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Trying to describe all primes of the form:


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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Now I'm curious. What book is this from? What examples have they done so far where they prove the primes are represented? This particular example is the easiest indefinite form with odd middle coefficient. –  Will Jagy Nov 13 '12 at 3:13
I don't think it's from a book. It was given as a homework question. The book I'm reading for the course is "A Concise INtroduction to the Theory of Numbers" by Alan Baker –  Thomas Nesbitt Nov 13 '12 at 3:15
I lied. I just checked the book. It's actually in there as Question 2 on page 41 –  Thomas Nesbitt Nov 13 '12 at 3:16
Have a look at my answer at math.stackexchange.com/questions/232631/… –  Gerry Myerson Nov 13 '12 at 5:23
@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. –  Will Jagy Nov 13 '12 at 5:31
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1 Answer

up vote 5 down vote accepted

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'.


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 ********* –  Thomas Nesbitt Nov 13 '12 at 6:26
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