How to solve this differential equation?

I prove this by induction. Suppose result is true for n = k. Then 14^(k) = p² + q² + r² for some integers p, q, r?14^(k + 2) = 14² x 14^k = (14p)² + (14q)² + (14r)².

Hence result is true for n = k? Result is true for n = k + 2. But result is true for n = 1?

Result is true for n= 3, 5, 7,.... Also result is true for n = 2? Result is true for n= 4, 6, 8,.... Hence result is true for n = 1, 2, 3, 4, 5, 6,..... So result is true for all integers n.

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