MathDB

Bounded integer function with prime quasiperiodicity

Source: 5th Memorial Mathematical Competition "Aleksandar Blazhevski - Cane" - Senior - Problem 3

January 29, 2024

functionnumber theory

Problem Statement

Find all functions

f: \mathbb{N} \rightarrow \mathbb{Z}

such that

|f(k)| \leq k

for all positive integers

k

and there is a prime number

p>2024

which satisfies both of the following conditions:

1)

For all

a \in \mathbb{N}

we have

af(a+p) = af(a)+pf(a),

2)

For all

a \in \mathbb{N}

we have

p|a^{\frac{p+1}{2}}-f(a).

Proposed by Nikola Velov