# Wall-Sun-Sun primes and Fibonacci divisibility

**Conjecture**For any prime , there exists a Fibonacci number divisible by exactly once.

Equivalently:

**Conjecture**For any prime , does not divide where is the Legendre symbol.

Let be an odd prime, and let denote the -adic valuation of . Let be the smallest Fibonacci number that is divisible by (which must exist by a simple counting argument). A well-known result says that unless divides , and . This conjecture asserts that for all . This has been verified up to at least . [EJ]

This conjecture is equivalent to non-existence of Wall-Sun-Sun primes.

## Bibliography

[EJ] Andreas-Stephan Elsenhansand and Jörg Jahnel, The Fibonacci sequence modulo p^2

[R] Marc Renault, Properties of the Fibonacci Sequence Under Various Moduli

*[W] D. D. Wall, Fibonacci Series Modulo m, American Mathematical Monthly, 67 (1960), pp. 525-532.

* indicates original appearance(s) of problem.