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a karaka quadruple (p, a, b, c) exists with p + 2 = (a + b + c)/3

Source: New Zealand NZMOC Camp Selection Problems 2016 p4

September 19, 2021

number theorydivisible

Problem Statement

A quadruple

(p, a, b, c)

of positive integers is a karaka quadruple if

\bullet

p

is an odd prime number

\bullet

a, b

and

c

are distinct, and

\bullet

ab + 1

,

bc + 1

and

ca + 1

are divisible by

p

. (a) Prove that for every karaka quadruple

(p, a, b, c)

we have

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

. (b) Determine all numbers

p

for which a karaka quadruple

(p, a, b, c)

exists with

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

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