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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) If p is a prime greater than five, then p divides u(p(p5)), where u(n) is the nth Fibonacci number and (ab) is the Legendre symbol (so (p5)=1 when p is a multiple of 5 plus either 1 or 4, and (p5)=1 when p is a multiple of 5 plus either 2 or 3). A prime p > 5 is a WallSunSun prime if p^{2} divides u(p(p5)). These are sometimes called FibonacciWieferich primes. No WallSunSun primes are known (and mathematicans have checked all primes below 100,000,000,000,000)! So why bother naming a type of prime when we know of no such numbers? For the following two reasons. First Sun and Sun showed in 1992 that if the first case of Fermat's Last Theorem (FLT) was false for the prime exponent p, then p is a WallSunSun prime. Before Wiles proved FLT, the search for WallSunSun primes was also the search for a counterexample to this theorem. Notice that this is the same reason that Sophie Germain primes first sparked mathematicians' interest. Second, heuristically it seems likely that there would be infinitely many such primes, but that they should be very rare (just as is conjectured for the Wilson primes and Wieferich primes). But we must admit this heuristic is based on the assumption that u(p(p5))/p behaves essentially randomly modulo p, and this assumption is made simply because we do not yet know otherwise.
See Also: WilsonPrime, WieferichPrime, SophieGermainPrime Related pages (outside of this work)
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Chris K. Caldwell © 19992019 (all rights reserved)
