On large Zsigmondy primes

Abstract

If a a and m m are integers greater than 1, then a large Zsigmondy prime is a prime l l such that l | a m − 1 , l ∤ a i − 1 l\left | {{a^m} - 1,l\nmid {a^i} - 1} \right . for 1 ⩽ i ⩽ m − 1 1 \leqslant i \leqslant m - 1 and either l 2 | a m − 1 {l^2}\left | {{a^m} - 1} \right . or l > m + 1 l > m + 1 . The main result of this paper lists all the pairs ( a , m ) \left ( {a,m} \right ) for which no large Zsigmondy prime exists.