For an integer k ≥ 2, a k-step Fibonacci function is a function f: Z → Z defined by f(n + k) = f(n + k − 1) + f(n + k − 2) + ⋯ + f(n) for any integer n. We mainly show the existence of primitive period of a k-step Fibonacci function in modulo m. Moreover, the explicit primitive period of a k-step Fibonacci function, when k = 2,3,4, under some conditions is also established.
