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PrintEstonian Mathematical Olympiad
Estonia number theory
Problem
Do there exist five distinct prime numbers for which the sum of any three of them is a prime number as well?
Solution
Answer: No.
Assume that there exist five such prime numbers. If there are three among them pairwise incongruent modulo , then by adding each of the three separately to the sum of the other two we get three distinct sums modulo . One of those three is modulo and therefore is not a prime number. However, if among the five we only get two distinct remainders when dividing by , then by the pigeonhole principle there must be three among them that are congruent modulo . The sum of those three, however, is divisible by and therefore not a prime number.
Assume that there exist five such prime numbers. If there are three among them pairwise incongruent modulo , then by adding each of the three separately to the sum of the other two we get three distinct sums modulo . One of those three is modulo and therefore is not a prime number. However, if among the five we only get two distinct remainders when dividing by , then by the pigeonhole principle there must be three among them that are congruent modulo . The sum of those three, however, is divisible by and therefore not a prime number.
Final answer
No
Techniques
Modular ArithmeticPigeonhole principle