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Indian Team Selection Test 2010 ST4 P3

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May 23, 2010
number theory proposednumber theory

Problem Statement

Prove that there are infinitely many positive integers mm for which there exists consecutive odd positive integers pm<qmp_m<q_m such that pm2+pmqm+qm2p_m^2+p_mq_m+q_m^2 and pm2+m⋅pmqm+qm2p_m^2+m\cdot p_mq_m+q_m^2 are both perfect squares. If m1,m2m_1, m_2 are two positive integers satisfying this condition, then we have pm1≠pm2p_{m_1}\neq p_{m_2}
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