As stated in the title, I am trying to prove a statement by minimum counterexample involving modular arithmetic. My problem is producing the contradiction, but I feel so close!(adsbygoogle = window.adsbygoogle || []).push({});

(The contradiction is [itex]p^m | (1 + p)^{p^{m - 1}} - 1[/itex])

1. The problem statement, all variables and given/known data

Let [itex]p[/itex] be an odd prime and let [itex]n[/itex] be a positive integer. Show that[itex](1 + p)^{p^{n - 1}} \cong 1 \mod p^n[/itex]

2. Relevant equationsThe professor said as a hint, "Use the binomial theorem."

[itex](a+b)^n = \displaystyle\sum\limits_{i=0}^n {n \choose i}a^{n - i}b^i[/itex]

3. The attempt at a solution

I took a picture of the latex .pdf file since the latex code would not work here.

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# Homework Help: Incredibly close to a modular arithmetic proof by minimum counterexample!

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