# Prime divisors of 4n^2+4n-1

1. May 30, 2010

### TTob

I need to prove that p congruent modulo 8 to $$\pm 1$$ for every prime divisor p of $$4n^2+4n-1$$.
$$4n^2+4n-1$$ is odd so we have
$$p \equiv \pm 1,3,5 \pmod{8}$$

I don't know how to continue from here... I need some hint.

Thanks.

Last edited: May 31, 2010
2. May 30, 2010

### Martin Rattigan

Congruent modulo what? $7|(4.5^2+4.5-1)$, but $7\neq\pm 1(5)$, so presumably the answer is not $n$.

3. May 31, 2010

### TTob

I need to prove that p congruent modulo 8 to $$\pm 1$$.

4. May 31, 2010

### Martin Rattigan

$4n^2+4n-1=(2n+1)^2-2=0(p)$ only if $2$ is a quadratic residue mod $p$ which is true only if $p=\pm 1(8)$. (This is proved in any number theory text in the section on quadratic reciprocity.)

5. May 31, 2010

### TTob

Thank you !

6. May 31, 2010

### robert Ihnot

$$p \equiv \pm 1,3,5 \pmod{8}$$

Boy! that was a confusing bunch of stuff. Lot of times, the trouble with a problem is the writer did not correctly organize what he wants to solve. But I see, TTob got it all correct!

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