that goes in line (in series) before the water goes to the shower head.

The standardized connector also allows easy creation of a valved T junction sending shower output to a garden hose, which also has a standardized connector.

The moral of the story is, interesting things become possible with standardization.

[icbyldfi] Regions have volume and surface area

We consider games of conquering regions, e.g., generalizations of the game of go 囲碁, perhaps higher dimensions or on a general graph.  Any connected group (a chain) has a volume (number of stones) and surface area (neighboring points not part of the group).  What else?  Much graph theory: diameter, vertex connectivity, edge connectivity.

Between two disconnected groups of the same color, the difficulty of cutting them or joining them.  Min-cut, max-flow.

Inspired by the difficulty of depicting on a 2D screen who owns what within a 3D space conquest game.  The graph theory statistics listed above could be provided to the player by a computer UI.  The game could be designed so that those graph theory statistics are relevant.

[zjrqfsdh] Adjacent Delaunay graphs

Points wander around the plane and the Delaunay triangulation induced between them changes.  Define two Delaunay triangulations to be adjacent if they are induced by two point configurations which differ by only an infinitesimal change in the position of one point.  Define two graphs to be adjacent if they are respectively isomorphic to adjacent Delaunay triangulations.

Adjacent graphs will probably differ by a convex quadrilateral of points divided by a diagonal in two different ways.

Not sure what this is useful for.  One could explore a space of graphs (maybe for optimization) by only considering transitions of adjacency.

Also, as other possible small changes to a Delaunay graph: a node splitting into two infinitesimally separated nodes, or the reverse; nodes meeting and merging.

Inspired by political districting amidst populations moving, growing, or shrinking.

[gukadmey] Sending a message to Santa, part 2

It is almost straightforward to use a famous large unfactored composite N to RSA-encrypt a message so that it will only become readable in the future when the composite gets factored (previously part 1: such a message would currently be readable only by a higher power): just raise (modulo N) the plaintext to an encryption exponent of your choosing, and package N and the encryption exponent with the ciphertext.

Interesting candidate N are the Fermat numbers, because people will keep working on factoring those.  However, I don't know whether the special form of Fermat numbers might allow breaking RSA without factoring.  (Probably not.)

The message could be a cryptocurrency prize encouraging efforts to factor Fermat numbers.  Though such a prize might discourage cooperation.

This is kind of a cryptographic time capsule, or time-lock encryption.

It's only "almost" straightforward because the encryption exponent should be relatively prime to eulerphi(N), but we can't compute eulerphi(N) without knowing the full factorization of N.  A workaround is to encrypt the same message with different exponents, producing multiple ciphertexts, hoping at least one of them satisfies gcd(e,phi)=1.  Also, choosing a large prime number as the exponent can make the probability of a common factor with phi astronomically low.

It might be dangerous to encrypt the same plaintext with different encryption exponents.  Prepend nonces to make the plaintexts different.  This is already a standard operation in padding, e.g., OAEP.

If a partial factorization of N is discovered (or already known), can it be used to crack the whole ciphertext?  ("Security of multi-prime RSA")  If using a partially factored Fermat number as the modulus N, can one elegantly still use the whole Fermat number as the modulus, or does one need to use only the currently unfactored cofactor?

[ajzhdbqv] Continuous go

Remove the discrete grid from go 囲碁 and allow placement of pieces anywhere.

A piece is a unit disc that can touch but not overlap any other discs.  A chain is a collection of discs all of the same color connected by tangency.

An eye is an empty area into which at least one unit disc could fit.  A chain is alive if it borders at least one eye.

Not sure how scoring would work.  Maybe Voronoi.

Variants: a player can play a disc that overlaps already played discs of the player's color.  A player can play a disc of any size equal to or smaller than a unit disc.  These moves probably get penalized in scoring.  A player can play any convex shape with area less than or equal to the unit disc.  The requirement of convexity hopefully avoids shapes from becoming too bizarre.

Previously.

Friday, December 14, 2018

[flzvnqeo] Points scored by a team against good defense

Individual scoring records, e.g., Wilt Chamberlain's 100 and Kobe Bryant's 81 points in a game (coincidentally 10^2 and 9^2), are not very interesting because they involve unnatural behavior: teammates feeding the star the ball without good reason.

Points scored by a team is more interesting because basketball is a team sport.  However, if the opposing team isn't trying very hard to play defense, then points scored is less interesting.

Which team had the best defense, say, over a given season?  Who scored the most points in a game against them?  Or over multiple meetings in a season?  Or a broader statistic of points a team scored over opponents' average points allowed?

Also consider discounting by probability points scored during garbage time.

Thursday, December 13, 2018

[xtzldyuy] Some notes on BOTW armor

What armor should one wear in Zelda BOTW?

We assume (radically) that you have every armor available (except Of The Wild), fully upgraded, as well as any food buffs and elixirs.

For everyday wear, just exploring, incorporate stealth, because that allows collecting critters and avoiding enemies.  Normally this would be the stealth set.  However, if you are spending a long time in a harsh environment, use an environment-appropriate set and eat stealth food.  This is marginally better than wearing the stealth set and eating environment-appropriate food because the defense on the stealth set is low.

Obviously, for specific missions one should wear armor suited for the mission.

Some of the mutually exclusive choices for combat: the ancient set with guardian++ weapons with level 3 attack food is the most potent.  Shock-proof (rubber set), fire-proof (flamebreaker set), unfreezable (snowquill set) are useful against specific enemies.  The thunder helm might be a good alternative to the rubber set, but I haven't investigated it.

If you are doing combat in a harsh environment (heat, cold, fire), it still might be better to go with the ancient set for its attack potency, and replace your hearts lost to the environment with food or other means.  Though fire removes hearts very quickly, so maybe not.

Stealth, either via the stealth set or via food, is useful for sneakstrike (8x attack multiplier).

Is the barbarian set useful for anything?  It seems one can always do better with some other set with more defense and an attack buff from food.  What situations is its set bonus, decreased stamina needed for a charge attack, useful?

Here is a nice video analyzing the climbing set.  It turns out climbing gear is only good for the impatient or when you are in a hurry: it doesn't let you climb further; it's all about faster (which includes climbing jumps).  The first piece, easily available at Ree Dahee shrine early in the game, gets you most of the speed up for normal climbing.  I kind of regret the effort I expended for getting the other two pieces, which are kind of difficult.

For farming, speed is probably important -- no need to explore; you've seen this scenery many times already.  Stealth set offers night speed up, but night also has stal enemies who might slow things down.  The climbing set also offers speed ups as mentioned above.

Interacting with NPCs while "naked" is interesting.

The highest defense possible is fully upgraded Champion's Tunic, Diamond Circlet, and Soldier's Greaves.  There are many other 28-defense alternatives for the latter two, but those two are (subjectively) the easiest to obtain and upgrade.  Upgrading the diamond circlet does not require fighting.  I'm not sure when you would ever want max defense; it seems always better to attack (with attack armor) to remove the threat: offense is the best defense.  Maybe it's fun wandering around with max defense armor combined with a max defense buff from food.  Maybe for a self-imposed policy of using up (typically weak) weapons instead of throwing them away.

It would have been interesting (or frustrating) if armor also had durability.

Monday, December 10, 2018

[zorvtcvn] The real Apu

According to legend, Kwik-E-Mart on The Simpsons and its proprietor Apu Nahasapeemapetilon were created by Simpson's writer (and later talk-show host) Conan O'Brien based on the real convenience store across the street from Mather House at Harvard University.  O'Brien lived in Mather while an undergraduate student at Harvard, and both frequented the convenience store and often looked at it from his dorm room window.

According to Google Maps, the convenience store is Louie's Superette, 26 Surrey St., Cambridge.

The character of Apu has recently been eliminated from The Simpsons to satisfy political correctness.  Has the owner of Louie's Superette been informed that his or her character has been eliminated as a gesture of respect?  How does he or she feel about it?

Sunday, December 09, 2018

[jeuofkrj] Sphere with one point missing

A sphere (2-sphere) and a sphere with one point removed are topologically very different objects: the latter can be deformed to a disc or a plane.  What a difference a point makes!

Conversely, it also suggests that a plane plus one additional point ("a point at infinity") is "equivalent" to a sphere.  We see this in the stereographic map projection, which must omit one point.

Higher dimensions?

[wktaebsg] Distortion on a large geodesic dome

Consider the dual of a high-order geodesic dome: 12 pentagons (a dome based on the regular dodecahedron) and lots and lots of hexagons.  Because it is large, small regions of the surface are nearly flat, easy to project into a plane.  What does a small region in the neighborhood of a pentagon look like?  The hexagons immediately adjacent to the pentagon are highly distorted compared to a regular hexagon.  What about the next ring of hexagons?  How quickly does the distortion dissipate?  On different sized domes, is it a function of hexagon count from a pentagon, or a function of angle measured from the center of the sphere?  The region has 5-fold symmetry.  If you were an ant on such a dome, how easy would it be to determine where you are (modulo symmetry) based on local measurements (lengths, angles) of the distortion of the hexagons in your neighborhood?

[rmufaypp] Something's got a hold on me

Create a music video of the song with scenes from Jaws, Gandalf getting caught by the Balrog in Lord of the Rings, etc.

"It must be love."

Saturday, December 08, 2018

[towqnolb] Fuzzy chess puzzle

Compose a chess position which encodes two chess problems: 1. White to play and win; 2. Black to play and win.

In combinatorial game theory, such a position is called fuzzy or incomparable with zero.  The star game is the simplest fuzzy position.

As is the custom with chess problems, both problems should be elegant: economical, no dual solutions, etc.

This has almost certainly already been done.  Such puzzles exhibit an additional form of economy: not having to specify who is to move.

Kings on h1 and a8, each behind their own pawns on g2 h2, a7 b7.  White rook h3; black rook on a6.

Friday, December 07, 2018

[uohbrdbd] Keep people believing in a lie

To verify a statement when direct science is not available, people often turn to their friends, their trust network.

To keep people believing in a lie, you need to manipulate their trust networks.  Examine how this is done.

Of course, keeping people believing in a lie is hugely important for profit and power.

Wednesday, December 05, 2018

[keouzjxw] Smooth under powers of 2

We list some primes p with smooth p-1 which are slightly less than nice powers of 2.  More precisely, primes of the form k*2^m+1 with the first priority to maximize m and second priority to maximize k, all while obeying the constraint to be less than 2^n.  The nice exponents n are of the form {1,3,5}*2^e.

