A note on some relations among special sums of reciprocals modulo <Emphasis Type="Italic">p</Emphasis>

In this note the sums s(k, N) of reciprocals are investigated, where p is an odd prime, N, k are integers, p does not divide N, N ≥ 1 and 0 ≤ k ≤ N − 1. Some linear relations for these sums are derived using “logarithmic property” and Lerch’s Theorem on the Fermat quotient. Particularly in case N = 10 another linear relation is shown by means of Williams’ congruences for the Fibonacci numbers. Published results were acquired using the subsidization of the Ministry of Education, Youth and Sports of the Czech Republic, research plan MSM 0021630518 “Simulation modeling of mechatronic systems”.