Number Theory

Number theory is a natural place to come across large numbers. Fermat’s Last Theorem (conjecture) states that there are no positive solutions to a^n+b^n = c^n for n > 2 which is now proven to be true ¹ . Leonhard Euler extended the conjecture in the late 1700s, proposing, among other examples, that there were no solutions to a^4+b^4+c^4 = d^4 in positive integers. This was disproved by Noam Elkies in 1988. The smallest counterexample is: (95800)^4 + (217519)^4 + (414560)^4 = (422481)^4. Is this natural or unnatural? The question, itself, certainly seems natural and involving small numbers. Nevertheless Elkies gave theoretical arguments for why the smallest counterexample is so large (involving 20-digit numbers on the two sides of the equation above) and seemingly unnatural. One may speculate that the unnatural numbers in physics arise from natural questions in number theory.