Previously, we computed primes of that form slightly greater than powers of 2.

We furthermore constrain the primes to have 3 as its least primitive root (generator).  2 seems to never be a primitive root for primes of this form, if the exponent m is large.  (Why?)

Primes with smooth p-1 are useful because it is one of the forms for which it can easily be checked whether a given number is a primitive root of the prime.  If primitive roots are likely to be used with these primes, we might as well make sure it has a nice one, namely 3.

If k*2^m is not smooth enough, consider Pierpont primes.

Pari/GP source code:

? f(bits)=local(g,m,x);found=0;for(db=1,bits,for(mx=1,2^db-1,m=2^db-mx; x=m*2^(bits-db)+1; if(ispseudoprime(x),g=lift(znprimroot(x));if(g==3,print(m,"*2^",bits-db,"+1 < 2^",bits);break(2)))))

? for(i=2,100,f(2^i);f(2^(i-2)*5);f(2^(i-1)*3))

3*2^1+1 < 2^4
1*2^4+1 < 2^5
1*2^4+1 < 2^6
7*2^4+1 < 2^8
1*2^8+1 < 2^10
13*2^8+1 < 2^12
5*2^13+1 < 2^16
1*2^16+1 < 2^20
7*2^20+1 < 2^24
13*2^28+1 < 2^32
113*2^33+1 < 2^40
29*2^43+1 < 2^48
95*2^57+1 < 2^64
29*2^75+1 < 2^80
7*2^92+1 < 2^96
7*2^120+1 < 2^128
167*2^151+1 < 2^160
13*2^188+1 < 2^192
467*2^247+1 < 2^256
13*2^316+1 < 2^320
667*2^374+1 < 2^384
127*2^504+1 < 2^512
101*2^633+1 < 2^640
373*2^758+1 < 2^768
1331*2^1013+1 < 2^1024
613*2^1270+1 < 2^1280
193*2^1528+1 < 2^1536
19*2^2038+1 < 2^2048
1385*2^2549+1 < 2^2560
179*2^3063+1 < 2^3072
305*2^4087+1 < 2^4096
269*2^5109+1 < 2^5120
1733*2^6133+1 < 2^6144
553*2^8182+1 < 2^8192
575*2^10229+1 < 2^10240
9637*2^12274+1 < 2^12288
5717*2^16371+1 < 2^16384
3457*2^20468+1 < 2^20480
1961*2^24565+1 < 2^24576
2561*2^32755+1 < 2^32768

[cochbzdy] Why people are the way they are

If you are proposing a method to change people, first explain why you believe they are the way they are.  Such a model should explain why they will stay the way they are in the absence of your proposed stimulus to change them.  Then, explain how your method works within your model.

These models ought to have names so they can be referred to concisely.

Inspired by seemingly willful ignorance, an attitude of "I don't want to know what's going inside the minds of those disgusting people whom I wish to change”, and likely wishful thinking about why people are the way they are.

I strongly suspect the reason people typically stay the way they are is because of some feedback system, probably involving society / social interaction; that is, their behavior is rationally optimal within the feedback system they are in.  Most attempts to directly change a person away from this optimal behavior aren't going to work.  Instead you need to change, (though of course first understand), the feedback system they are in.  But there's the willful ignorance mentioned above regarding studying such feedback systems.

Inspired by Rat Park: if drug addiction is a response to society having abandoned a person, then changing the person to stop addiction isn't going to help much: they still have to face a society that has abandoned them.  Willful ignorance and wishful thinking come into play when you have ugly and unpleasant-to-think-about reasons (e.g., class and racial prejudices) for which you might believe a person deserves to be abandoned by society.

[hooitehb] Go cuts

Given two go 囲碁 chains (connected groups), can you connect them in 1 move?  If so, how many consecutive free (unanswered) moves beforehand by your opponent would eliminate your ability to connect in one move?  This gives a measure of how "almost connected" two chains are.

For example, the bamboo joint and diagonal connection are two examples of connections that require the opponent to make two free moves to cut.

Given two go 囲碁 chains, can you connect them in at most 2 free moves?  If so, how many consecutive free moves beforehand by your opponent would eliminate your ability to connect in at most two moves?  This gives another measure of how "almost connected" two chains are.

Obviously this can be generalized to more moves, yielding a table (specific to that initial position).  Can we then compute some interesting statistic over all the (infinite) entries in the table?

It feels a little like quantum mechanics, summing over all possible paths between two points.

Ko threats sometimes yield free moves.

Generalizing to some amount of alternation between moves: define a template of length N, and a color assigned to each number 1 through N.  There are 2^N possible templates of length N.  The template gives who gets to move on that move number (so not necessarily strictly alternating).  Given a template, determine whether two chains can be connected.  Figure out families of templates which result in connection or cut.

Unclear whether these exercises yield anything useful for the actual game of go 囲碁.  It might be more relevant to Hex, which ultimately tries to connect the two "groups" of opposite sides.

[eqnpqhfi] Two or more sleeps a day

I suspect many people regularly get their requisite amount of sleep through two or more sleeping sessions per day, perhaps a night time sleep and a daytime nap.

Reporting just their night time sleep seems to give a scary small number of hours of sleep per day, but the nap brings the total up to a normal amount.

Also occasional days getting more sleep.  Outliers affect the mean quite a bit.

It's interesting that the body's circadian rhythm can adapt to such a sleep schedule.

Spanish culture had notably given us the word siesta.

Monday, December 03, 2018

[oiqnkhyx] Many basketball point zones

Given a population of players and their shooting accuracies from different locations, assign point values to the court so that shot selection of the population is uniform across the whole area.  Each player of course chooses to shoot from where they have the highest expected point outcome.

Goal is to spread the game over the whole court (not necessarily a good idea).

The point value map could be determined iteratively: at the end of each season, where ever there was high number of shots attempted, decrease the point value from there; where ever there was a low number, increase.

Simpler: solo "linear" shooting contest, no defense.  Each player can choose to shoot from any point along the line down the middle of the court (the line from one basket to the basket on the other end of the court).  Point values vary along the line.  Spectators can be much closer.

[jjjppcgb] Chess4

Roll a d4 die at the start of the game (or two coin flips), randomizing white's first move among c4 d4 e4 Nf3.

If white really doesn't like the randomly selected move, he or she can instead play some other (unusual) opening move not among those 4.

This is a conservative alternative to Chess960.

Inspired by some players, e.g., Nakamura, who regularly play all 4 opening moves as white, making it difficult for opponents to deeply prepare openings against them (which was the point of Fischer Random).  Carlsen rotated among 3 in the recent World Championship.

[avpsclrh] Scrambling center squares of a Rubik's cube

In a solved 3x3 Rubik's cube, if each center square could be independently rotated 4 different ways, then there would be 4^6=4096 visually identical solved states.  However, only half of them, 2048, are possible.  This can be computed by comparing the number of possible states of a supercube versus regular cube.

For 4x4, if everything were independent, the 4 center squares of 1 face could be permuted and rotated in 6144 = 2^11 * 3 different ways, yielding a hypothetical total of 6144^6 ~= 5.4e22.  However, the actual total number is 95551488 = 2^17 * 3^6 = (2^3 * 3)^6/2, so less by a factor of 2^49.

Which center face states are possible, and what are some algorithms to reach them?

[guqblqkv] Azimuthal equidistant on a manifold

The azimuthal equidistant map projection can easily be applied to any 2D manifold.  Are any other projections so generally applicable?  Probably many azimuthal ones.

From the center point, the map extends in a given straight-line direction, traveling along a geodesic, until it encounters points for which that geodesic ray isn't the shortest path to those points.  (Instead, perhaps go around the other way.)  Can really weird manifolds have concave maps?

There are probably also regions unreachable by any geodesic from some starting points.

[uqhrqusi] Construction of Stormbreaker / Mjolnir

Avengers Infinity War depicting the construction of Thor's new weapon at a neutron star is a cute and almost scientifically plausible idea.  Let's assume that Mjolnir and Stormbreaker are both made of neutronium mined from neutron stars.  They are extremely heavy, which explains why no one other than Thor is able to lift them.  (Thor has god-like strength.)

(How do you say "Stormbreaker" in Old Norse or Icelandic?)

The weapons need to have a few additional imaginary technologies incorporated into them:

The outer shell must be a superstrong pressure vessel, able to keep its contents in the neutronium state despite being removed from the intense gravity of the neutron star.  The shell also needs to withstand the impact of the weapon hitting things.

Tangentially, assuming such superstrong materials exist, how might neutron star mining realistically work?  Maybe just scoop it out; far less dramatic than the movie.  Dig an actual mine into the neutron star with tunnel walls built out of the pressure-resistant material.  Alternatively, could an active support structure resist pressures inside a neutron star?

Does the density of the neutronium differ according to depth?  Maybe the more powerful weapon (Stormbreaker) is made of denser neutronium from deeper in the star, or from a more massive neutron star.  Maybe so dense that the material is even more exotic than neutronium, e.g., quark matter from a quark star.

Each weapon is probably massive enough to exert significant gravitational attraction on its environment, so it would be annoyingly destructive even when not being used.  It needs to have some sort of antigravity generator the cancel out the gravity its mass generates.  The weapons would be paradoxical objects, having gravitational masses different from their respective inertial masses.  (They keep their huge inertial masses to be potent as weapons.)  Maybe the antigravity generator can be turned off briefly on impact with a target, allowing its gravity to inflict additional damage.

Which attack involved more mass: Thanos throwing a moon at Iron Man, or Thor throwing Stormbreaker at Thanos?

Superheavy superdense objects tend to sink immediately to the core of the earth, undeterred by the dirt and rock in the way, which might as well be air given how relatively not dense they are compared to neutronium.  The antigravity generator therefore also needs to allow the weapons to hover, not sinking, not destroying tables they are placed on.

However, once we posit this all-powerful antigravity generator that's allowing Thor and his hammer to be moved by (say) a conventional elevator, Thor doesn't have to be very strong, just good at operating the controls of the antigravity generator.

