Congruence Consequences I’d Forgotten


My integer factorization study has me looking a great deal at differences of perfect squares, one even and one odd. This is an easily justified path to take, since powers of two will be apparent and can divide out as soon as we start, regardless of the numerical base we are using. But I took a diversion to look at even positive integers that have expressions as the difference of perfect squares, both even or both odd. This is where I realized something simple, which perhaps merits mention only because I had either forgotten or failed to realize the principles that led to it before.

Since the squares of odd integers are always congruent to 1 modulo 8, the difference between two odd integers’ perfect squares will always have 8 as a divisor. For the difference of squares of even integers, the largest power of 2 that divides them might be 2 and not 3, but for odd ones, the principle holds true. So if an even integer is divisible by 4 but not by 8, it will have a difference-of-two-squares expression whose terms must be even. And if it is divisible by 2 but not 4, as I explained in my post some time ago, it has no compact expression as the difference of two squares.

The even residues modulo 8 have squares congruent to either 0 or 4, by the way, which means differences of even perfect squares can be congruent to 0 or 4 modulo 8 as well. That’s essentially almost explained above, but I’m spelling it out for completeness.

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