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Number Theory Odd composite property 1

yourskadhir

New member
Dec 26, 2012
1
Hi,

For any odd composite 'N', let u = (N-1)/2, v = u+1, then u^2(mod p) = v^2(mod p) if and only if 'p' is a factor of 'N'.
 
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Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
Re: Odd composite property1

suppose that:

$u^2 \equiv v^2\ (\text{mod }p)$

since $v = u+1$ this is the same as saying:

$0 \equiv 2u + 1\ (\text{mod }p)$

that is:

$2u + 1 = kp$ for some integer $k$.

recalling that $u = \frac{N-1}{2}$, we see that:

$2u + 1 = N = kp$, in which case $p$ divides (is a factor of) $N$.

on the other hand, suppose that $N = kp$.

then:

$u^2 = \frac{N^2 - 2N + 1}{4}$ while:

$v^2 = \left(\frac{N+1}{2}\right)^2 = \frac{N^2 + 2N + 1}{4}$

so:

$v^2 - u^2 = \frac{4N}{4} = N = kp$ thus:

$u^2 \equiv v^2\ (\text{mod }p)$

(it might be instructive to see why p cannot be 2).