Baltic Way 1992 · Problem 1

Number Theory

Let $p$ and $q$ be two consecutive odd prime numbers. Prove that $p+q$ is a product of at least three positive integers greater than 1 (not necessarily different).

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Review

Topics

Divisibility and factorization · Primes

Solutions

Solution

Solution: Since $q-p=2k$ is even, we have $p+q=2(p+k)$ . It is clear that $p < p+k < p+2k = q$ . Therefore $p+k$ is not prime and, consequently, is a product of two positive integers greater than 1.

Contest context

Results from Baltic Way 1992

8 teams

Mean score: 4.6 / 5
Scores of 4 or 5: 7 / 8
Estonia: 2 / 5

Score distribution

All team scores

Team	Score
Denmark	5 / 5
St. Petersburg	5 / 5
Poland	5 / 5
Latvia	5 / 5
Iceland	5 / 5
Lithuania	5 / 5
Estonia	2 / 5
Sweden	5 / 5

Elsewhere

AoPS · Baltic Way 1992
Official results