We show that no Fibonacci number (larger than 1) divides the sum of its divisors.

We describe an algorithm to determine whether or not a given system of congruences is satisfied by Cullen numbers. We use this algorithm to prove that there are infinitely many Cullen numbers which are both Riesel and Sierpiński. (Such numbers should be discarded if you are searching prime numbers with Proth's theorem.)

Here, we show that for most primes p, every residue class modulo p can be represented as a sum of 32 Fibonacci numbers.