SUR QUELQUES CONSÉQUENCES
DE LA CONJECTURE (abc)
EN ARITHMÉTIQUE ET EN LOGIQUE

MICHEL LANGEVIN

Abstract:

We recall the following problem of P. Erdos and A. Woods whose solution would be of interest in number theory and in logic:

Does there exist an integer k>2 with the following property: ``If x and y are positive integers such that, for tex2html_wrap_inline26 , the two numbers x+i and y+i have the same prime factors, then x=y''? (A. Woods showed the equivalence of this problem with an open question of J. Robinson about definability of arithmetic by comprimeness and successor function.)

We show how to deduce from the (abc) conjecture of J. Oesterlé and D. Masser a solution for the following extension of this open problem (the result can be improved when d and tex2html_wrap_inline38 are fixed): if x, y, d and tex2html_wrap_inline38 are positive integers (with tex2html_wrap_inline48 , tex2html_wrap_inline50 such that, for tex2html_wrap_inline52 , (x+id) and tex2html_wrap_inline56 have the same prime factors, then tex2html_wrap_inline58 belongs to a finite set.