But for smaller e[s] one can show that Abs[m[s]]  (1 - Sinh[2 β ] -4 ) 1/8 where β can be deduced from e[s]  -(Coth[2 β ](1 + 2 EllipticK[4 Sech[2 β ] 2 Tanh[2 β ] 2 ] (-1 + 2 Tanh[2 β ] 2 )/ π )) This implies that just below the critical point e 0 = - √ 2 (which corresponds to β = Log[1 + √ 2 ]/2 ) Abs[m] ~ (e 0 - e) 1/8 , where here 1/8 is a so-called critical exponent. … One marginally more complicated case effectively involving 13 neighbors is IsingEvolve[list_, t_Integer] := First[Nest[IsingStep, {list, Mask[list]}, t]] IsingStep[{a_, mask_}] := {MapThread[ If[#2  2 && #3  1, 1 - #1, #1]&, {a, ListConvolve[ {{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}, a, 2], mask}, 2], 1 - mask} where Mask[list_] := Array[Mod[#1 + #2, 2]&, Dimensions[list]] is set up so that alternating checkerboards of cells are updated on successive steps. … (The slow approach to this limit can be viewed as being a consequence of smallness of finite size scaling exponents in Ising-like systems.)

In 1909 Axel Thue showed that any equation of the form p[x, y]  a , where p[x, y] is a homogeneous irreducible polynomial of degree at least 3 (such as x 3 + x y 2 + y 3 ) can have only a finite number of integer solutions. … (For quadratic equations Hasse's Principle implies that if no solutions exist for any n then there are no solutions for ordinary integers—but a cubic like 3x 3 + 4 y 3 + 5 z 3  0 is a counterexample.) … If one wants to enumerate all possible Diophantine equations there are many ways to do this, assigning different weights to numbers of variables, and sizes of coefficients and of exponents.