Factors of generalized Fermat numbers

Abstract:

A search for prime factors of the generalized Fermat numbers $F_n(a,b)=a^{2^n}+b^{2^n}$ has been carried out for all pairs $(a,b)$ with $a,b\leq 12$ and GCD$(a,b)=1$. The search limit $k$ on the factors, which all have the form $p=k\cdot 2^m+1$, was $k=10^9$ for $m\leq 100$ and $k=3\cdot 10^6$ for $101\leq m\leq 1000$. Many larger primes of this form have also been tried as factors of $F_n(a,b)$. Several thousand new factors were found, which are given in our tables.—For the smaller of the numbers, i.e. for $n\leq 15$, or, if $a,b\leq 8$, for $n\leq 16$, the cofactors, after removal of the factors found, were subjected to primality tests, and if composite with $n\leq 11$, searched for larger factors by using the ECM, and in some cases the MPQS, PPMPQS, or SNFS. As a result all numbers with $n\leq 7$ are now completely factored.