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Iran TST

Source: Iranian TST 2019, second exam day 2, problem 6

June 24, 2019
number theory

Problem Statement

For any positive integer nn, define the subset SnS_n of natural numbers as follow Sn={x2+ny2:x,y∈Z}. S_n = \left\{x^2+ny^2 : x,y \in \mathbb{Z} \right\}. Find all positive integers nn such that there exists an element of SnS_n which doesn't belong to any of the sets S1,S2,…,Sn−1S_1, S_2,\dots,S_{n-1}.
Proposed by Yahya Motevassel
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