Roots of Identity of Boolean 3x3 matrices

see also Ken's blog: Inverses of Binary 3x3 matrices

The 22 square roots of unity (identity) are 124² = 174² = 241² = 247² = 326² = 351² = 412² = 421² = 423² = 425² = 427² = 431² = 456² = 461² = 465² = 471² = 521² = 531² = 621² = 623² = 712² = 721² = 421

Other roots... There appear to be 1st, 2nd, 3rd, 4th, and 7th roots. There are 168 total. The distribution is 1 1st root, 21 2nd roots, 56 3rd roots, 42 4th roots, 48 7the roots.

124 421
125 524 421
126 623 324 421
127 724 421
134 471 164 421
135 564 421
136 652 273 345 517 764 421
137 745 516 672 253 364 421
142 214 421
143 317 736 675 562 254 421
146 615 567 753 372 234 421
147 712 274 421
152 235 573 367 746 614 421
153 326 654 421
156 674 421
157 763 342 215 576 634 421
162 276 657 735 543 314 421
163 375 542 216 637 754 421
164 471 134 421
165 574 421
172 247 714 421
173 354 421
174 421
175 534 421
214 142 421
215 157 576 763 634 342 421
216 163 637 375 754 542 421
217 174 742 421
234 372 753 567 615 146 421
235 367 614 152 573 746 421
236 351 546 421
237 346 421
241 421
243 427 245 421
245 427 243 421
247 421
251 531 341 421
253 516 137 364 672 745 421
254 562 675 736 317 143 421
256 547 421
261 641 421
263 645 421
265 647 421
267 643 421
271 741 421
273 764 652 517 136 345 421
274 712 147 421
276 735 314 162 657 543 421
314 543 735 657 276 162 421
315 456 362 421
316 762 421
317 675 254 143 736 562 421
324 623 126 421
325 726 421
326 421
327 526 421
341 531 251 421
342 634 763 576 157 215 421
345 136 517 652 764 273 421
346 237 421
351 421
352 715 421
354 173 421
357 241 537 421
361 751 421
362 456 315 421
364 253 672 516 745 137 421
367 152 746 235 614 573 421
371 651 421
372 567 146 234 753 615 421
375 216 754 163 542 637 421
376 124 673 421
412 421
413 432 421
416 465 452 421
417 472 421
421
423 421
425 421
427 421
431 421
432 413 421
435 461 475 421
436 457 421
452 465 416 421
453 476 421
456 421
457 436 421
461 421
463 425 467 421
465 421
467 425 463 421
471 421
472 417 421
475 461 435 421
476 453 421
512 721 612 421
513 632 421
516 364 745 253 137 672 421
517 273 136 764 345 652 421
521 421
523 621 723 421
524 125 421
526 327 421
531 421
532 713 421
534 175 421
537 241 357 421
542 754 375 637 163 216 421
543 657 162 314 735 276 421
546 351 236 421
547 256 421
561 431 571 421
562 736 143 254 675 317 421
564 135 421
567 234 615 372 146 753 421
571 431 561 421
573 614 235 746 152 367 421
574 165 421
576 342 157 634 215 763 421
612 721 512 421
613 732 421
614 746 367 573 235 152 421
615 753 234 146 567 372 421
621 421
623 421
625 423 627 421
627 423 625 421
631 521 731 421
632 513 421
634 576 215 342 763 157 421
637 542 163 754 216 375 421
641 261 421
643 267 421
645 263 421
647 265 421
651 371 421
652 345 764 136 273 517 421
654 326 153 421
657 314 276 543 162 735 421
672 137 253 745 364 516 421
673 124 376 421
674 156 421
675 143 562 317 254 736 421
712 421
713 532 421
714 247 172 421
715 352 421
721 421
723 621 523 421
724 127 421
726 325 421
731 521 631 421
732 613 421
735 162 543 276 314 657 421
736 254 317 562 143 675 421
741 271 421
742 174 217 421
745 672 364 137 516 253 421
746 573 152 614 367 235 421
751 361 421
753 146 372 615 234 567 421
754 637 216 542 375 163 421
756 412 765 421
762 316 421
763 215 634 157 342 576 421
764 517 345 273 652 136 421
765 412 756 421

67424 sudoku puzzles

computer generated sudoku puzzles. The format is one puzzle per line, rows separated by pluses, suitable for string input into Sudoku Solver by Logic.

