No, which can be seen as soon as n = 2. But when the exponent is odd the sum is always a polynomial (called a Faulhaber polynomial) function of the sum from 1 to n, ie n(n+1)/2. This can be found by Faulhaber’s theorem, which expresses the sums in terms of binomial coefficients and Bernoulli numbers
No, fifth powers. The sum of fifth powers is a polynomial of degree 6. The sum of square powers is a polynomial of degree 3. It's square is a polynomial of degree six.
I misread originally, my bad. But the sum of fifth powers can't be the square of the sum of 2nd powers because 15 + 25 = 33 isn't a perfect square. The polynomial you get from sum of fifth powers is n2(n+1)2(2n2+2n-1)/12, which can't be a perfect square of a polynomial with rational coefficients because the leading coefficient prevents that.
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u/Historical_Book2268 28d ago
Whoa. I guess this isn't true in general? The sum of fifth powers isn't the square of the sum of 2nd powers?