MathDB

Let

p = 101

and let

S

be the set of

p

-tuples

(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p

of integers. Let

N

denote the number of functions

f: S \to \{0, 1, \dots, p-1\}

such that
[*]

f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}

for all

a, b \in S

, and [*]

f(a) = f(b)

whenever all components of

a-b

are divisible by

p

.
Compute the number of positive integer divisors of

N

. (Here addition and subtraction in

\mathbb{Z}^p

are done component-wise.)
Proposed by Ankan Bhattacharya

Problem Statement