My late father, Moisey Bogomolny, was an inveterate human calculator. During the 1930s at the height of the New Economic Policy (NEP) that allowed in the Soviet Union a degree of post-revolution entrepreneurship, he made a living by giving on-stage mental math performances. He became an electrical engineer when the NEP was curtailed.
Browsing through his notes I have recently come across an observation concerning the fifth powers of integers and its relevance to the absence of integer solutions of - Fermat's equation for .
The story of the FLT is well known. Sometime, probably in the 1630s, Fermat left a note in the margins of his copy of Diophantus' Arithmetica, where he claimed to have found a proof for the impossibility of integer solutions to , for . His proof for where he introduced the Method of Infinite Descent has been eventually recreated. In 1753 Euler has adapted the infinite descent to proving the case of .
In her correspondence with F. Gauss, Sophie Germain introduced the class of odd primes (now known as Germain's primes), for which is also prime. For these she proved that would imply that at least one of is divisible by .
Since both and are prime, Germain's theorem applies to . In 1825 the theorem has inspired Gustav Lejeune Dirichlet and Adrien-Marie Legendre to tackle that equation. The 20 year old Dirichlet proved that, for no solution is possible in which one of is both even and divisible by . (Even for a primitive solution, one of must be even.) Dirichlet had confessed that he was unable to prove the second case wherein one of was divisible by and another by . A couple of months later, Legendre (73 years old at the time) gave a complete proof. [Edwards, p. 70] remarks that the circumstance serves a counterexample to the common notion that only young men can do important work in mathematics. However, he adds that Legendre's proof of the case that defeated Dirichlet's first attempts was rather artificial and involved a great deal of unmotivated manipulation, perhaps a symptom of his great age and long experience. Later same year (1825) Dirichlet submitted a proof of the remaining case which was simpler and shorter than Legendre's.
Now, what has all this history to do with my father's notes. My father observed that, for , it is possible to establish Germain's theorem by elementary means.
The fifth power of the integers has a property that it produces only remainders modulo . In other words, the set of two-digit numbers with which the fifth powers of the integers may terminate contains only 15 elements:
The second table shows that the last two digits of the sums of two fifth powers may be the fifth power only if one of the addends is divisible by or when the sum is divisible by .
This part is indeed elementary. To complete the proof for you may now follow either Dirichlet or Legendre, or come up with your own ideas.