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Consider a prime number

p\geqslant 11

. We call a triple

a,b,c

of natural numbers suitable if they give non-zero, pairwise distinct residues modulo

p{}

. Further, for any natural numbers

a,b,c,k

we define

f_k(a,b,c)=a(b-c)^{p-k}+b(c-a)^{p-k}+c(a-b)^{p-k}.

Prove that there exist suitable

a,b,c

for which

p\mid f_2(a,b,c)

. Furthermore, for each such triple, prove that there exists

k\geqslant 3

for which

p\nmid f_k(a,b,c)

and determine the minimal

k{}

with this property.
Călin Popescu and Marian Andronache

Problem Statement