Saturday, September 12, 2020

[rysiotfu] Easiest-to-state unsolved mathematical problem

Find 8 positive integers, none of which are 1, which multiply to yield 2^4096+1.  This requires explaining what multiplication and exponentiation are.  If we were to multiply 2^4096+1 out, then the statement of the problem would look a lot more complicated, so we don't.

Find 2 positive integers, neither of which is 1, which multiply to yield 2^1277-1.  Other currently completely unfactored Mersenne number exponents below 4096: 1619 1753 2137 2267 2273 2423 2521 2713 2719 2851 3049 3673 3691 3847 3881 3919 4007 4049.

Find 2 positive integers, neither of which is 1, which multiply to yield 2^2^20+1.  (This requires explaining that chained exponentiation is evaluated right to left.)

Collatz conjecture.  This requires explaining what even integers, division by 2, multiplication by 3, and adding 1 are.

Any conjecture directly involving prime numbers requires explaining what a prime number is.  Sometimes there are indirect ways of stating one of the many unsolved problems around primes in a way that avoids mentioning prime numbers.  Above, for 12th Fermat number 2^4096+1, we avoided mentioning prime factors by requiring 8 factors: currently 6 prime factors are known, so the remaining 7th composite needs to be factored into 2 to get to 8.

What regular polygons can be constructed using only compass and straightedge?  (Whether the list of known Fermat primes is complete is not yet known.)  This requires explaining the rules of a compass and straightedge construction.

What convex shape has the worst packing density?

In general, any problem involving geometry requires explaining what geometry is, which might be difficult (e.g., parallel postulate, Tarski's circle-squaring problem) depending on how rigorously you do that.  Of course, explaining integer arithmetic, including place-value notation, rigorously is probably also difficult.

Unsolved problems around simple games might be relatively easy to state.

Who wins 3D infinite (ordinal) Chomp?  This requires explaining the rules of Chomp, and ordinal numbers.

Who wins go 囲碁?  By how many stones?

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