Previously on superheroes interacting with astronomical objects: (1) (2) (3)

[tjndzmpj] Censorship of entertainment

Entertainment is a common target of censorship.  Examine which people were consuming the censored entertainment, i.e., the people who have been most affected by the censorship.  Hypothesize that we will see boundaries and classes of society reflected in whose entertainment is getting censored.

Obviously, those who are politically not powerful will not be able to fight to maintain the legitimacy of their preferred entertainment.

Saturday, December 01, 2018

[xoaeaiqu] Avoiding Armageddon

Chess match tiebreaks often end with pairs of blitz games followed by an Armageddon game if necessary.  Modify it so that instead of going into Armageddon, the players can mutually agree to continue to play another pair of blitz.  After each tied pair of blitz games, offer the choice again.  If either player doesn't want to continue pairs of blitz, then Armageddon.

They can also mutually agree for the outcome of the match to be decided by coin toss, perhaps as a joint political statement about their dislike of Armageddon or even of pairs of blitz.

[lkiccdly] Feel like exercising

People tend to exercise when they feel like exercising.  When do they feel like exercising?  Probably self-qi.  Fitness, the result of exercising, becomes a reflection of self-qi.

[syjibxqb] Probability of comeback

Team is X points behind with Y minutes remaining in the game.  What is the probability of comeback, the score becoming (at least) tied before the end of the game?

Draw a scatter plot of X and Y colored by whether a comeback occurred.

Goal is to demarcate where "garbage time" begins at the end of a basketball game, whose play ought not count in statistics.

[yafmvelb] Spherical potential functions

We imagine the experience of a ghostlike neutrino traveling around a universe populated by a single spherical mass of uniform density.

Outside the spherical mass, it feels a gravitational potential energy function proportional to -1/r.  Explanation: Newton's universal law of gravitation: force ~= 1/r^2.  Work: integral (force * dr) = -r^(-1).

Assuming a 3D universe, inside the spherical mass (within which the neutrino moves unimpeded), the potential energy function is a parabola, quadratic.  Newton again: Force=GMm/r^2 = G (4/3) Pi r^3 rho m / (r^2) ~= r.  Work = integral r dr ~= r^2.  Incidentally, the quadratic potential function induces simple harmonic motion, assuming you don't exit the well.  This yields the famous problem of a ball oscillating within a tube drilled through the center of the earth.

Together, you get a piecewise potential function of a broad sheet of -1/r in empty space capped by a r^2+C bowl inside the spherical mass.  The bowl prevents -1/r from going to negative infinity.

Generalizing Newton's law to other dimensions, the potential function remains 1/r outside the mass.  However, inside, it changes.  2D: U = abs(r). 1D: U = log(abs(r)).  In 1D, we weirdly have a singularity at the center of the 1D ball, the midpoint of a line segment.  4D: U = abs(r)^3.  Depict test masses moving within these potential wells.  It's no longer simple harmonic motion.

[jywthyvb] Satire of police killing blacks

Create satire in which police kill black men in ridiculous and uncalled-for ways.  This has almost certainly already been done.

The Avengers have just finished defeating the supervillain.  The police show up and kill Black Panther for no reason, making the world a safer place as is their mission, then ask for high-fives all around.  Inspired by Jemel Roberson.

Shooting a cup of black coffee, because shoot anything black.

Video game.  We need to avoid the game becoming a celebration of killing African-Americans.  Maybe the playable character, a police officer, automatically tries to kill African-Americans unless the player acts to prevent it, and you do well by preventing the most deaths.  The characters could be presented extremely abstractly: a blue dot approaches a black dot, turning it red.

[eurtnluy] Repairing Vah Ruta

The true ending to BOTW has Zelda and Link heading off to investigate a malfunctioning Divine Beast Vah Ruta.  This suggests a sequel: an engineering game in which the heroes (A woman in engineering!  Working with a man who isn't sexually harassing her!  Zelda would be the lead engineer as she has studied Ancient technology, much to the dismay of her father.) study, disassemble, repair, and reassemble a very complicated contraption.  Probably many iterations of this as real engineering really does.

Also crafting replacement parts.  Before that, crafting the tools needed to craft the replacement parts.  Crafting tools needed to craft tools.  Maybe some world-trotting adventures to acquire materials.

Nintendo will probably never make such a sequel; engineering is too boring.  However, an independent developer can easily do it and avoid any intellectual property issue: just change the names.  Or keep the names and defend the work as parody.

The hard part of the game design would be inventing an internally consistent complicated fictional technology.  Or maybe the characters come to understand the technology without the player doing so, though that would be lame.

[nyvuhumi] Attack an undergroup

A terrorist attacks a group, e.g., Americans, in an act of terrorism.  Sometimes such an attack counterintuitively makes the attacked group stronger, e.g., causing them to band together against a common enemy, so it would have been better for the terrorist not to have attacked in the first place.

Hypothesize that the terrorist can avoid this problem by first studying the social stratifications within the group to be attacked, then attacking the lowest subclass within the group.  Despite being in the same group, others in the group may feel, "those are the people I despise anyway; I have no qualms at seeing them be destroyed".  We assume the terrorist's political goal remains achieved even if the targeted group thinks this way.

If a terrorist wanted to attack America, our lower classes are conveniently color-coded (though with some exceptions like "white trash" who are white but low class).  We should decrease racial discrimination to make effective target selection more difficult for terrorists.

It's not just terrorists who do this.  A government can select a group, for example African-Americans, for greater scrutiny: racial profiling.  (This violates the constitutional right to equal protection.)  However, it only attacks (e.g., convicting them of crimes detected because of the greater scrutiny) the worst members of the group.  The civil rights of the whole group have been violated by the greater scrutiny and probably increased fear in their lives that the government is watching them, but no one complains because no one politically wants to defend the worst members of the group, those who are being sent to jail.

[ambdtzgq] Precisely estimating the number of prime powers

How many numbers below a given upper bound are primes or prime powers? It's asymptotically the same as the number of primes, but can we be more precise?

This question might be slightly easier than coming up with approximations to the normal prime counting function which only counts first powers.

Inspired by Reed-Solomon codes and finite fields in general, which need an alphabet whose size is a prime or power of a prime.

[lnnykmvg] Banana ice cream

Mix some banana pieces into ice cream.  (Or cheat further and just mix banana flavor into ice cream.)

The traditional banana split has an awkward shape, with the cylindrical banana an unstable base for ice cream and toppings stacked on it, with the whole thing threatening to roll when attacked with a spoon.

One might need to mix the banana pieces in only shortly before serving, in the style of Cold Stone Creamery, because letting the banana freeze might make it too hard to eat.

One could incorporate many traditional banana split toppings into the ice cream also, but it further loses nice visual presentation.

Banana splits were invented during a previous variety of bananas (Gros Michel), before that variety was wiped out due to monoculture and replaced with Cavendish.  How did the banana change affect banana splits?

[ppkdjuny] Encrypted QB communication

Is the radio communication between coach and quarterback in the NFL encrypted?  Encrypting voice in real time with low latency is a neat technical problem.  There is huge incentive for the opposing team to break the encryption and eavesdrop.  If the encryption is strong (a big if), then do practical attacks like rubber-hose and side-channel.  (Sounds like something the Patriots would do.)  Having well-funded attacks and defendings of cryptosystems is good for the field of cryptography.  Maybe it can be a testbed for cutting-edge cryptography.

However, for entertainment, I think the NFL should revert back to the old days and ban radios, and do communication via large sideline placards.  This gives fans additional interesting things to watch during the game.  Teams can also engage in practical battles of cryptography and cryptanalysis, stealing signs like baseball.  It needs to be simple but secure, a difficult tradeoff.  Is the QB permitted to have a decryption computer as part of the uniform?

[mtdmhbii] Legitimancy

Legitimancy is a spell that alters a target's notions of ethics: change something they believe is wrong to right.  (More generally, alter in either direction.)  Perhaps the target is an entire society or culture.  It's not just messing with their mind; it's messing with their moral backbone.

Is this a class that would only be taught at Hogwarts, or is it done in the muggle world?

[fwdgrrnc] Primes near powers of 3

Here are some prime numbers of the forms 3^n-k, 3^n+k, and k*3^n+1.  The exponents grow as round(10^(i/25)).  The growth rate was arbitrarily chosen.  Each k is minimal for a given exponent and form.

These might be useful as compactly expressible primes which don't look funny when written in binary.  For example, they will have average-case behavior if used as an exponent in the standard algorithm of exponentiation by repeated squaring, in contrast to numbers like 2^k+n which behave especially well, or 2^k-n which behave especially poorly.

Many previous similar, doing with powers of 2 instead of 3: (1) (2) (3)

