MathDB

Let

p

be an odd prime number and

\mathbb{Z}_{>0}

be the set of positive integers. Suppose that a function

f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}

satisfies the following properties:
[*]

f(1,1)=0

. [*]

f(a,b)+f(b,a)=1

for any pair of relatively prime positive integers

(a,b)

not both equal to 1; [*]

f(a+b,b)=f(a,b)

for any pair of relatively prime positive integers

(a,b)

.
Prove that

\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.

Problem Statement