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Trying to describe all primes of the form:

$x^2+5xy+5y^2$

A hint was given with the question to show all primes $p$ for which 5 is a quadratic residue mod $p$. I've been able to show that all primes $\pm1$ mod 5 satisfy this... but I don't know how this helps. Any next-step pointers would be appreciated! Thanks!

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    @Gerry, I see what you mean. The OP for that question seems to answer them under a different name and accept those. The common act is the extra ! characters while typing.2012-11-13

1 Answers 1

5

Um, look, any (positive) odd prime with $(5|p) = (p | 5) = 1$ can be used to produce a binary quadratic form $f(x,y) = a x^2 + b x y + c y^2,$ or $\langle a,b,c \rangle,$ with particular coefficients $\langle p,\beta,\gamma \rangle$ with discriminant $\Delta = \beta^2 - 4 p \gamma = 5.$ This can then be reduced to $\langle 1,1,-1 \rangle$ or the equivalent $\langle -1,1,1 \rangle,$ showing that we can write $p = u^2 + u v - v^2.$ A small change shows $p = x^2 + 5 x y + 5 y^2,$ maybe you can do that part.

See Numbers representable as $x^2 + 2y^2$

Note that reduction for indefinite forms is a little different from positive. We can arrange $\langle a,b,c \rangle,$ such that $0 < b < \sqrt \Delta$ and $ \sqrt \Delta - b < 2 |a| < \sqrt \Delta + b$ and there are more than one, a finite number of reduced forms in each equivalence class.

Reduction can be arranged by a finite sequence of these steps: take the "Hessian" matrix of the form $\langle a,b,c \rangle,$ as $ H = \left( \begin{array}{cc} 2a & b \\ b & 2c \end{array} \right). $ Now take the matrix $P \in SL_2 \mathbb Z$ given by $ P = \left( \begin{array}{cc} 0 & -1 \\ 1 & \delta \end{array} \right) $ and calculate $ G = P^T H P. $ Now, $G$ is the Hessian matrix of an "equivalent" form (think about how to go back from a Hessian matrix to a form). A correct choice of the integer $\delta$ takes the form closer to reduced, after a few such steps the form is reduced, and further steps take the form through a cycle of equivalent forms, back to the first reduced one. The absolute values of the $\delta$'s (once reduced) are the digits for the repeated part of the continued fraction for a certain quadratic irrational, tied up with Pell's equation. The continued fraction with all "digits" equal to $1$ is the Golden Ratio. I'm just sayin'.

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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2     1       5      5      0  form              1           5           5  delta      0   1  form              5          -5           1  delta     -2   2  form              1           1          -1             -1           2            0          -1  To Return             -1          -2            0          -1  0  form   1 1 -1   delta  -1 1  form   -1 1 1   delta  1 2  form   1 1 -1 minimum was   1rep 1 0 disc   5 dSqrt 2.2360679775  M_Ratio  5 Automorph, written on right of Gram matrix:   -1  -1 -1  -2  Trace:  -3   gcd(a21, a22 - a11, a12) : 1 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$  

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    merci *********2012-11-13