[seqfan] Re: Trying to compute A002052 (Prime determinants of forms with class number 2)
Rick Shepherd
rlshepherd2 at gmail.com
Fri May 1 21:05:19 CEST 2015
Hi Sean and Seqfans,
You mean "except for d=79" below. Your post inspired me to make some
additions to A002052, but I also cannot make the test work as stated (or
with 3 or 4 other interpretations). Notice that both "discriminant" and
"determinant" are used in the paper.
If someone could clarify Chowla's test (in the seq), I'd appreciate it.
Regards and thanks,
Rick
On Apr 21, 2015 5:21 PM, "Sean A. Irvine" <sairvin at gmail.com> wrote:
> In my quest to improve some of the early OEIS sequences, I have been
> trying to reproduce A002052 (Prime determinants of forms with class
> number 2). Suryanarayana's paper (which is only 2 pages) describes a
> test for membership as:
>
> "Let d = 3 (mod 4), d > 0. Let p denote any prime factor of d - x^2 (x
> < sqrt(d)), p < sqrt(d). Then h(d) = 2 [i.e. d is in this sequence] if
> p occurs as a "partial quotient" in the simple continued fraction for
> sqrt(d)."
>
> It is not clear to me, if the test must hold for all prime factors of
> d - x^2 or just one of them. But either way, I have been unable to
> reproduce the sequence. It is likely that I'm misunderstanding
> something about the test. Perhaps someone else could have a crack at
> it?
>
> Suryanarayana's paper is available here:
>
> http://www.ias.ac.in/j_archive/proca/2/2/178-179/viewpage.html
>
> Some of my data: In the following, the first column corresponds to d,
> "primes" is a list of possible p satisfying condition above, and
> "cfrac" is the continued fraction for sqrt(d), all these d should be
> in the sequence except for d=83:
>
> 7 primes [2] cfrac [2, 1, 1, 1, 4]
> 11 primes [2] cfrac [3, 3, 6]
> 19 primes [2, 3] cfrac [4, 2, 1, 3, 1, 2, 8]
> 23 primes [2] cfrac [4, 1, 3, 1, 8]
> 31 primes [2, 3, 5] cfrac [5, 1, 1, 3, 5, 3, 1, 1, 10]
> 43 primes [2, 3] cfrac [6, 1, 1, 3, 1, 5, 1, 3, 1, 1, 12]
> 47 primes [2] cfrac [6, 1, 5, 1, 12]
> 59 primes [2, 5] cfrac [7, 1, 2, 7, 2, 1, 14]
> 67 primes [2, 3, 7] cfrac [8, 5, 2, 1, 1, 7, 1, 1, 2, 5, 16]
> 71 primes [2, 5, 7] cfrac [8, 2, 2, 1, 7, 1, 2, 2, 16]
> 79 primes [2, 3, 5, 7] cfrac [8, 1, 7, 1, 16]
> 83 primes [2] cfrac [9, 9, 18]
> 103 primes [2, 3] cfrac [10, 6, 1, 2, 1, 1, 9, 1, 1, 2, 1, 6, 20]
> 107 primes [2, 7] cfrac [10, 2, 1, 9, 1, 2, 20]
> 127 primes [2, 3, 7] cfrac [11, 3, 1, 2, 2, 7, 11, 7, 2, 2, 1, 3, 22]
> 131 primes [2, 5] cfrac [11, 2, 4, 11, 4, 2, 22]
> 139 primes [2, 3, 5] cfrac [11, 1, 3, 1, 3, 7, 1, 1, 2, 11, 2, 1, 1,
> 7, 3, 1, 3, 1, 22]
> 151 primes [2, 3, 5, 7] cfrac [12, 3, 2, 7, 1, 3, 4, 1, 1, 1, 11, 1,
> 1, 1, 4, 3, 1, 7, 2, 3, 24]
>
> Regards,
> Sean.
>
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