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Prove that k is composite and 1984 divides k

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October 13, 2010
modular arithmeticnumber theorynumber theory unsolved

Problem Statement

Let a,b,ca, b, c be natural numbers such that a+b+c=2pq(p30+q30),p>qa+b+c = 2pq(p^{30}+q^{30}), p > q being two given positive integers. (a)(a) Prove that k=a3+b3+c3k = a^3 + b^3 + c^3 is not a prime number. (b)(b) Prove that if a⋅b⋅ca\cdot b\cdot c is maximum, then 19841984 divides kk.
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