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Problem 1003

AMC 12 late, AIME early

Number theory Difficulty 4.9 Prove it Estonian Mathematical Olympiad · Estonia

Do there exist five distinct prime numbers for which the sum of any three of them is a prime number as well?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Answer: No.

Assume that there exist five such prime numbers. If there are three among them pairwise incongruent modulo $3$ , then by adding each of the three separately to the sum of the other two we get three distinct sums modulo $3$ . One of those three is $0$ modulo $3$ and therefore is not a prime number. However, if among the five we only get two distinct remainders when dividing by $3$ , then by the pigeonhole principle there must be three among them that are congruent modulo $3$ . The sum of those three, however, is divisible by $3$ and therefore not a prime number.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.