3^1-0 3^1+0 2*3^1+1
3^2-2 3^2+2 2*3^2+1
3^3-4 3^3+2 4*3^3+1
3^4-2 3^4+2 2*3^4+1
3^5-2 3^5+8 2*3^5+1
3^6-2 3^6+4 2*3^6+1
3^7-8 3^7+16 8*3^7+1
3^8-8 3^8+2 6*3^8+1
3^9-2 3^9+4 2*3^9+1
3^10-20 3^10+2 8*3^10+1
3^11-16 3^11+20 28*3^11+1
3^12-58 3^12+16 10*3^12+1
3^13-22 3^13+8 12*3^13+1
3^14-8 3^14+2 4*3^14+1
3^16-98 3^16+26 2*3^16+1
3^17-10 3^17+34 2*3^17+1
3^19-14 3^19+56 20*3^19+1
3^21-4 3^21+56 24*3^21+1
3^23-20 3^23+32 48*3^23+1
3^25-20 3^25+14 34*3^25+1
3^28-22 3^28+26 18*3^28+1
3^30-26 3^30+4 2*3^30+1
3^33-52 3^33+70 14*3^33+1
3^36-10 3^36+2 16*3^36+1
3^40-38 3^40+118 62*3^40+1
3^44-34 3^44+8 70*3^44+1
3^48-44 3^48+56 32*3^48+1
3^52-124 3^52+50 8*3^52+1
3^58-220 3^58+158 8*3^58+1
3^63-28 3^63+2 18*3^63+1
3^69-16 3^69+76 70*3^69+1
3^76-370 3^76+28 8*3^76+1
3^83-140 3^83+236 34*3^83+1
3^91-20 3^91+100 340*3^91+1
3^100-10 3^100+148 90*3^100+1
3^110-110 3^110+2 90*3^110+1
3^120-82 3^120+56 176*3^120+1
3^132-28 3^132+382 2*3^132+1
3^145-110 3^145+314 14*3^145+1
3^158-350 3^158+20 188*3^158+1
3^174-128 3^174+4 200*3^174+1
3^191-38 3^191+70 124*3^191+1
3^209-874 3^209+188 136*3^209+1
3^229-364 3^229+560 40*3^229+1
3^251-346 3^251+32 496*3^251+1
3^275-106 3^275+104 456*3^275+1
3^302-538 3^302+140 174*3^302+1
3^331-694 3^331+136 8*3^331+1
3^363-134 3^363+2 56*3^363+1
3^398-98 3^398+764 14*3^398+1
3^437-46 3^437+256 412*3^437+1
3^479-608 3^479+52 198*3^479+1
3^525-380 3^525+928 760*3^525+1
3^575-266 3^575+2860 630*3^575+1
3^631-284 3^631+644 50*3^631+1
3^692-28 3^692+566 162*3^692+1
3^759-1690 3^759+464 664*3^759+1
3^832-832 3^832+652 98*3^832+1
3^912-502 3^912+1018 736*3^912+1
3^1000-1084 3^1000+968 102*3^1000+1
3^1096-1528 3^1096+632 1196*3^1096+1
3^1202-820 3^1202+1024 448*3^1202+1
3^1318-1360 3^1318+32 1550*3^1318+1
3^1445-406 3^1445+2524 156*3^1445+1
3^1585-3836 3^1585+3016 1560*3^1585+1
3^1738-1990 3^1738+2102 380*3^1738+1
3^1905-1234 3^1905+40 312*3^1905+1
3^2089-230 3^2089+1090 2572*3^2089+1
3^2291-6320 3^2291+1132 2008*3^2291+1
3^2512-1244 3^2512+412 2752*3^2512+1
3^2754-2860 3^2754+542 1150*3^2754+1
3^3020-6188 3^3020+2066 492*3^3020+1
3^3311-3158 3^3311+6130 786*3^3311+1
3^3631-5704 3^3631+934 4238*3^3631+1
3^3981-6134 3^3981+1568 202*3^3981+1
3^4365-1312 3^4365+4994 2126*3^4365+1
3^4786-8060 3^4786+5752 272*3^4786+1
3^5248-12320 3^5248+3368 4430*3^5248+1
3^5754-2410 3^5754+1510 8340*3^5754+1
3^6310-4472 3^6310+7750 4248*3^6310+1
3^6918-18850 3^6918+6718 7194*3^6918+1
3^7586-1736 3^7586+10732 5580*3^7586+1
3^8318-4946 3^8318+3914 1308*3^8318+1
3^9120-14878 3^9120+2368 110*3^9120+1
3^10000-1372 3^10000+10466 496*3^10000+1
3^10965-5434 3^10965+8378 2280*3^10965+1
3^12023-20554 3^12023+2402 1160*3^12023+1
3^13183-1678 3^13183+170 19740*3^13183+1
3^14454-7900 3^14454+3064 4362*3^14454+1
3^15849-12890 3^15849+25246 3126*3^15849+1
3^17378-4816 3^17378+6314 15154*3^17378+1
3^19055-2950 3^19055+260 4208*3^19055+1
3^20893-12334 3^20893+19930 3422*3^20893+1
3^22909-111106 3^22909+48196 7270*3^22909+1

The final entry 3^22909-111106 required more than a 64 metric megabyte stack size in Pari/GP.  (I don't understand why precprime needs a large stack.)  After restarting with 256 megabytes, it took 24 hours.

[molyjmda] Spherical flag

We revisit the idea of not planting a country's flag on the moon.

What would be a nice flag representing the whole earth?  Obviously a depiction of our planet, but there's the standard problem of map projection onto a flat flag.

Easy solution: Don't plant a flag, but instead a globe hanging from a pole.  Design this.  There is a mechanical challenge of designing a globe that would have fit in the Apollo lunar lander, something that stores compactly but is easily assembled into 3D.  "Inflatable" probably does not work so well in the vacuum of the moon.

[jopvhvwa] Pierpont primes

Here is a list of the 25394 Pierpont primes less than 2^10000, in order by magnitude.  Pierpont primes are primes of the form 2^x * 3^y + 1.

We also looked for prime powers, but did not find any large ones.

Previously.

Other than Fermat primes, Pierpont primes have the smoothest possible factorization of p-1, so they are aesthetically the opposite of safe primes.

Friday, November 30, 2018

[qgzjdvne] Continued fraction of log(3)/log(2)

Below are 21151 terms of the simple continued fraction expansion of the base 2 logarithm of 3, stopping at a record large term.  The record large terms are 1 2 3 5 23 55 100 964 2436 3308 4878 8228 24477 59599.

Truncating before 23: which is larger, 2^1054 or 3^665?

Truncating before 964: which is larger, 2^2822415852915495005567037610945253111955180281114106828540887882071899903182549413194498201674121306786342728992705 or 3^1780746138556148123471309323598980060001250637409759635946474946986610184795331066338871761432700674308476219853074 (approximately 2^2^380.196 and 3^3^239.458)?

This technique would have been cool, but it was not used.

Motivation was, how many digits of floating point precision are needed to sort logarithms of Pierpont primes?

