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Old 2022-07-03, 02:55   #1
carpetpool
 
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"Sam"
Nov 2016

5·67 Posts
Post Is it possible for an odd perfect number to have the form 8*k+5?

From Euler's Criterion of an odd perfect number n, we know that n = p^m*s^2 for a prime p = 1 mod 4 and exponent m = 1 mod 4.

Consequently, n is the sum of two squares.

Is n necessarily the sum of a square and two times a square?

This would eliminate the possibility of odd perfect numbers congruent to 5 modulo 8.

Conversely, if n is congruent to 1 mod 8, then n = a^2 + 2*b^2 for nonnegative integers a and b.

Last fiddled with by carpetpool on 2022-07-03 at 02:56
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