Order of an element modulo $n$ divides $varphi(n)/2$.

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Let $n$ be an integer different from $2,4,p^alpha$ and $2p^alpha$; ($p$ is odd prime).



Using just elementary number theory (not group isomorphism), prove that
$$a^varphi(n)/2=1 mod n$$
(I have proved it using group isomorphism and order of elements, but i want an elementary proof).




linear-algebra number-theory elementary-number-theory




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    $n = 12$, I suppose, $varphi(12) = 4$, but $2^2 neq 1 mod 12$
    – dEmigOd
    Jul 15 at 11:47











  • math.stackexchange.com/questions/114841/…
    – lab bhattacharjee
    Jul 15 at 11:53






  • 1




    To me it looks like the point of the hypotheses is that n can be written as xy where gcd(x,y) = 1 and where phi(x) and phi(y) are both even. This implies phi(n)/2 is divisible by phi(x) and is divisible by phi(y). Then use the Chinese Remainder Theorem and Euler's Theorem. (I don't think the linear-algebra tag is appropriate.)
    – CJD
    Jul 15 at 11:54










  • I mean by $a$ is an element of the multiplicatif group $left(mathbbZ/nmathbbZright)^*$; so in the example $a=2$ modulo $12$ will not work. @dEmigOd
    – C.S.
    Jul 15 at 12:16















up vote
0
down vote

favorite
1












Let $n$ be an integer different from $2,4,p^alpha$ and $2p^alpha$; ($p$ is odd prime).



Using just elementary number theory (not group isomorphism), prove that
$$a^varphi(n)/2=1 mod n$$
(I have proved it using group isomorphism and order of elements, but i want an elementary proof).




linear-algebra number-theory elementary-number-theory




share|cite|improve this question















  • 1




    $n = 12$, I suppose, $varphi(12) = 4$, but $2^2 neq 1 mod 12$
    – dEmigOd
    Jul 15 at 11:47











  • math.stackexchange.com/questions/114841/…
    – lab bhattacharjee
    Jul 15 at 11:53






  • 1




    To me it looks like the point of the hypotheses is that n can be written as xy where gcd(x,y) = 1 and where phi(x) and phi(y) are both even. This implies phi(n)/2 is divisible by phi(x) and is divisible by phi(y). Then use the Chinese Remainder Theorem and Euler's Theorem. (I don't think the linear-algebra tag is appropriate.)
    – CJD
    Jul 15 at 11:54










  • I mean by $a$ is an element of the multiplicatif group $left(mathbbZ/nmathbbZright)^*$; so in the example $a=2$ modulo $12$ will not work. @dEmigOd
    – C.S.
    Jul 15 at 12:16













up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





Let $n$ be an integer different from $2,4,p^alpha$ and $2p^alpha$; ($p$ is odd prime).



Using just elementary number theory (not group isomorphism), prove that
$$a^varphi(n)/2=1 mod n$$
(I have proved it using group isomorphism and order of elements, but i want an elementary proof).




linear-algebra number-theory elementary-number-theory




share|cite|improve this question











Let $n$ be an integer different from $2,4,p^alpha$ and $2p^alpha$; ($p$ is odd prime).



Using just elementary number theory (not group isomorphism), prove that
$$a^varphi(n)/2=1 mod n$$
(I have proved it using group isomorphism and order of elements, but i want an elementary proof).





linear-algebra number-theory elementary-number-theory





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 15 at 11:37









C.S.

134




134







  • 1




    $n = 12$, I suppose, $varphi(12) = 4$, but $2^2 neq 1 mod 12$
    – dEmigOd
    Jul 15 at 11:47











  • math.stackexchange.com/questions/114841/…
    – lab bhattacharjee
    Jul 15 at 11:53






  • 1




    To me it looks like the point of the hypotheses is that n can be written as xy where gcd(x,y) = 1 and where phi(x) and phi(y) are both even. This implies phi(n)/2 is divisible by phi(x) and is divisible by phi(y). Then use the Chinese Remainder Theorem and Euler's Theorem. (I don't think the linear-algebra tag is appropriate.)
    – CJD
    Jul 15 at 11:54










  • I mean by $a$ is an element of the multiplicatif group $left(mathbbZ/nmathbbZright)^*$; so in the example $a=2$ modulo $12$ will not work. @dEmigOd
    – C.S.
    Jul 15 at 12:16













  • 1




    $n = 12$, I suppose, $varphi(12) = 4$, but $2^2 neq 1 mod 12$
    – dEmigOd
    Jul 15 at 11:47











  • math.stackexchange.com/questions/114841/…
    – lab bhattacharjee
    Jul 15 at 11:53






  • 1




    To me it looks like the point of the hypotheses is that n can be written as xy where gcd(x,y) = 1 and where phi(x) and phi(y) are both even. This implies phi(n)/2 is divisible by phi(x) and is divisible by phi(y). Then use the Chinese Remainder Theorem and Euler's Theorem. (I don't think the linear-algebra tag is appropriate.)
    – CJD
    Jul 15 at 11:54










  • I mean by $a$ is an element of the multiplicatif group $left(mathbbZ/nmathbbZright)^*$; so in the example $a=2$ modulo $12$ will not work. @dEmigOd
    – C.S.
    Jul 15 at 12:16








1




1




$n = 12$, I suppose, $varphi(12) = 4$, but $2^2 neq 1 mod 12$
– dEmigOd
Jul 15 at 11:47





$n = 12$, I suppose, $varphi(12) = 4$, but $2^2 neq 1 mod 12$
– dEmigOd
Jul 15 at 11:47













math.stackexchange.com/questions/114841/…
– lab bhattacharjee
Jul 15 at 11:53




math.stackexchange.com/questions/114841/…
– lab bhattacharjee
Jul 15 at 11:53




1




1




To me it looks like the point of the hypotheses is that n can be written as xy where gcd(x,y) = 1 and where phi(x) and phi(y) are both even. This implies phi(n)/2 is divisible by phi(x) and is divisible by phi(y). Then use the Chinese Remainder Theorem and Euler's Theorem. (I don't think the linear-algebra tag is appropriate.)
– CJD
Jul 15 at 11:54




To me it looks like the point of the hypotheses is that n can be written as xy where gcd(x,y) = 1 and where phi(x) and phi(y) are both even. This implies phi(n)/2 is divisible by phi(x) and is divisible by phi(y). Then use the Chinese Remainder Theorem and Euler's Theorem. (I don't think the linear-algebra tag is appropriate.)
– CJD
Jul 15 at 11:54












I mean by $a$ is an element of the multiplicatif group $left(mathbbZ/nmathbbZright)^*$; so in the example $a=2$ modulo $12$ will not work. @dEmigOd
– C.S.
Jul 15 at 12:16





I mean by $a$ is an element of the multiplicatif group $left(mathbbZ/nmathbbZright)^*$; so in the example $a=2$ modulo $12$ will not work. @dEmigOd
– C.S.
Jul 15 at 12:16
















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