Saturday, December 4, 2010

Factorials expressed as sum and difference of squares

The factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n.



To show that

(1) it is possible to express n! as the difference of two perfect squares for all positive integers except n = 2 and n = 3, that is

n! = a^2 - b^2

(2) whether it is possible to express n! as the sum of two squares, that is

n! = a^2 + b^2


See sequence A138196 http://oeis.org/A138196


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