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Thread: x^n ≡ 3 (mod 4) only if x ≡ n ≡ 1 (mod 2)

  1. #1
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    x^n ≡ 3 (mod 4) only if x ≡ n ≡ 1 (mod 2)

    Show that x^n ≡ 3 (mod 4) only if x ≡ n ≡ 1 (mod 2)
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  2. #2
    Senior Member Dinkydoe's Avatar
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    You mean: "if and only if" ?

    Anyway: look at resctlasses modulo 4

    That is: $\displaystyle \mathbb{Z}/4\mathbb{Z} = \left\{0,1,2,3\right\}$

    Let $\displaystyle x\equiv 0 $ mod 4. Then $\displaystyle x^n\equiv 0$ mod 4
    Let $\displaystyle x\equiv 1$ mod 4. Then $\displaystyle x^n\equiv 1$ mod 4
    Let $\displaystyle x\equiv 2$ mod 4. Then $\displaystyle x^n\equiv 0$ mod 4 for $\displaystyle n\geq 2$.

    Let $\displaystyle x\equiv 3$ mod 4. Then $\displaystyle x^n\equiv 1$ mod 4 for even n. And $\displaystyle x^n\equiv 3$ mod 4 for odd n.

    Hence the conclusion follows.
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