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Fermat's little theorem

$a^{p} \equiv a\pmod p$ if p prime

I have noticed that the formula $a^{p} \equiv a\pmod {2p}$ (p prime and p>2) can be also written by using Fermat's little theorem

Proof: Let $a$ be any integer and $p>2$ be some prime number.

$a^{p} \equiv a\pmod p$

$a^{p}-a \equiv 0\pmod p$

$a(a^{p-1}-1) \equiv 0\pmod p$

$a(a^{p-1}-1) \equiv 0\pmod p$

$F(a)=(a^{p-1}-1)$ can be divided to $(a-1)$ because $F(1)=(1^{p-1}-1)=0$

$F(a)=(a^{p-1}-1)=(a-1).R(a)$ //R(a) is polinom that has degree p-2 and coefficients are Integer numbers.

$a (a-1) R(a) \equiv 0\pmod p$

$a (a-1)$ is always an even number so $a (a-1) R(a)$ will be always even number and also can be divided to $p$ because of Fermat's little theorem . Thus $a^{p} \equiv a\pmod {2p}$ if (p prime and p>2) .

I searched internet but I have not seen that relation in the internet.

Is it known formula?Please help if it is known relation(I would like to learn what the subject is )

Sorry if someone else asked the same question in here.

Thank you for answers and helps.

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If $a$ is any integer, and $n$ a positive integer, then $a^n \equiv a \mod 2$. Your equation is a combination of this fact and Fermat's little theorem. Both are known congruences, but I'm note sure there's a name for their combination. –  Joel Cohen Feb 4 '12 at 14:49
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3 Answers

up vote 1 down vote accepted

As Joel rightly observed :If $a$ is any integer, and $n$ a positive integer, then $a^n\equiv a \pmod2$

So : $a^p\equiv a \pmod2$ , and $a^p\equiv a \pmod p$

According to Chinese Remainder Theorem :

$a^p \equiv a \pmod { \operatorname{lcm} (2,p)} $

,and since $p$ is an odd prime it follows that : $\operatorname{lcm}(2,p) = 2p$ , therefore :

$a^p \equiv a \pmod {2p}$

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The answer, Maybe, can be perfect and if we add some additional steps to show how proof of Chinese Remainder Theorem. c,d,e are integers if $a^p\equiv a \pmod2$ then $a^p=a+2.c$ $a^p-a=2.c$ if $a^p\equiv a \pmod p$ then $a^p=a+p.d$ $a^p-a=p.d=2.c$ $d=2.e$ can be written because $2.c$ is even number Thus $a^p-a=p.d=2.c=2.p.e$ $a^p-a\equiv 0 \pmod{2p}$ $a^p\equiv a \pmod{2p}$ Thanks in advice –  Mathlover Feb 4 '12 at 19:32
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You are correct I think, but the way you write it down makes it look more difficult than it is. Also you should be more specific on whether you are talking about one particular $a$ or "$\forall a$".

So let me show how I would write this down. First, let $a$ be any integer and $p>2$ be some prime number. Then

$$ a^p \equiv a \pmod{2p} \iff 2p \mid a^p - a \iff 2\mid a^{p}-a \text{ and } p\mid a^p-a$$ The last equivalence holds because $2$ and $p$ are two different primes.

But $2\mid a^p-a$ for all $a$ and $p$: if $a$ is even, then clearly $a^p$ will be even as well; if $a$ is odd, then so is $a^p$ and hence $a^p-a$ will be even again. So this is indeed equivalent to

$$\dots \iff p\mid a^p-a \iff a^p \equiv a \pmod p.$$

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Thanks for answer. –  Mathlover Feb 4 '12 at 14:46
    
No actually it's a continuation of the chain of equivalences up above. So it should be read as $$ a^p \equiv a \pmod{2p} \iff \dots \iff a^p\equiv a \pmod{p}$$ –  Myself Feb 4 '12 at 14:51
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This is the special case $\rm\ m = 2\:p,\ p\:$ odd, of the following generalization of Fermat's little theorem, quoted from Bill Dubuque's post

THEOREM $\ $ For naturals $\rm\: k,m>1 $

$\qquad\rm m\ |\ a^k-a\ $ for all $\rm\:a\in\mathbb N\ \iff\ m\:$ is squarefree and prime $\rm\: p\:|\:m\: \Rightarrow\: p-1\ |\ k-1 $

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