1 1 1 2 2 3 1 5 2 23 2 2 1 1 55 1 4 3 1 1 15 1 9 2 5 7 1 1 4 8 1 11 1 20 2 1 10 1 4 1 1 1 1 1 37 4 55 1 1 49 1 1 1 4 1 3 2 3 3 1 5 16 2 3 1 1 1 1 1 5 2 1 2 8 7 1 1 2 1 1 3 3 1 1 1 1 5 4 2 2 2 16 8 10 1 25 2 1 1 1 2 18 10 1 1 1 1 9 1 5 6 2 1 1 12 1 1 1 6 2 12 1 1 12 1 1 2 12 1 12 3 1 5 1 14 1 1 14 2 3 1 2 2 1 4 1 4 8 1 1 1 3 5 1 1 1 1 2 1 4 3 7 5 3 1 32 1 1 1 18 1 3 2 5 2 1 3 1 8 1 1 1 2 6 6 5 33 2 2 3 1 1 1 1 29 1 3 2 1 21 1 6 52 1 8 1 4 14 9 7 1 4 18 2 2 1 1 2 100 39 1 2 1 1 19 1 5 9 1 3 964 5 1 1 1 39 1 1 1 1 5 3 1 88 1 2 1 3 1 11 1 23 11 1 1 1 2 1 1 4 3 1 5 1 4 2 1 75 1 2 1 11 17 2 5 3 1 3 34 1 10 2 4 7 1 1 23 1 6 3 1 7 1 17 2 1 24 1 1 1 10 1 4 1 1 5 3 2 1 2 1 1 3 6 8 1 8 2 1 1 4 2 7 9 2 2 2 1 7 12 2436 1 2 1 9 10 1 5 1 3 1 2 1 2 3 1 1 3 1 4 6 1 2 1 2 2 1 2 1 1 3 46 31 196 4 1 1 3 11 1 3 14 1 1 3 2 20 1 3 6 3 85 1 7 1 9 4 5 2 1 1 78 1 4 4 2 6 6 2 4 8 4 5 1 1 11 1 2 1 5 13 2 1 3 4 2 7 5 2 2 1 2 10 1 163 1 3 1 1 1 2 1 1 2 1 6 30 1 2 2 13 1 1 2 1 2 1 1 1 3 2 5 1 5 3 1 3 1 3 2 36 1 1 1 1 9 7 1 28 2 1 1 5 1 11 10 3 1 2 1 1 2 19 2 5 5 1 4 1 1 2 1 5 3 10 3 3 1 1 8 3 5 1 10 22 10 7 1 2 1 1 69 6 1 10 7 4 1 1 1 2 1 7 3 6 1 8 1 3308 1 4 1 7 1 1 2 1 1 4 4 47 9 1 1 1 2 1 1 1 1 7 1 1 1 1 1 1 1 9 1 4 71 1 1 1 1 1 31 1 31 3 1 13 36 1 92 2 2 4 1 1 93 1 17 2 2 5 1 3 1 1 7 20 3 2 47 1 2 1 1 2 1 1 19 54 2 1 4 1 4 3 75 2 21 1 1 1 1 1 2 8 3 1 2 1 1 5 2 6 1 1 40 5 23 1 4 1 2 21 1 6 1 2 1 20 5 5 3 1 2 2 12 2 41 1 24 13 4 1 2 5 3 3 2 1 3 2 1 2 73 3 1 11 9 1 7 2 1 7 14 1 2 2 1 1 3 2 1 19 15 1 45 1 4 1 6 1 3 1 6 2 4 1 3 7 10 5 1 1 1 4 1 2 1 1 26 1 1 77 24 1 1 1 3 3 4 2 1 1 5 10 2 34 1 4 1 1 1 1 2 10 1 6 16 2 1 10 1 1 3 1 1 1 1 2 2 2 21 1 3 1 1 4 16 1 2 1 24 2 6 3 2 3 1 14 1 1 1 8 1 1 15 52 3 3 2 4 11 1 2 1 2 1 1 1 3 3 43 2 1 6 1 1 1 1 1 8 4 1 1 1 57 1 4 1 2 3 14 1 1 9 1 1 3 8 1 2 4 1 1 2 1 13 4 1 4 1 3 2 5 4 3 1 4 2 2 1 4 2 3 2 2 3 1 13 1 1 1 9 9 2 2 1 6 5 2 3 1 1 5 12 1 10 1 46 81 2 5 1 3 1 1 11 1 3 2 2 6 488 1 16 8 27 1 4 67 1 8 14 1 2 1 17 7 2 12 1 12 2 1 15 1 1 1 8 4 4 1 9 1 2 3 1 4 5 2 6 1 3 3 3 1 6 1 3 3 3 2 1 1 10 4 1 6 1 1 2 1 1 2 3 240 1 2 1 1 4 1 4 1 4 2 4 1 5 3 1 6 3 5 2 1 19 1 1 24 6 11 77 1 1 1 11 3 3 3 6 5 4 1 21 1 3 1 14 2 5 1 12 1 1 1 14 1 5 4 3 2 1 3 2 2 4 3 2 6 3 1 1 2 1 4 2 1 1 8 1 8 2 1 1 1 3 9 5 1 3 10 16 5 1 2 1 29 4 2 4 1 5 1 1 1 1 2 1 3 3 1 1 1 1 2 1 3 3 1 23 1 2 2 1 6 1 2 1 3 4 19 2 2 1 3 1 9 1 1 2 5 2 2 4 7 5 7 2 1 3 89 1 2 2 4 13 1 1 8 1 1 1 1 1 3 1 1 1 6 14 1 1 8 1 35 1 2 1 3 1 1 1 41 1 3 1 6 1 15 10 1 1 1 1 271 1 1 2 3 1 2 2 1 1 1 4 1 1 1 25 8 2 2 3 1 1 1 1 1 87 1 2 1 2 33 1 3 1 1 1 21 10 3 565 10 1 18 5 2 4 2 1 1 1 2 5 1 1 1 1 2 1 9 6 12 1 7 1 1 3 18 1 2 1 3 1 3 1 2 3 3 2 1 9 2 2 1 11 6 3 5 1 1 1 34 1 7 5 7 3 1 4 1 1 1 7 4 2 4 4 3 1 1 6 3 1 1 1 1 19 1 3 2 33 1 14 3 4 3 1 3 7 90 7 9 4 10 2 8 1 2 1 139 3 7 2 1 10 11 1 91 13 1 1 3 1 3 1 1 1 1 1 13 1 8 7 3 5 75 4 1 1 1 2 3 295 1 1 2 2 1 3 1 1 1 10 6 2 1 2 1 1 6 8 3 2 1 1 1 5 4 1 3 3 2 83 5 1 16 8 343 3 1 6 2 29 1 2 3 1 6 1 2 3 1 1 1 1 8 1 16 2 1 1 1 1 1 1 2 1 2 5 2 1 2 1 5 1 3 1 4 1 4 1 2 1 2 3 1 1 2 2 70 2 1 14 1 3 1 5 5 2 1 4 4 1 3 1 1 1 3 5 1 101 7 1 1 2 2 25 1 13 1 6 1 1 2 2 2 1 1 1 1 2 3 5 1 1 2 151 2 3 5 2 2 4 1 4 1 2 1 16 1 1 9 1 4 1 6 1 1 1 1 12 33 1 5 2 28 1 1 4 1 1 12 5 94 1 1 1 6 3 1 1 1 37 2 2 3 1 12 1 1 6 1 4 1 4 3 37 6 1 3 1 3 24 3 1 2 12 22 1 39 1 4 1 1 11 1 4 1 2 2 1 1 2 1 1 1 3 1 497 1 1 1 1 2 1 5 1 4 3 1 12 1 1 1 3 6 1 1 1 1 1 11 9 10 1 5 31 1 3 3 2 1 1 8 23 2 1 1 1 1 2 30 1 1 1 1 14 1 2 2 6 16 14 2 1 2 2 1 3 1 3 4 3 1 1192 2 1 3 1 7 2 2 11 2 1 14 2 3 1 256 1 1 1 1 1 7 1 7 4 1 1 1 2 1 1 2 1 1 2 17 1 2 1 1 6 3 2 22 2 3 1 1 3 4 2 9 2 1 11 7 1 262 1 2 1 32 1 43 1 8 1 1 1 8 6 5 1924 1 6 1 3 3 14 1 1 11 1 1 2 1 2 2 1 1 2 1 3 1 4 1 7 1 19 3 2 1 2 1 5 4 8 2 1 5 4 4 1 1 6 2 2 1 1 4 1 1 7 9 1 1 15 3 3 2 1 1 1 4 2 1 1 2 1 1 69 5 11 2 2 4 2 56 2 4 1 2 1 3 24 10 1 2 5 5 1 4 3 1 1 1 8 2 3 2 2 1 87 8 2 1 4 1 1 1 4 1 5 1 35 2 2 35 9 1 3 2 7 1 1 23 2 1 13 1 1 3 1 1 2 1 184 8 1 1 3 4 1 9 2 6 1 44 3 11 26 1 6 3 1 1 7 4 2 1 1 1 8 28 9 4 2 2 7 1 1 2 4 1 1 3 4 2 28 5 2 1 3 6 1 50 11 1 2 2 7 2 1 1 3 2 3 1 3 1 1 9 57 2 9 1 8 1 7 49 9 2 2 9 1 1 32 3 1 1 3 1 1 1 1 1 1 4 3 1 3 1 2 1 4 1 2 2 3 2 2 1 4 10 5 2 1 15 1 7 6 4 2 2 33 1 2 2 1 2 1 3 5 1 26 6 2 2 3 4 2 1 1 2 2 2 2 2 1 9 2 1 50 3 4 1 1 22 2 13 1 4 9 2 1 7 5 1 3 2 2 3 55 1 1 9 4 1 2 1 4 3 4 3 13 1 1 2 27 3 1 6 1 1 14 7 1 1 1 1 2 1 3 2 1 1 43 1 1 2 3 1 1 6 1 1 11 1 4 8 6 1 1 1 7 1 29 3 1 13 1 33 10 1 2 5 6 5 1 4 1 7 1 4 1 1 2 1 14 1 2 2 2 1 1 3 1 2 1 5 1 33 2 1 2 1 9 1 1 2 2 2 31 3 1 1 1 3 2 1 1 3 1 1 2 5 2 11 4 2 7 1 366 3 11 1 1 1 5 1 1 1 3 4 64 1 3 1 2 1 1 2 3 1 2 2 1 1 4 1 1 10 35 1 3 1 1 26 1 20 22 2 1 4 12 2 1 1 2 7 1 1 5 1 50 1 1 2 1 2 6 1 1 3 12 1 1 2 1 1 1 19 1 2 1 19 1 1 1 3 2 2 4 1 1 5 2 13 1 5 2 4 11 1 12 3 1 30 2 5 1 1 1 1 4 1 7 1 1 5 5 1 7 3 15 1 45 1 1 5 2 1 2 1 1 6 1 10 4 3 2 1 2 2 32 1 1 6 1 4 2 2 2 2 20 16 2 1 20 1 1 5 1 9 1 1 1 1 2 1 1 1 3 1 2 5 1 1 2 4 7 15 4 14 4 2 1 21 1 20 2 35 6 1 2 2 1 21 1 11 1 2 21 1 2 1 13 1 3 6 4 2 1 2 5 7 4 1 19 1 3 4 29 10 1 1 28 1 1 1 3 5 2 1 2 2 1 2 2 9 181 1 1 3 4 1 2 1 1 1 8 1 1 1 13 2 1 1 3 2 3 2 7 3 3 1 1 31 1 1 9 4 1 2 5 5 1 2 1 1 1 1 1 4 5 4 2 2 3 26 2 4 1 4 3 37 64 2 3 2 7 1 2 2 3 1 1 1 2 1 20 1 1 1 6 3 2 2 11 1 1 2 1 1 11 1 4 4 2 1 1 1 2 1 1 5 8 10 1 2 5 1 2 1 6 5 1 1 1 5 1 1 1 1 2 60 1 3 4 6 1 17 5 4 5 2 21 1 1 6 12 1 19 1 1 11 4 1 3 4 1 5 4 2 1 1 3 3 33 1 1 5 6 2 2 8 1 7 1 1 1 4 1 1 1 48 3 1 7 4 2 1 1 2 20 1 21 1 2 2 22 2 2 1 1 3 1 5 1 1 3 2 3 1 2 1 2 9 1 99 25 1 2 11 4 1 1 2 95 1 1 2 1 6 4 2 3 1 1 1 1 1 2 1 4 3 2 1 6 2 2 7 1 1 1 1 1 7 2 182 8 1 5 4 11 1 1 1 3 1 1 11 1 5 8 1 2 1 2 11 2 1 1 2 2 2 2 2 1 7 73 1 4 1 4 3 1 9 2 1 1 4 5 2 5 5 31 1 20 1 1 6 3 3 17 5 5 1 1 1 9 3 17 1 73 1 3 1 117 2 4 1 1 13 1 1 1 1 1 7 1 3 3 1 1 1 1 61 3 3 5 5 1 1 3 5 5 3 1 1 13 1 2 1 8 1 1 5 1 2 1 2 2 7 2 1 1 6 1 1 1 8 19 1 3 5 1 4 10 6 11 3 3 1 23 1 4 1 2 1 1 3 2 4 8 1 11 9 1 4 2 1 1 2 1 1 3 1 11 4 17 1 2 1 8 2 1 1 3 5 1 2 3 1 30 4 2 8 4 1 31 1 4 1 1 6 2 1 2 4 16 7 3 17 1 1 2 1 8 2 4878 1 20 2 3 7 1 10 1 7 1 1 2 20 1 1 1 1 6 2 11 6 2 1 2 7 1 3 8 1 1 24 1 3 2 16 1 2 1 2 2 1 2 10 1 2 2 2 1 1 1 1 6 2 6 2 2 1 1 2 3 4 2 38 1 1 5 3 1 1 3 3 8 5 19 1 8 1 202 1 1 7 18 1 2 1 1 1 3 4 1 7 1 17 3 3 1 1 9 20 2 2 2 217 32 2 2 2 4 1 2 14 3 2 3 1 4 3 13 12 3 3 1 7 1 26 1 4 4 1 1 6 2 1 1 7 4 1 1 2 1 9 3 1 3 2 1 1 4 8 2 15 2 18 5 4 13 18 2 1 3 2 82 6 1 1 7 1 1 3 3 25 1 19 1 2 1 2 1 1 2 12 8 3 1 1 1 13 1 2 1 1 8 2 3 1 2 2 1 5 3 5 1 2 6 8 1 1 18 1 1 1 1 1 2 79 3 46 1 2 1 2 2 1 3 4 1 7 4 1 1 1 7 13 1 4 17 1 50 1 1 7 1 5 2 1 8 10 1 126 3 3 1 1 3 5 1 1 2 1 1 5 3 7 1 2 127 2 41 1 1 1 4 14 92 1 1 1 5 2 14 2 1 2 6 1 297 5 5 2 18 3 1 4 2 1 9 7 4 54 3 2 1 1 1 2 13 1 3 1 1 4 5 4 1 1 7 5 1 7 6 1 5 1 2 5 1 1 1 46 2 1 1 4 1 1 4 24 1 2 8 1 2 12 1 13 1 102 2 1 3 1 1 197 2 3 2 2 1 4 2 4 2 1 1 2 1 10 1 5 1 8 1 3 3 3 1 8 2 7 2 1 1 10 4 1 2 5 1 30 