1
$\begingroup$

Evidently Legendre showed that, for positive primes, if $p \equiv 3 \pmod 8$ there is an integral solution to $x^2 - p y^2 = -2.$ Next, if $q \equiv 7 \pmod 8$ there is an integral solution to $x^2 -q y^2 = 2.$

What I would like, and seems to be true, is $x^2 - 2 p y^2 = -2$ for $p \equiv 3 \pmod 8,$ and $x^2 - 2 q y^2 = 2$ for $q \equiv 7 \pmod 8.$ It is probably in Mordell's book, which I do not have here.

Mordell does $x^2 - r y^2 = -1$ for any prime $r \equiv 1 \pmod 4,$ I do remember that. Anyway, I am writing up something and this issue came up.

P.S. Note these are the same as $2x^2 - p y^2 = -1$ if $p \equiv 3 \pmod 8,$ while if $q \equiv 7 \pmod 8$ there is $2x^2 -q y^2 = 1.$

$\endgroup$
2
  • $\begingroup$ Dirichlet generalized Legendre's technique to composite values of m in $x^2 - my^2 = d$; whether he actually treated the cases you are interested in is irrelevant since the method of proof can be transferred easily. I might have given a few references in my papers on descent on Pell conics. $\endgroup$ Jul 2, 2012 at 18:49
  • $\begingroup$ @Franz thanks. I looked at some of your homework solution pdfs and did not see this. I will look at the Pell conics items. $\endgroup$
    – Will Jagy
    Jul 2, 2012 at 19:01

1 Answer 1

2
$\begingroup$

According to Dickson (History of numbers Vol. 2, Ch. XII, p.376), Göpel (Jour. für Math. 45, 1853, 1-14) proved your conjectures "by use of continued fractions".

Actually Jour. für Math. stands for Crelle's journal, and Göpel's paper (which is his 1835 doctoral dissertation) is available online here.

$\endgroup$
5
  • $\begingroup$ Thanks. I have Dickson's history at home, I did not think to check there. Also, this means Dickson was definitely aware of this when he wrote the book Studies in 1930. I did not find the indefinite ternary quadratic form I wanted in the tables on pages 150-151, then I realized that it would all work out if the binary forms I mentioned behaved as Gopel proved. Here we go, Crelle's is nowadays Journal für die reine und angewandte Mathematik. Very good. $\endgroup$
    – Will Jagy
    Jul 2, 2012 at 21:37
  • $\begingroup$ It's in Latin. Wow. $\endgroup$
    – Will Jagy
    Jul 2, 2012 at 21:40
  • $\begingroup$ I tried to find a more readable account, but no luck. It also took me a bit of time to realize that "Jour. für Math." meant "Journal für die reine und angewandte Mathematik". I think the title of the journal never changed, "Crelle's journal" has always been folklore. $\endgroup$
    – GH from MO
    Jul 2, 2012 at 21:47
  • $\begingroup$ According to en.wikipedia.org/wiki/Adolph_Göpel, Gopel died in 1847, 34 years old, and "after his death some of his works were published in Crelle's Journal." More onformation about Gopel at www-history.mcs.st-andrews.ac.uk/Biographies/Gopel.html. If you go to Google Scholar and type in gopel quadratic you'll find a few papers that mention his work. I haven't checked to see whether any of them give his, or other, proofs of the results in question. $\endgroup$ Jul 2, 2012 at 23:26
  • $\begingroup$ In volume 3, page 19, Dickson gives a bit more detail, but he seems to be reporting on writing $p$ and $2p$ as $x^2 \pm 2 y^2.$ Anyway, continued fractions are fine by me, I am happy using "reduced" indefinite binary quadratic forms, and demonstrating equivalence by finding the cycle of neighboring forms. This is a disguise for continued fractions. So if $p \equiv 3 \pmod 8,$ then $ p x^2 - 2 y^2 \equiv x^2 - 2 p y^2, $ and if $q \equiv 7 \pmod 8,$ then $ 2 x^2 - q y^2 \equiv x^2 - 2 q y^2. $ $\endgroup$
    – Will Jagy
    Jul 2, 2012 at 23:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.