MathDB

The set of numbers

(p, a, b, c)

of positive integers is called Sozopolian when:
* p is an odd prime number
*

a

b

and

c

are different and
*

ab + 1

bc + 1

and

ca + 1

are a multiple of

p

.
a) Prove that each Sozopolian set satisfies the inequality

p+2 \leq \frac{a+b+c}{3}

b) Find all numbers

p

for which there exist a Sozopolian set for which the equality of the upper inequation is met.

Problem Statement