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Part of 2014 EGMO

Problems(1)

On existence of infinitely many positive integers satisfying

Source: European Girls' Mathematical Olympiad-2014 - DAY 1 - P3

4/12/2014
We denote the number of positive divisors of a positive integer mm by d(m)d(m) and the number of distinct prime divisors of mm by ω(m)\omega(m). Let kk be a positive integer. Prove that there exist infinitely many positive integers nn such that ω(n)=k\omega(n) = k and d(n)d(n) does not divide d(a2+b2)d(a^2+b^2) for any positive integers a,ba, b satisfying a+b=na + b = n.
quadraticsnumber theoryDivisorsCombinatorial Number TheoryEGMOEGMO 2014