20 45 1 13 5 1 1 1 1 3 11 94 3 1 2 2 8 1 34 2 3 2 1 12 1 1 5 1 9 1 15 1 6 8 2 2 2 1 179 2 5 1 1 13 3 16 1 2 1 1 1 1 1 2 7 2 3 1 2 6 1 6 1 1 1 4 1 1 1 2 1 1 1 1 1 4 1 14 1 8 2 17 1 1 3 102 2 306 14 4 1 5 1 2 11 1 2 2 1 8 2 5 8 1 2 1 1 1 22 2 2 1 18 1 3 1 1 1 7 2 14 1 2 3 4 1 2 4 2 1 1 2 5 1 2 2 3 1 1 2 1 3 1 6 1 1 1 31 3 1 1 1 1 3 2 1 4 188 1 1 85 2 1 1 10 2 2 1 88 1 10 1 1 3 3 2 1 70 2 1 2 5 4 11 2 3 8 4 1 1 1 4 2 1 1 2 3 1 3 1 1 8 1 1 1 10 8 1 11 10 1 4 1 9 1 2 8 2 30 1 140 1 1 7 2 2 1 1 5 1 1 3 4 1 1 1 3 1 3095 5 1 4 4 3 1 15 1 9 1 1 2 1 1 2 1 4 5 1 2 317 1 2 1 5 1 2 3 2 1 2 1 7 1 1 1 18 1 7 1 1 1 1 1 3 2 3 4 4 1 1 4 1 1 3 2 1 2 6 1 9 3 6 1 6 1 9 1 3 1 2 579 2 2 11 1 10 1 3 1 10 11 66 3 1 3 2 1 41 7 1 5 882 1 3 3 1 3 3 3 1 48 41 7 1 1 4 1 3 4 2 1 9 5 2 26 1 13 1 1 18 1 1 1 1 2 8 1 2 14 1 1 1 3 1 1 1 2 1 1 23 1 1 3 3 2 1 4 2 1 1 5 1 1 4 3 1 2 1 1 1 1 1 3 3 2 9 1 2 3 1 2 3 14 4 1 23 1 1 10 1 3 2 1 1 1 12 1 9 1 1 2 486 1 2 1 6 1 1 7 10 1 1 2 1 2 5 1 3 1 3 13 1 4 1 2 5 35 1 15 46 10 18 4 6 3 5 9 17 2 2 4 1 1 1 1 1 11 68 3 9 1 1 2 2 2 13 13 2 3 4 1 1 3 1 2 1 6 3 26 3 2 2 1 1 3 1 1 22 1 5 10 3 1 1 5 13 18 1 20 1 9 2 2 2 1 1 1 2 3 2 2 1 6 2 18 1 1 1 17 1 3 1 10 1 9 4 4 1 1 2 1 1 2 11 1 2 1 221 1 1 5 8 26 2 3 6 1 4 1 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10 1 1 1 1 6 1 6 1 1 1 23 1 2 2 2 2 4 4 3 13 2 1 2 1 1 6 1 12 1 1 1 3 1 1 23 2 8 3 1 9 8 3 6 1 1 16 1 2 1 1 7 1 7 3 1 2 38 1 1 1 2 6 2 1 2 1 7 5 1 8 1 3 2 3 1 3 17 6 1 1 5 2 16 2 1 2 2 1 2 2 3 4 1 1 1 3 8 1 4 5 1 7 6 2 1 2 5 1 1 2 1 10 1 10 1 1 9 1 3 1 2 4 2 4 15 2 1 15 21 1 2 10 4 24 1 1 1 3 1 31 1 13 141 1 1 2 1 8 2 1 4 3 1 1 36 2 1 1 1 2 1 5 2 28 2 4 1 2 5 1 2 1 4 21 1 3 1 2 1 2 48 2 1 7 4 1 2 1 1 2 1 1 1 1 11 2 12 3 3 3 2 2 3 2 2 1 25 1 1 37 1 8 1 3 3 2 1 7 1 1 1 2 1 1 1 1 3 1 2 2 6 4 3 2 1 2 1 1 2 49 1 3 1 2 1 1 12 2 3 11 1 15 9 13 2 4 15 1 1 2 8 2 1 1 1 7 2 1 4 3 1 15 1 2 22 1 10 6 1 2 2 3 1 2 4 10 1 1 2 1 5 2 2 8 1 14 1 7 2 1 17 1 1 1 149 1 4 2 1 4 1 2 1 1 2 1 1640 4 2 17 1 1 1 1 1 1 17 2 5 1 1 1 3 9 1 1 4 1 2 2 2 7 4 1 9 1 5 1 52 1 2 1 2 1 3 1 2 9 1 39 1 1 2 1 1 1 3 2 2 35 7 2 1 5 2 3 1 1 2 1 10 3 1 24 71 1 7 1 1 3 2 9 1 3 1 3 1 11 1 2 2 1 1 1 1 5 1 24 1 4 5 13 5 1 1 11 6 296 1 19 12 2 3 1 1 8 74 7 3 3 1 3 8 3 2 1 10 2 1 1 1 3 1 6 4 1 5 1 19 5 1 6 1 3 1 18 4 7 2 2 1 3 3 1 3 1 2 1 2 1 1 4 1 1 4 3 1 1 4 1 1 1 1 1 1 2 1 3 15 1 1 2 1 4 2 2 62 5 1 18 2 4 1 2 3 40 1 1 1 1 3 12 26 1 1 19 2 3 1 5 3 1 1 2 3 4 2 1 5 2 1 3 1 6 1 1 1 3 1 2 1 1 1 5 5 1 3 1 1 3 140 2 3 1 2 15 6 3 1 1 3 2 6 1 1 2 10 1 7 2 2 6 1 14 5 2 4 2 1 1 2 1 1 1 24 2 23 2 5 1 1 2 1 6 1 1 1 95 6 2 11 1 3 2 1 23 1 45 14 1 7 2 3 1 17 7 1 2 5 3 201 20 48 2 1 1 3 2 23 1 3 1 1 6 1 26 2 3 10 1 1 3 1 10 17 1 1 2 1 8 1 6 25 15 1 3 3 2 2 1 2 1 17 1 19 1 1 1 2 5 1 1 2 1 6 11 1 1 4 1 1 1 2 16 1 2 1 1 28 1 14 3 5 12 1 25 5 1 1 1 1 14 1 3 3 1 1 1 1 2 5 1 1 2 4 64 2 6 2 1 9 1 3 1 6 32 1 1 5 1 2 30 2 1 30 1 1 1 1 4 1 2 1 1 5 1 4 3 1 5 2 2 1 1 2 3 2 77 1 1 1 2 1 19 2 1 1 46 5 25 1 1 1 1 6 5 2 1 7 1 37 9 1 7 2 14 33 4 1 1 1 1 1 1 11 1 9 2 6 1 1 1 2 1 1 2 12 27 2 2 5 1 5 2 1 3 3 3 1 40 1 2 8 1 5 1 23 2 4 1 2 2 4 3 5 2 4 1 2 1 10 1 3 11 15 2 4 2 4 1266 2 6 2 1 21 1 5 2 14 3 5 1 8 1 21 135 2 3 1 1 1 2 2 1 1 8 5 1 1 55 1 2 1 5 2 2 3 65 2 1 4 1 1 19 2 2 1 2 1 2 1 2 1 2 1 2 3 1 3 3 6 1 3 7 4 1 50 2 2 80 3 20 3 2 5 1 7 1 17 8 2 2 1 1 3 1 9 2 7 2 2 4 1 1 7 2 74 3 1 2 1 7 5 20 3 1 134 3 1 1 8 3 1 2 7 8 4 2 1 1 3 2 2 2 75 1 10 2 3 2 1 3 1 7 1 26 2 9 12 429 9 3 2 7 2 1 2 1 1 4 1 5 3 1 5 1 4 3 1 1 3 1 1 1 1 2 1 4 1 10 1 3 6 1 9 3 1 2 1 10 1 2 3 2 4 5 57 3 4 1 2 1 1 1 7 4 1 43 2 3 1 1 3 1 2 1 1 1 4 6 1 1 11 2 1 9 1 27 3 1 270 9 2 1 3 1 46 1 10 5 1 7 1 1 1 9 1 14 1 1 4 1 3 1 1 4 1 19 6 15 3 2 6 2 1 1 8 1 17 1 3 1 1 3 1 1 1 1 1 2 6 1 3 1 2 4 10 1 2 1 256 2 2 2 1 3 1 2 35 1 13 1 4 1 1 4 1 20 2 3 28 1 1 13 2 2 2 2 1 3 5 1 32 1 6 2 4 2 6 11 57 2 6 14 2 3 3 1 2 26 1 1 1 18 8 1 2 2 15 2 1 1 1 4 1 2 2 128 1 5 1 1 3 1 1 2 17 1 1 1 1 1 1 6 1 1 4 1 5 3 1 2 6 1 1 1 4 1 3 2 2 3 190 1 1 89 1 9 2 11 2 1 13 4 2 1 9 1 5 5 1 5 1 2 1 4 3 1 1 488 1 3 2 5 1 11 3 3 1 10 4 2 2 1 1 6 2 1 1 3 1 12 12 51 2 1 5 12 3 3 1 2 5 2 1 11 1 5 1 1 1 8 4 4 1 1 8 1 6 28 1 11 1 19 4 4 120 8 5 3 7 1 4 5 2 2 1 2 1 1 2 2 4 3 18 1 2 5 4 1 1 2 16 6 1 2 1 1 74 8 1 1 3 1 1 61 20 1 1 21 2 1 2 9 2 57 9 8 4 1 29 26 1 121 1 1 1 4 14 1 1 19 1 7 1 2 1 1 1 4 1 5 6 1 1 2 2 2 4 1 2 2 2 4 1 3 11 4 1 2 4 5 1 4 4 10 1 24 1 2 1 2 11 1 2 1 1 1 1 12 1 12 2 1 1 2 4 7 1 4 1 58 5 3 6 1 6 3 5 1 1 3 2 7 1 2 5 1 1 3 1 10 1 12 62 10 1 1 5 3 1 1 2 1 2 1 10 15 1 35 2 27 1 1 12 2 5 8 4 1 1 2 15 1 3 1 1 1 9 6 1 1 1 8 3 1 1 3 2 6 15 2 29 4 8 1 1 1 55 2 3 4 1 11 15 3 1 1 3 1 5 1 3 1 1 2 5 2 1 6 2 2 1 5 1 1 2 4 4 1 1 1 1 1 2 1 11 1 1 4 723 13 2 2 1 4 5 2 1 1 2 3 4 5 1 2 1 2 10 1 13 1 1 1 1 1 4 1 1 2 9 3 1 1 1 1 8 1 2 1 2 1 4 9 6 232 3 2 3 1 1 2 2 1 2 6 2 1 2 1 11 13 2 6 18 1 1 1 1 1 1 2 1 2 2 28 1 3 4 4 1 6 1 3 1 2 1 9 46 1 1 1 1 1 5 1 25 1 7 14 1 5 1 54 3 2 1 1 3 1 1 1 1 5 1 57 1 2 1 5 1 4 18 1 1 1 2 3 3 1 5 14 1 2 5 2 2 6 2 2 64 9 2 2 1 1 4 2 1 1 1 3 1 3 2 6 1 2 1 5 2 4 1 3 7 2 1 4 4 6 6 1 8 2 3 2 2 5 2 1 101 2 1 3 1 1 1 1 1 3 2 2 2 3 1 4 3 1 20 11 1 3 2 1 36 2 2 3 1 30 1 5 3 4 1 2 1 1 2 2 1 1 8 7 21 1 4 3 1 5 8 1 1 3 1 1 3 1 4 3 11 1 1 1 4 1 8 21 1 1 9 1 8 1 2 1 2 55 1 1 1 4 2 12 2 1 13 3 1 11 3 1 3 1 1 1 17 1 2 3 1 1 1 66 4 9 2 3 108 1 117 1 282 2 4 1 2 29 1 6 1 1 1 3 2 1 19 1 6 3 1 1 7 1 12 8 2 1 1 1 37 1 6 19 1 9 2 1 5 1 5 22 1 3 1 2 10 1 5 12 3 1 3 84 1 2 1 2 28 1 1 33 7 1 1 1 3 4 1 1 1 2 2 3 4 269 1 3 1 7 1 4 1 262 6 2 1 22 6 1 8 1 3 6 2 1 5 3 4 6 1 47 1 3 1 2 1 1 2 10 1 30 1 2 10 1 1 1 1 1 2 2 1 1 1 2 7 2 1 1 1 2 1 7 1 2 1 1 2 1 16 1 6 4 2 1 9 1 7 1 7 5 34 3 2 4 5 1 1 2 2 3 71 2 75 16 3 6 1 1 1 1 1 1 4 3 3 3 1 1 3 2 8 1 5 37 9 37 2 2 2 30 16 19 1 6 4 1 1 2 1 25 2 1 1 7 1 6 1 1 4 19 2 2 1 1 2 2 1 1 3 1 1 3 5 1 1 3 1 2 6 3 1 1 1 4 2 3 1 2 11 1 2 1 1 11 7 1 6 2 1 1 1 1 4 2 1 2 126 2 6 49 1 19 1 6 2 2 7 4 1 1 3 6 4 7 6 2 1 1 18 2 22 1 9 3 28 2 4 1 9 1 5 1 8 2 3 2 4 2 607 1 6 10 2 1 2 217 2 7 1 6 1 1 2 1 5 3 1 1 1 14 4 1 1 5 1 2 2 3 2 75 1 28 3 1 1 9 7 6 1 1 1 8 1 8 1 1 5 11 1 1 1 1 3 2 701 1 3 5 7 2 2 2 1 1 1 47 2 1 3 5 5 4 4 6 1 1 1 1 1 1 1 2 1 1 1 1 2 4 1 3 17 1 7 4 1 3 1 1 1 31 3 1 8 3 1 1 1 2 1 41 388 1 4 1 1 10 1 1 26 2 17 5 3 16 1 1 5 2 2 2 4 15 3 1 1 4 1 2 11 38 2 3 2 6 2 32 5 1 1 3 1 3 2 8 1 5 2 2 3 2 3 4 1 3 5 16 1 3 1 1 11 6 20 1 2 1 1 4 33 1 2 2 1 2 7 1 1 1 3 4 2 1 1 1 2 1 3 2 4 1 5 1 4 1 2 4 1 1 1 3 10 8 3 14 1 1 1 1 34 7 1 1 37 2 1 5 1 75 131 3 1 11 1 56 21 1 1 1 1 14 1 1 7 1 1 1 3 12 3 3 2 6 1 29 3 192 1 3 1 4 2 1 4 6 1 3 6 1 3 32 2 3 3 10 1 8 3 1 1 2 3 1 2 1 2 39 1 4 3 1 12 19 1 2 1 2 1 1 3 3 1 1 3 1 1 2 1 1 5 17 1 4 2 13 3 1 4 1 9 6 1 6 1 1 1 12 3 1 4 10 15 2 7 5 1 13 2 1 1 5 5 3 4 2 1 1 1 1 1 3 5 3 1 15 1 1 2 3 7 5 1 5 3 3 1 1 1 3 1 2 339 29 1 2 2 4 1 1 1 2 6 1 1 8 1 1 10 6 2 1 5 7 2 5 2 1 2 1 1 59599

