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sums of first n primes

Source: Problem 8 of Russian Regional Olympiad 2010 Grade 9

September 25, 2011
quadraticsnumber theoryprime numbersnumber theory proposed

Problem Statement

For every positive integer nn, let SnS_n be the sum of the first nn prime numbers: S1=2,S2=2+3=5,S3=2+3+5=10S_1 = 2, S_2 = 2 + 3 = 5, S_3 = 2 + 3 + 5 = 10, etc. Can both SnS_n and Sn+1S_{n+1} be perfect squares?
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