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Infinite number of primes expressible in a given form

Source: Romania TST 2013 Test 2 Problem 3

April 26, 2013
quadraticsinequalitiesinductionnumber theory proposednumber theory

Problem Statement

Let SS be the set of all rational numbers expressible in the form (a12+a1−1)(a22+a2−1)…(an2+an−1)(b12+b1−1)(b22+b2−1)…(bn2+bn−1)\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)} for some positive integers n,a1,a2,…,an,b1,b2,…,bnn, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n. Prove that there is an infinite number of primes in SS.
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