NFSNET large factorization

NFSNET should attempt a very large sieving portion of a factorization, leaving the linear algebra to the future: just leave the sieving results posted for someone else, with different resources, to tackle.

It may be more data than the project may wish to keep on hand, so some distributed storage system of volunteers is needed. BitTorrent is close, but not exactly what we want: we want to tell bittorrent to please download upto (say) 50GB of this file, and a randomly chosen 50GB, and stop, but keep "seeding" the partial file.

Rubik's cube center face rotates

As I suspected, the center square of each face of a rubik's cube can rotate, leaving an identically colored cube. So even when a cube appears solved, it may not be in the original state. Therefore, a cube with an image painted on each (or even just one) side is more difficult than the regular cube.

What center square rotations are possible? Are all 4⁶ rotations possible, or are they linked? What are the quickest algorithms from one center square rotation to another, leaving other squares untouched? What states are the furthest distance apart?

105th power

105=3*5*7; the cyclotomic polynomial has a 2 coefficient.

? for(v=2,105,print(v," ",factor((v^105-1)/gcd(v^105-1,(v^35-1)*(v^21-1)*(v^15-1)*(v^7-1)))))
2 [29191, 1; 106681, 1; 152041, 1]
3 [421, 1; 6301, 1; 1616161, 1; 26751945361, 1]
4 [211, 1; 29191, 1; 106681, 1; 152041, 1; 664441, 1; 1564921, 1]
5 [1736701, 1; 119461537021, 1; 21226783250214361, 1]
6 [211, 1; 35281, 1; 58171, 1; 61921104791950322094158011, 1]
7 Mat([42693162668620904426304495389707999425601, 1])
8 [870031, 1; 983431, 1; 29728307155963706810228435378401, 1]
9 [211, 1; 421, 1; 1051, 1; 6301, 1; 24151, 1; 1616161, 1; 3369031, 1; 3454081, 1; 26751945361, 1]
10 [30703738801, 1; 625437743071, 1; 57802050308786191965409441, 1]
11 [421, 1; 540751, 1; 599551, 1; 2598121, 1; 126713791, 1; 2373141440024702184811, 1]
12 [2521, 1; 126001, 1; 9562853581, 1; 2268301966722094170973637436845641, 1]
13 [995277238201, 1; 320553148774196624469658887398942218735081, 1]
14 [20161, 1; 416104481521, 1; 1325735305253077646619978407581074191011, 1]
15 [24596480723711374110433501, 1; 12340745958492257916284855626741, 1]
16 [211, 1; 421, 1; 29191, 1; 106681, 1; 152041, 1; 664441, 1; 1564921, 1; 146919792181, 1; 1041815865690181, 1]
17 [1471, 1; 704761, 1; 872761, 1; 8208901, 1; 245614111, 1; 67096559666323873912471273981, 1]
18 [3361, 1; 26881, 1; 50451031, 1; 6391293850256281, 1; 65081321032032689150122343401, 1]
19 [116131, 1; 41888491, 1; 5206520255936910609072540845913496243195666334401, 1]
20 [211, 1; 1051, 1; 143677768081, 1; 338649469410070561, 1; 27455958930629469508499288821, 1]
21 [81090871, 1; 3987476221, 1; 5124240195826801224421, 1; 1855106691418101398774071, 1]
22 [8878834021, 1; 496997570388443929351, 1; 6482477075260707069597975952254601, 1]
23 [21001, 1; 4558141981, 1; 8829545269167938682361, 1; 285259002489570164883253468981, 1]
24 [211, 1; 15331, 1; 573771004175140372915323230675033255524409369213889481220161, 1]
25 [421, 1; 1736701, 1; 5236141, 1; 119461537021, 1; 21226783250214361, 1; 1354224218968567573270561, 1]
26 [421, 1; 2311, 1; 691049040256641389611, 1; 128273925551743698644742928265028665062111, 1]
27 [1571221, 1; 335422140063430947448491304902193015341719353456516541680805341, 1]
28 [421, 1; 6313861, 1; 98336245890572745925217555761, 1; 11536371289790796513662140395301, 1]
29 [14071, 1; 1153456110611495859334316274353558275930447581299596591411479220711, 1]
30 [211, 1; 391061376751449733631793465114261320107553845297768849234837044581521, 1]
31 [421, 1; 39849234924648425393072876341, 1; 23707462166621201930562854073708336673561, 1]
32 [4201, 1; 7351, 1; 181165951, 1; 325985508875527587669607097222667557116221139090131514801, 1]
