Type: Article

On congruences involving products of variables from short intervals

Garaev, M. Z.;

Abstract:

We prove several results which imply the following consequences. For any ? >0 and any sufficiently large prime p, if I'1,..,I' 13 are intervals of cardinalities |I' j|>p1/4+? and abc 0(modp), then the congruence ax1..x6+bx7..x13 c(modp) has a solution with xj I' j. There exists an absolute constant n0 I• such that for any 0<? <1 and any sufficiently large prime p, any quadratic residue modulo p can be represented in the form x1..xn0(modp),xi I•,xi?p1/(4e2/3)+?. For any ? >0, there exists n=n(?) I• such that for any sufficiently large m I• the congruence x1..xn 1(modm),xi I•,xi?m? has a solution with x1â ‰ 1.