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January 11th, 2019, 11:18 PM   #1
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A stronger fomulation of Fermat's Last Theorem

We consider only positive integers.

The formulation is as follows:

Every squared integer can be expressed as the difference of two squared integers; for powers greater than two, there is not a single integer for which an analogous statement is true.

In short, every integer is a member of a Pythagorean triple.

Note that for prime numbers and even semi-primes (2*odd prime), the expression is unique and consists in the former case of consecutive integers, in the latter of integers which differ by two. In other cases, there are as many expressions as there are factorisations of the squared integer into two unequal integer factors of the same parity (where 1 is considered a legitimate factor).
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January 12th, 2019, 03:07 AM   #2
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Examples:

Prime: 17×17 = 145×145 - 144×144

Even semi-prime: 22×22 = 122×122 - 120×120

Other: 12×12 = 37×37 - 35×35 = 20×20 -16×16 = 15×15 - 9×9 = 13×13 - 5×5
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