In this work, we state and prove Lerch’s theorems for Fermat and Euler quotients over function fields defined analogously to the number fields. 1. Results The Fermat’s little theorem states that if p is a prime and a is an integer not divisible by p, then ap−1 ≡ 1 mod p. This gives rise to the definition of the Fermat quotient of p with base a, q(a, p) = ap−1 − 1 p , which is an integer. This q...
