Remarks concerning possible connections between Fermat's last theorem and integral p×p circulants

Abstract Let p be a prime and x, y integers. Then x p + y p occurs as the determinant of an integral p × p circulant. Some problems concerning the set of values assumed by det C as C runs through the set of integral p × p circulants are given. The problem of finding sufficient conditions on an integral p × p circulant C which ensure that C can be factored in the form C 1 C ' 1 (where ' denotes transpose) with C 1 an integral circulant is referred to, and it is conjectured that such a factorization is possible if C is unimodular and positive definite symmetric and 1 2 ( p –1) is also a prime.