The index of small length sequences


Let n ≥ 2 be a fixed integer. Define (x)n to be the unique integer in the range 0 ≤ (x)n < n which is congruent to x modulo n. Given x1,⋯,x, let (x1,⋯,x)Z1 = min{(ux1)n + ⋯ + (uxR)n:u Z;, gcd(u,n) = 1} and define Ind(x1,⋯,xR) = 1 nZ(x1,⋯,xR)Z1 to be the index of the sequence (x1,⋯,xR). If x1,⋯,x4 have α [1,4]xα 0 (modn) but α Ixα 0 for all proper, non-empty subsets I [1, 4], then a still open conjecture asserts that Ind(S) = 1 provided that gcd(n, 6) = 1. We give an alternative proof, that does not rely on computer calculations, verifying this conjecture when n is a product of two prime powers.

Publication Title

International Journal of Algebra and Computation