Functions which are PN on infinitely many extensions of $$\mathbb {F}_p,\,p$$ F p , p odd

Jedlicka, Hernando and McGuire proved that Gold and Kasami functions are the only power mappings which are APN on infinitely many extensions of $$\mathbb {F}_2$$ F 2 . For $$p$$ p an odd prime, we prove that the only power mappings $$x\mapsto x^m$$ x ? x m such that $$m\equiv 1\mod p$$ m ? 1 mod p which are PN on infinitely many extensions of $$\mathbb {F}_p$$ F p are those such that $$m=1+p^l$$ m = 1 + p l , l positive integer. As Jedlicka, Hernando and McGuire, we prove that $$\frac{(x+1)^m-x^m-(y+1)^m+y^m}{x-y}$$ ( x + 1 ) m - x m - ( y + 1 ) m + y m x - y has an absolutely irreducible factor by using Bezout's theorem.