  ***   Warning: MPQS: the factorization of this number will take several hours.
  ***   user interrupt after 33mn, 9,863 ms.
33 [403964075517082772293915365361, 1; 19755054605825307891112987294773170648106001, 1]
34 [176689034091151, 1; 321659534033041, 1; 587943929701845737685493450745728795510977721, 1]
35 [28657329822614221, 1; 4532176487551678210228760701, 1; 1033481205046232340291587468041, 1]
36 [211, 1; 35281, 1; 58171, 1; 71191, 1; 61921104791950322094158011, 1; 271613602977153099649378586566681, 1]
  ***   Warning: MPQS: the factorization of this number will take several hours.
37 [211, 1; 604227625372300046512884458502732151, 1; 15138762145219690961962450760690572381, 1]
38 [71562541, 1; 82398331, 1; 1176474239209348636348557351624067262849540520502788065743861, 1]
39 [631, 1; 39171091, 1; 58072141, 1; 16803972285393341346109651158716862478166907695439650629681, 1]
40 [1471, 1; 3585774151, 1; 15405385590389057603824293644619918311352163800508046297118171121, 1]
41 [101487961, 1; 9768978220172011, 1; 10880555194899391, 1; 2788122069477512041, 1; 8833161559392115261, 1]
  ***   Warning: MPQS: the factorization of this number will take several hours.
42 [19234605792687574641676801, 1; 43887638956844058654097430702733105025130615171946031, 1]
43 [23188215751, 1; 50271319771, 1; 4239813168584421248716609561, 1; 528159661714709770316955649021, 1]
44 [211, 1; 1051, 1; 1471, 1; 4621, 1; 75391, 1; 12343265131, 1; 2915628987996025141, 1; 1923056117158327113038993356492711, 1]
45 [211, 1; 7351, 1; 931981, 1; 9783661231, 1; 51146053902293083861, 1; 31959962063191228529620779850412740291, 1]
46 [211, 1; 421, 1; 30228241, 1; 24713789741966327389039366423788447503493258050203408693878591280861, 1]
47 [211, 1; 820681, 1; 6637023991133761, 1; 828843276245460525211, 1; 195492626587979268263464507123040761, 1]
48 [974469721, 1; 95525875237061641, 1; 13537746567872078581, 1; 405772948213010063185435016024571181, 1]
49 [211, 1; 338640865331157691, 1; 450798894542150330401, 1; 42693162668620904426304495389707999425601, 1]
  ***   Warning: MPQS: the factorization of this number will take several hours.
50 [23311, 1; 7880534122660344205512149931241, 1; 19733953405281613784871717440438356213119123601, 1]
51 [211, 1; 93871, 1; 17885219802451, 1; 63198678486961711, 1; 418744552822050715296333996416764939471201561, 1]
52 [211, 1; 27251095508101501, 1; 4139273766349324247633692510538201503430289333680550996595088851, 1]
53 [211, 1; 18481, 1; 15223111428372564346683550329362640515883425513618463091596333098764813473451, 1]
  ***   Warning: MPQS: the factorization of this number will take several hours.
54 [13441, 1; 116131, 1; 839200442022969151211731, 1; 3353171631789531190661401, 1; 33137090098857649151825131, 1]
55 [152969041, 1; 199860547097401, 1; 11482720617732652469340768313683772852152750941829881152898041, 1]
56 [211, 1; 914800273801, 1; 1348525214401, 1; 1623309657426313921, 1; 1972327250907702154284670941298761138091, 1]
57 [12601, 1; 24221906101, 1; 9869526083562601, 1; 19436009551855957201, 1; 33278276387002226014349151966235501, 1]
  ***   Warning: MPQS: the factorization of this number will take many hours.
  ***   Warning: MPQS: Gauss elimination will require more than 32MBy of memory.
  ***   user interrupt after 17h, 56mn, 49,598 ms.
? for(v=59,105,print(v," ",factor((v^105-1)/gcd(v^105-1,(v^35-1)*(v^21-1)*(v^15-1)*(v^7-1)))))
59 [211, 1; 3622696675111749953551, 1; 297692120869614337146281279671, 1; 44795737776903417541988307453331, 1]
60 Mat([22832698711357981223057247374907077389765848479071322392852181540132857844696246403661, 1])
61 [271960822349610241, 1; 185567708800758872704787206596113693351252154841008941966679239455201, 1]
  ***   Warning: MPQS: the factorization of this number will take many hours.
  ***   Warning: MPQS: Gauss elimination will require more than 32MBy of memory.
  ***   user interrupt after 4h, 38mn, 45,583 ms.

