[seqfan] Re: Overview of sequence arising from binary quadratic forms
Lang, Wolfdieter
wolfdieter.lang at partner.kit.edu
Sat Jun 28 15:14:37 CEST 2014
Dear Neil (and the other seqfans),
nice binary quadratic forms OEIS wiki page!
This reminds me of my old Maple on-line program
http://www.itp.kit.edu/~wl/BinQuadForm.html
for the indefinite case.
I used the book
A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973.
Best regards,
Wolfdieter
________________________________________
Von: SeqFan [seqfan-bounces at list.seqfan.eu] im Auftrag von Neil Sloane [njasloane at gmail.com]
Gesendet: Samstag, 28. Juni 2014 04:39
An: Sequence Fanatics Discussion list
Betreff: [seqfan] Overview of sequence arising from binary quadratic forms
Dear Seq Fans, For the past month I've been surveying the sequences in the
OEIS that arise from binary quadratic forms ax^2+bxy+cy^2. Example: A033207
gives primes of the form x^2+7y^2.
The result can be seen here: <a href="
https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic
Forms and OEIS</a>
Here's a summary.
A binary quadratic form f(x,y) = ax^2+bxy+cy^2 has discriminant d =
b^2-4ac, and is positive definite if d < 0, or indefinite if d > 0.
f(x,y) represents an integer n if there are integers x and y such that
f(x,y)=n.
This web page gives an index to the following sequences in the OEIS:
- numbers (or primes) represented by positive definite binary quadratic
forms (Section 3),
- numbers (or primes) represented by indefinite binary quadratic forms
(Section 4),
This page also lists programs for computing these sequences (in Section 5)
and references and links (in Section 6).
There is a lot still to be done:
- there are several cases where a sequence appears to arise in several
different ways, not all of which have been proved to give the same sequence
(see for example the multiple meanings listed in A033212 and A141184). Once
these have been resolved, the entries can probably be merged.
- not all the sequences presently in the OEIS have been included, and
- there are many more that could be added (both to the OEIS and here).
- Furthermore, hundreds of the sequences mentioned here were computed using
a program QuadPrimes (see Section 5 and A106856) that unfortunately
contains a bug, which can occasionally cause it to produce incorrect
answers. These sequences are listed in Section 7 and need to be rechecked.
- Many of the sequences arising from indefinite quadratic forms (see
Section 4) were computed by "brute force", which is a notoriously
unreliable method (as one knows from studying Pell's equation). These also
need to be checked.
Neil
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