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O 59

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May 25, 2007
inductiongreatest common divisornumber theoryrelatively prime

Problem Statement

Let a1<a2<a3<⋯a_{1} < a_{2} < a_{3} < \cdots be an infinite increasing sequence of positive integers in which the number of prime factors of each term, counting repeated factors, is never more than 19871987. Prove that it is always possible to extract from AA an infinite subsequence b1<b2<b3<⋯b_{1} < b_{2} < b_{3} < \cdots such that the greatest common divisor (bi,bj)(b_i, b_j) is the same number for every pair of its terms.
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