Someone else did this

Inverses of Binary 3x3 matrices

A 3x3 boolean matrix may be expressed as 3 octal digits, each row corresponding to a digit. The identity matrix is 421, and here are the 168 possible invertible matrices: 124*124 = 125*524 = 126*324 = 127*724 = 134*164 = 135*564 = 136*764 = 137*364 = 142*214 = 143*254 = 146*234 = 147*274 = 152*614 = 153*654 = 156*674 = 157*634 = 162*314 = 163*754 = 164*134 = 165*574 = 172*714 = 173*354 = 174*174 = 175*534 = 214*142 = 215*342 = 216*542 = 217*742 = 234*146 = 235*746 = 236*546 = 237*346 = 241*241 = 243*245 = 245*243 = 247*247 = 251*341 = 253*745 = 254*143 = 256*547 = 261*641 = 263*645 = 265*647 = 267*643 = 271*741 = 273*345 = 274*147 = 276*543 = 314*162 = 315*362 = 316*762 = 317*562 = 324*126 = 325*726 = 326*326 = 327*526 = 341*251 = 342*215 = 345*273 = 346*237 = 351*351 = 352*715 = 354*173 = 357*537 = 361*751 = 362*315 = 364*137 = 367*573 = 371*651 = 372*615 = 375*637 = 376*673 = 412*412 = 413*432 = 416*452 = 417*472 = 421*421 = 423*423 = 425*425 = 427*427 = 431*431 = 432*413 = 435*475 = 436*457 = 452*416 = 453*476 = 456*456 = 457*436 = 461*461 = 463*467 = 465*465 = 467*463 = 471*471 = 472*417 = 475*435 = 476*453 = 512*612 = 513*632 = 516*672 = 517*652 = 521*521 = 523*723 = 524*125 = 526*327 = 531*531 = 532*713 = 534*175 = 537*357 = 542*216 = 543*276 = 546*236 = 547*256 = 561*571 = 562*317 = 564*135 = 567*753 = 571*561 = 573*367 = 574*165 = 576*763 = 612*512 = 613*732 = 614*152 = 615*372 = 621*621 = 623*623 = 625*627 = 627*625 = 631*731 = 632*513 = 634*157 = 637*375 = 641*261 = 643*267 = 645*263 = 647*265 = 651*371 = 652*517 = 654*153 = 657*735 = 672*516 = 673*376 = 674*156 = 675*736 = 712*712 = 713*532 = 714*172 = 715*352 = 721*721 = 723*523 = 724*127 = 726*325 = 731*631 = 732*613 = 735*657 = 736*675 = 741*271 = 742*217 = 745*253 = 746*235 = 751*361 = 753*567 = 754*163 = 756*765 = 762*316 = 763*576 = 764*136 = 765*756 = 421

The 22 square roots of unity (identity) are 124² = 174² = 241² = 247² = 326² = 351² = 412² = 421² = 423² = 425² = 427² = 431² = 456² = 461² = 465² = 471² = 521² = 531² = 621² = 623² = 712² = 721² = 421

There are also 22 square roots of zero: 000² = 002² = 004² = 006² = 010² = 033² = 040² = 044² = 050² = 077² = 100² = 110² = 200² = 202² = 300² = 333² = 505² = 555² = 660² = 666² = 707² = 770² = 000

