I have a question; Let m have a prime factor [itex] p \equiv 1 (mod 4)[/itex]. Then Euler function [itex]\varphi(m)[/itex] is divisible by 4. Let [itex]x = r^{\varphi(m)}[/itex], then [itex]m|(x^4-1)[/itex] and [itex]x^4-1=(x^2-1)(x^2+1)[/itex]. As [itex]gcd(x^2-1,x^2+1)|2[/itex], either [itex]x^2-1[/itex] or [itex]x^2+1[/itex] is divisible by m. My book says here because of the nature of a prime factor, and that [itex]x^2-1[/itex] isn't divisible by m, so that [itex]x^2+1[/itex] is divisible by m. I cannot understand why [itex]x^2-1[/itex] isn't divisible by m because of "a nature of a prime factor".(adsbygoogle = window.adsbygoogle || []).push({});

Will anyone help me?

TIA

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# Primitive root question

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