[upqphxne] No associativity for non-associative operators

Consider providing in a programming language a feature in which the programmer can optionally disable acceptability of expressions that use subtraction or division twice or more in a row, for example a - b - c - d or a/b/c/d, because they are potentially confusing because you might forget that the operations must be performed left to right.  Tricky formatting could induce confusion:

/* this is actually equal to 1/4 */
one=1/2
     /
    1/2;

The programmer must explicitly parenthesize: ((a - b) - c) - d, ((a/b)/c)/d, or a - (b + c + d), a/(b*c*d), or a + (-b) + (-c) + (-d), a * (1/b) * (1/c) * (1/d).

Multiple additions or multiplications in a row are not confusing (and therefore remain acceptable) because those operations are associative (ignoring issues like precision loss and overflow).

Other operators we may want to disable their use in series: a <= b <= c, a == b == c, because those expressions don't mean what they mean in mathematics.  The remainder operator (mod): a%b%c.  Exponentiation a^b^c because people often forget it is right associative.

We probably don't want to disable the use of all non-associative operators in series.  Here are some we likely want to keep: list construction (:) and function application ($) in Haskell.  (Previously on expressions of this form: syntactic fold.)  (Incidentally, function composition (.) is associative.)  Stream input (>>) and output (<<) in C++.  However, we may want to disable the same C++ operators when used in series in arithmetic expressions, e.g., 1 << 1 << 3, for the same reason as doing so for subtraction and division: it's potentially confusing.  Left shift feels similar to exponentiation, but it is left associative.

Tuesday, November 27, 2018

[yprxrewc] Buses and property values

Does putting in a bus line through a neighborhood increase or decrease property values?

Increase, because more convenient travel for residents in the neighborhood.

Decrease, because buses are noisy.  NIMBY rears its ugly head, ultimately giving us global warming through lots of individual cars.

The net result probably reflects car ownership in the neighborhood, then probably wealth.

[nlgsjgxg] RootOf for a calculator

After all the functions, the next most useful feature a calculator could provide is RootOf.  However, to specify the function to be solved, it needs to provide a general-purpose programming language, with all its syntax and UI complexities.

Though once you bite that bullet, lots of other things become possible as well, e.g., numerical integration, automatic differentiation, optimization, summing finite series.

Monday, November 26, 2018

[llsnfuxb] Example finite fields

Here are the multiplication, reciprocal, and exponentiation tables for a few small finite fields (Galois fields) of composite order (size).  A composite order necessarily means a prime raised to a power two or greater.  We don't give addition tables because they are not very interesting, just modular addition digit by digit without carry.

We follow the format of a similar presentation of a finite field of size 13.

Computations were done with Pari/GP whose Mod() notation can do finite field arithmetic reducing by a polynomial.  Source code.  The ffinit() function finds an irreducible polynomial (one of many possible irreducible polynomials) for a finite field of given size.  The values are given as the coefficients to the polynomial representation of elements, in big-endian order:

mffdisp(x,digits)=for(j=1,digits,i=digits-j;print1(lift(polcoeff(lift(x),i))));

In the multiplication table, we highlight in bold instances where the product is one, the multiplicative identity: these mark reciprocals.

In the exponentiation table, we again highlight outputs which are one.  Rows which have the minimal number of ones are primitive polynomials, generators of the multiplicative group.  In rows which aren't generators, the ones occur periodically, so the pattern depends on the factorization of (order-1).  We observe a phenomenon similar to Fermat's Little Theorem: the penultimate column is always one.  0^0 is undefined, so we mark it with a question mark.

Here is another example multiplication table for GF(3^2), though using a different reduction polynomial than below.

Future task: create an images of GF(p^3) tables, mapping each 3-component element to an RGB color.  This could be done for lower powers, too.