There 260 squares: 000 001 002 003 004 005 006 007 010 011 013 017 020 021 022 025 030 032 033 036 040 044 050 053 055 057 060 061 065 066 070 072 076 077 100 101 105 107 110 111 113 115 121 124 125 127 132 135 136 137 142 143 145 146 152 155 156 157 162 163 165 166 171 173 174 175 200 202 213 214 215 216 220 221 222 227 232 234 235 237 241 247 253 254 255 256 260 261 262 263 264 265 266 267 270 271 273 276 300 303 305 307 313 314 316 317 321 325 326 327 330 332 333 334 342 343 345 346 351 352 354 356 360 361 364 367 370 371 372 375 400 401 403 404 411 412 413 417 420 421 422 423 424 425 426 427 430 431 432 436 440 441 444 447 451 453 456 457 461 465 471 472 474 476 500 504 505 506 513 514 516 517 520 521 524 526 530 531 532 534 542 543 544 547 550 552 554 555 562 563 564 567 573 574 576 577 600 601 603 606 613 614 615 616 621 623 632 634 635 637 640 641 642 643 644 645 646 647 650 651 652 657 660 666 672 674 675 677 700 704 706 707 711 712 713 715 721 722 724 726 732 735 736 737 740 741 745 746 750 751 753 754 762 763 764 767 770 773 775 777 .

Abstract Value Simulation

It would be nice if there were a framework for abstract value simulation at compile time (static). Things a complete programming language ought to have as libraries: parser, syntax table (what identifiers are in scope and where are they defined), type inferer, pretty printer, interpreter, abstract value analyzer,...

GMP ECM Memory use (Step 2)

A graph of step 2 memory use in kilobytes of a 135-digit number, B1=63million

2nd letters

What are the most common second occurence letters in words?

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Sound Flux

You can measure rainfall intensity with a microphone. By causing turbulence, you measure flux of any liquid or gas similarly.

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Solar Wind Average Proton Flux

On the average, how many protons (per unit of time) does the sun expel as solar wind?

135-digit progress

did the 135-digit to B1=63036659.

Body odor

One can measure changes in health and certainly dietary habits by the changes in body odor. This could be a simple way for an agent, perhaps at a residential care facility, to automatically monitor someone who needs care. One could design a shower-head which initially sucks in air for a second, that is, "takes a sniff", has a bunch of sensors which might tell you or your care provider if you haven't been getting enough vitamins or something, and then proceeds to dispense water like a normal shower head. Assuming people take showers at a regular time (this can be checked with a clock) it serves as a relatively constant reference to which delta changes can be measured day-to-day.

webserver zip

There should be a web server (perhaps apache module?) frob that'll let you browse a bunch of files in a directory, typically pictures, one by one, or on-the-fly compress them into a single archive for downloading.

Terraforming Venus

Venus is a better target for human exploration than Mars. The distance to it is shorter, Earth and Venus come into the proper planetary alignment for journey there and back more frequently. It is of more similar mass as Earth than Mars, so consequently has more similar gravity.

There is the problem that it is really hot, 460 degrees C, there. But it can be solved! The temperature is caused by a greenhouse effect caused by the excess carbon dioxide. But, there are plants and cyanobacteria which eat carbon dioxide. There are also bacteria that can survive at very high temperatures, for example near hydrothermal vents at 380 degrees C. There are organisms that can eat rock, like lichen. The lack of water is troublesome, though there are of course desert organisms that live with very little water.

No single organism on earth possesses all these characteristics, but we could breed one. Breeding bacteria is a well studied art. We can trigger horizontal transfer using viruses and other bacteria-bacteria interactions to get all the right genes into a single organism. Finally we can force evolution from Earth conditions to Venus conditions of many bacteria generations to breed a suitable Venus bacteria.

We only need to deliver a small payload of the bacteria (a small payload in space transprt is good!) to Venus to start the terraforming process. The lack of natural predators means the bacteria will multiply exponentially and quickly get their work done.

In the meantime, we can figure out how to get humans to Venus. We'll arrive to find it with a oxygen atmosphere covered in a sludge of dead bacteria.

slideshow images

I wish there were a way to create a web-based slide show of images such that while viewing one image, the next image loads over the network and decodes (jpeg2000 is slow).

Chains

Catenary animation, with moving endpoints and changing length.

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Mauna Kea

Where are the points on a surface accessible by some path of always nonnegative gradient, from some "floor" region?

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