Vol. 3, No. 3, 2010

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Distinct solution to a linear congruence

Donald Adams and Vadim Ponomarenko

Vol. 3 (2010), No. 3, 341–344

Given n,k and a1,a2,,ak n, we give conditions for the equation a1x1 + a2x2 + + akxk = 1 in n to admit solutions with all the xi distinct.

A sufficient condition is that k ϕ(n) and ai be invertible in n for all i.

If n > 2 is prime, the following conditions together are necessary and sufficient: k n, each ai is nonzero, and either k < n or not all of the ai are equal.

linear congruence, minimal zero-sum sequence, property B
Mathematical Subject Classification 2000
Primary: 11B50, 11D79
Received: 7 July 2010
Revised: 29 September 2010
Accepted: 29 September 2010
Published: 16 October 2010

Proposed: Scott Chapman
Communicated by Scott Chapman
Donald Adams
Arizona State University
Tempe, AZ 85287-1804
United States
Vadim Ponomarenko
San Diego State University
Department of Mathematics and Statistics
5500 Campanile Dr.
San Diego, CA 92182-7720
United States