Hello,
thus the numbers can never be the same above one
when the mersenne number mod 4 is always 3 and
the odd square mod 4 is always 1.
Your analysis is correct but your conclusion is wrong.
You've proved that all Mersenne numbers greater than 1 cannot be odd
squares but you haven't proved that they can't be square free.
From your analysis you can conclude that if Mn is *not* square-free,
then it must have an odd number of factors that are congruent to 3
(mod 4).
With M21, for example:
M21 = 2^21-1 = 7 * 7 * 127 * 337
Breaks down as:
7 * 7 = 49 == 1 (mod 4)
127 == 3 (mod 4)
337 == 1 (mod 4)
The previous conclusion can be generalized to simply: All Mersenne
numbers greater than one must have an odd number of factors that are
congruent to 3 (mod 4).
-Don Leclair
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