Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Nearby powers [and integer equations]

One can potentially find integer equations with large solutions but small coefficients by looking say for pairs of integer powers close in value. The pictures below show what happens if one computes x^m and y^n for many x and y, sorts these values, then plots successive differences. The differences are trivially zero when x=s^n, y=s^m. Often they are large, but surprisingly small ones can sometimes occur (despite various suggestions from the so-called ABC conjecture). Thus, for example, 5853886516781223^3 - 1641843 is a perfect square, as found by Noam Elkies in 1998. (Another example is 55^5-22434^2==19.)

From Stephen Wolfram: A New Kind of Science [citation]