1977 IMO Problems/Problem 3
Let be a given number greater than 2. We consider the set of all the integers of the form with A number from is called indecomposable in if there are not two numbers and from so that Prove that there exist a number that can be expressed as the product of elements indecomposable in in more than one way. (Expressions which differ only in order of the elements of will be considered the same.)
Lemma: there are many prime numbers .
Assume that there are only finitely many of them and let their product be . Then all prime factors of must be since this number is coprime with . But that would mean that which is impossible for .
Now we can tackle the problem: There are only finetely many residue classes , thus we can find two primes with and coprime with . Let be the smallest positive integer with , then the same property holds for too. Now we have that are all indecomposable since all their nontrivial divisors are . But the product gives a number that is represented in two ways.
The above solution was posted and copyrighted by ZetaX. The original thread for this problem can be found here: 
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