Lagrange's Four-Square Theorem states that every positive integer can be written as the sum of at most four squares.  For example, 6 = 2² + 1² + 1² is the sum of three squares.  Given this theorem, prove that any positive multiple of 8 can be written as the sum of eight odd squares.

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By Lagrange's Theorem, for any non-negative integer n we have n = a² + b² + c² + d², where a, b, c, and d are non-negative integers.

So 8n = (2a + 1)² + (2a − 1)² + (2b + 1)² + (2b − 1)² + (2c + 1)² + (2c − 1)² +(2d + 1)² + (2d − 1)² − 8,
and thus 8(n + 1) is the sum of eight odd squares.

Therefore, any positive multiple of 8 can be written as the sum of eight odd squares.

By Fermat's Little Theorem, the number x = (2^p−1 − 1)/p is always an integer if p is an odd prime.  For what values of p is x a perfect square?

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The only values of p for which (2^p−1 − 1)/p is a perfect square are 3 and 7.

Compute the infinite product

[sin(x) cos(x/2)]^1/2 · [sin(x/2) cos(x/4)]^1/4 · [sin(x/4) cos(x/8)]^1/8 · ... ,

where 0 x 2.

Answer(select the area below in mouse for seeing answer)

[sin(x) cos(x/2)]^1/2 · [sin(x/2) cos(x/4)]^1/4 · [sin(x/4) cos(x/8)]^1/8 · ... = ½|sin(x)|,0 x 2. for