Hellenic Mathematical Olympiad

Question

(α) Write the expression A = k^4 + 4, where k is a positive integer, as a product of two factors each of them being a sum of two squares of integers. (β) Simplify the expression K = (2^4 + 14)(4^4 + 14)(6^4 + 14) ((2n)^4 + 14)(1^4 + 14)(3^4 + 14)(5^4 + 14) ((2n-1)^4 + 14) and write it as a sum of the squares of two successive integers.

Accepted Answer

(α) We have aligned k^4 + 4 &= (k^2)^2 + 4k^2 + 2^2 - 4k^2 = (k^2 + 2)^2 - (2k)^2 &= (k^2 + 2 - 2k)(k^2 + 2 + 2k) = [(k-1)^2 + 1^2][(k+1)^2 + 1^2]. aligned (β) We multiply both terms of the fraction by (2^4)^n, to receive: aligned K &= (2^4 + 14)(4^4 + 14)(6^4 + 14) [(2n)^4 + 14](1^4 + 14)(3^4 + 14)(5^4 + 14) [(2n-1)^4 + 14] &= (3^2 + 1)(5^2 + 1)(7^2 + 1)(9^2 + 1)(11^2 + 1) [(4n-3)^2 + 1][(4n-1)^2 + 1][(4n+1)^2 + 1](1^2 + 1)(3^2 + 1)(5^2 + 1)(7^2 + 1)(9^2 + 1)(11^2 + 1)(13^2 + 1) [(4n-3)^2 + 1][(4n-1)^2 + 1] &= (4n+1)^2 + 11^2 + 1 = 8n^2 + 4n + 1 = 4n^2 + 4n^2 + 4n + 1 = (2n)^2 + (2n+1)^2. aligned