1975 IMO Problems/Problem 2
Problem
Let be an infinite increasing sequence of positive integers. Prove that for every there are infinitely many which can be written in the formwith positive integers and .
Solution
If we can find such that , we're done: every sufficiently large positive integer can be written in the form . We can thus assume there are no two such . We now prove the assertion by induction on the first term of the sequence, . The base step is basically proven, since if we can take and any we want. There must be a prime divisor which divides infinitely many terms of the sequence, which form some subsequence . Now apply the induction hypothesis to the sequence .
The above solution was posted and copyrighted by grobber. The original thread for this problem can be found here: [1]
See Also
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