GF(2^2)

Order: 4
Reduction polynomial: Mod(1, 2)*x^2 + Mod(1, 2)*x + Mod(1, 2)

Multiplication

*00011011
0000000000
0100011011
1000101101
1100110110

Multiplicative inverse (reciprocal)

x011011
x^-1011110

Raising to a power (exponentiation)

^01234
00?00000000
010101010101
100110110110
110111100111


GF(2^3)

Order: 8
Reduction polynomial: Mod(1, 2)*x^3 + Mod(1, 2)*x^2 + Mod(1, 2)

Multiplication

*000001010011100101110111
000000000000000000000000000
001000001010011100101110111
010000010100110101111001011
011000011110101001010111100
100000100101001111011010110
101000101111010011110100001
110000110001111010100011101
111000111011100110001101010

Multiplicative inverse (reciprocal)

x001010011100101110111
x^-1001110100011111010101

Raising to a power (exponentiation)

^012345678
000?000000000000000000000000
001001001001001001001001001001
010001010100101111011110001010
011001011101010110111100001011
100001100111110010101011001100
101001101110100011010111001101
110001110011111101100010001110
111001111010011100110101001111


GF(2^4)

Order: 16
Reduction polynomial: Mod(1, 2)*x^4 + Mod(1, 2)*x^3 + Mod(1, 2)*x^2 + Mod(1, 2)*x + Mod(1, 2)

Multiplication

*0000000100100011010001010110011110001001101010111100110111101111
00000000000000000000000000000000000000000000000000000000000000000000
00010000000100100011010001010110011110001001101010111100110111101111
00100000001001000110100010101100111011111101101110010111010100110001
00110000001101100101110011111010100101110100000100101011100011011110
01000000010010001100111110110111001100010101100111011110101001100010
01010000010110101111101111100001010010011100001101100010011110001101
01100000011011001010011100011011110111101000001001001001111101010011
01110000011111101001001101001101101001100001100011110101001010111100
10000000100011110111000110011110011000101010110101010011101111000100
10010000100111010100010111001000000110100011011111101111011000101011
10100000101010110001100100110010100011010111011011000100111011110101
10110000101110010010110101100100111101011110110001111000001100011010
11000000110001111011111000101001010100111111010010001101000110100110
11010000110101011000101001111111001010110110111000110001110001001001
11100000111000111101011010000101101111000010111100011010010010010111
11110000111100011110001011010011110001001011010110100110100101111000

Multiplicative inverse (reciprocal)

x000100100011010001010110011110001001101010111100110111101111
x^-1000111111010100001100101100101000111001111101101110010110010

Raising to a power (exponentiation)

^012345678910111213141516
0000?0000000000000000000000000000000000000000000000000000000000000000
000100010001000100010001000100010001000100010001000100010001000100010001
001000010010010010001111000100100100100011110001001001001000111100010010
001100010011010111111110110110000111100101001100101100100110101000010011
010000010100111100101000000101001111001010000001010011110010100000010100
010100010101111010001001110000101010001111111101011101001011011000010101
011000010110101101000111110111110011101000101100100110001110010100010110
011100010111101010000110110100101110101111111100010101000011100100010111
100000011000001011110100000110000010111101000001100000101111010000011000
100100011001001101000101110011111011111000101101011010001010011100011001
101000011010011000101011110001001001011110001101111011110101001100011010
101100011011011111111010110010000101011001001101001100101001111000011011
110000011100110100011100110100011100110100011100110100011100110100011100
110100011101110000011101110000011101110000011101110000011101110000011101
111000011110100100100011110101000110010110001100101011110111101100011110
111100011111100001000010000111111000010000100001111110000100001000011111


GF(3^2)

Order: 9
Reduction polynomial: Mod(1, 3)*x^2 + Mod(1, 3)*x + Mod(2, 3)

Multiplication

*000102101112202122
00000000000000000000
01000102101112202122
02000201202221101211
10001020210111122202
11001122011220021021
12001221112002220110
20002010120222211101
21002112221001110220
22002211022110012012

Multiplicative inverse (reciprocal)

x0102101112202122
x^-10102111021221220

Raising to a power (exponentiation)

^0123456789
00?000000000000000000
0101010101010101010101
0201020102010201020102
1001102122022012110110
1101111220022221100111
1201120221011202210112
2001202111021012220120
2101210212012102120121
2201221210021121200122


GF(3^3)

Order: 27
Reduction polynomial: Mod(1, 3)*x^3 + Mod(1, 3)*x^2 + Mod(1, 3)*x + Mod(2, 3)

Multiplication

*000001002010011012020021022100101102110111112120121122200201202210211212220221222
000000000000000000000000000000000000000000000000000000000000000000000000000000000000
001000001002010011012020021022100101102110111112120121122200201202210211212220221222
002000002001020022021010012011200202201220222221210212211100102101120122121110112111
010000010020100110120200210220221201211021001011121101111112122102212222202012022002
011000011022110121102220201212021002010101112120211222200012020001122100111202210221
012000012021120102111210222201121100112211220202001010022212221200002011020122101110
020000020010200220210100120110112102122012002022212202222221211201121111101021011001
021000021012210201222120111102212200221122110101002020011121112100001022010211202220
022000022011220212201110102121012001020202221210122111100021010002211200222101120112
100000100200221021121112212012022122222210010110101201001011111211202002102120220020
101000101202201002100102200001122220021020121222221022120211012110112210011010111212
102000102201211010112122221020222021120100202001011110212111210012022121220200002101
110000110220021101211012122202210020100201011121222002112120200010111221001102212022
111000111222001112220002110221010121202011122200012120201020101212021102210022100211
112000112221011120202022101210110222001121200012102211020220002111201010122212021100
120000120210121211001212002122101221011222012102010100220202022112020110200111201021
121000121212101222010202020111201022110002120211100221012102220011200021112001122210
122000122211111200022222011100001120212112201020220012101002121210110202021221010102
200000200100112012212221121021011211111120020220202102002022222122101001201210110010
201000201102122020221211112010111012210200101002022220121222120021011212110100001202
202000202101102001200201100002211110012010212111112011210122021220221120022020222121
210000210120212122002121001211202112022111021201020200110101011221010220100222102012
211000211122222100011111022200002210121221102010110021202001212120220101012112020201
212000212121202111020101010222102011220001210122200112021201110022100012221002211120
220000220110012202122021211101120010200102022212111001221210100020222112002201121011
221000221112022210101011202120220111002212100021201122010110001222102020211121012200
222000222111002221110001220112020212101022211100021210102010202121012201120011200122

Multiplicative inverse (reciprocal)

x001002010011012020021022100101102110111112120121122200201202210211212220221222
x^-1001002111202120222210101122022112212010102012220100211221011021200110121201020

Raising to a power (exponentiation)

^0123456789101112131415161718192021222324252627
000?000000000000000000000000000000000000000000000000000000000000000000000000000000000
001001001001001001001001001001001001001001001001001001001001001001001001001001001001001
002001002001002001002001002001002001002001002001002001002001002001002001002001002001002
010001010100221022220012120121101201122111001010100221022220012120121101201122111001010
011001011121222221210122200012102010110101002022212111112120211100021201020220202001011
012001012111220122022201221101100121010120001012111220122022201221101100121010120001012
020001020100112022110012210121202201211111002010200221011220021120212101102122222001020
021001021111110122011201112101200121020120002012222220211022102221202100212010210001021
022001022121111221120122100012201010220101001022121111221120122100012201010220101001022
100001100022012121201111010221220120101122001100022012121201111010221220120101122001100
101001101220010201012100122120221111121022001101220010201012100122120221111121022001101
102001102120011010211121110100222101021221002201210022020122212220200111202012112001102
110001110201200120222022202010021122112121002220102100210111011101020012211221212001110
111001111122201101121120012220022221100010001111122201101121120012220022221100010001111
112001112012202111200220212122020022210201002221021101222100110121211010011120102001112
120001120010121100101221201022122220111012001120010121100101221201022122220111012001120
121001121221122012010101022111120100201220001121221122012010101022111120100201220001121
122001122101120220221010111201121012022100001122101120220221010111201121012022100001122
200001200022021121102111020221110120202122002100011012212201222010112220210101211001200
201001201120022010122121220100111101012221001201120022010122121220100111101012221001201
202001202220020201021100211120112111212022002101110010102012200122210221222121011001202
210001210010212100202221102022211220222012002120020121200101112201011122110111021001210
211001211101210220112010222201212012011100002122202120110221020111102121021022200001211
212001212221211012020101011111210100102220002121112122021010202022222120200201110001212
220001220201100120111022101010012122221121001220201100120111022101010012122221121001220
221001221012101111100220121122010022120201001221012101111100220121122010022120201001221
222001222122102101212120021220011221200010002111211201202121210012110022112100020001222


GF(5^2)

Order: 25
Reduction polynomial: Mod(1, 5)*x^2 + Mod(1, 5)*x + Mod(1, 5)

Multiplication

*00010203041011121314202122232430313233344041424344
0000000000000000000000000000000000000000000000000000
0100010203041011121314202122232430313233344041424344
0200020401032022242123404244414310121411133032343133
0300030104023033313432101311141240434144422023212422
0400040302014044434241303433323120242322211014131211
1000102030404404142434334303132322324202121121314101
1100112233440410213243031420314202132430410112233440
1200122431431421334002233042041132440113204103102234
1300132134422432400311430114223012203341043144021023
1400142332413443021120132231400442011024332130440312
2000204010303303234313113101214144143404242242123202
2100214213344314300122310223441024401132031233042041
2200224411330320421431012340123404214310320224411330
2300234114321331042240214412300334022043114210330124
2400244312312342113004411034032214330221403201204413
3000301040202202321242442404341411412101313313432303
3100311243243213442001144021023341220334102304301142
3200321441234224013310341143200221033012441340220431
3300331144220230134124043210432101341240230331144220
3400341342211241200433240332114031104423024322013014
4000403020101101413121221202423233231303434434241404
4100413223142112034430423324100113044031223420110243
4200423421133123100244120441332043302214012411034032
4300433124124134221003322013014423110442301402403321
4400443322110140342312024130241303423120140443322110

Multiplicative inverse (reciprocal)

x010203041011121314202122232430313233344041424344
x^-1010302044440322131221320434133141230421124342310

Raising to a power (exponentiation)

^012345678910111213141516171819202122232425
00?00000000000000000000000000000000000000000000000000
010101010101010101010101010101010101010101010101010101
020102040301020403010204030102040301020403010204030102
030103040201030402010304020103040201030402010304020103
040104010401040104010401040104010401040104010401040104
100110440110440110440110440110440110440110440110440110
110111100444400111100444400111100444400111100444400111
120112331340410331443420230443224210140224112130320112
130113033404420221011303340442022101130334044202210113
140114201311430332103433240441304244120223402122310114
200120110310330430440240220120110310330430440240220120
210121024204340313012102420434031301210242043403130121
220122400244300433100311200122400244300433100311200122
230123303411310241101322120432202144240314404233430123
240124223440320243441330140431332110230312114220410124
300130110210220420440340330130110210220420440340330130
310131222140230212444230410424333410320343111320140131
320132302111240214104222430423203444310341401333120132
330133400344200422100211300133400344200422100211300133
340134021304210342013402130421034201340213042103420134
400140440410110140440410110140440410110140440410110140
410141204211120323102133310414301344430232403422240141
420142032104130234014203210413023401420321041302340142
430143334240140324442120320412221310410231113430230143
440144100144100144100144100144100144100144100144100144