# comment to A002523 n^4 + 1, and/or new sequence?

Jonathan Post jvospost3 at gmail.com
Wed Jun 11 02:41:42 CEST 2008

```Should this be added as a comment to A002523 n^4 + 1?

Mabkhout (1993) proved that every number x^4 + 1, for x > 3, has a
prime factor greater than or equal to 137. Størmer's theorem is an
important part of his proof, in which he reduces the problem to the
solution of 128 Pell equations.

Mabkhout, M. (1993). "Minoration de P(x4+1)". Rend. Sem. Fac. Sci.
Univ. Cagliari 63 (2): 135–148.

Should that factor => 137 of A002523(n) = n^4 + 1, for n>3, be a
sequence on its own?

257, 313, 1297, 1201, 241, 193, 137, 7321, 233, 14281, 937, 1489,
65537, 41761, 929, 3833,  160001, 97241, 3209, 139921, 331777, 11489,
26881, 6481, 614657, 353641, and now we get an ambiguity:

30^4 + 1 = 810001 = 241 * 3361 and so we are free to define either the
seq given below (A096172), OR to define a new seq:

Least prime factor => 137 of n^4+1 {with offset 4,1}.

The same Mabkhout comment and citation should be, (shouldn't it?) in
A096172  Largest prime factor of n^4+1.

The next times that the proposed new seq differs from A096172 are for
n = 31, 40, 41, 43, 45, 58, 59, 60, ... (itself a possible new
sequence):

31^4 + 1  = 923522 = 2 * 409 * 1129

40^4 + 1 = 2560001 = 769 * 3329

41^4 + 1 = 2825762 = 2 * 137 * 10313 (a lovely border-line case)

43^4 + 1 = 3418802 = 2 * 17 * 193 * 521

45^4 + 1 = 4100626 = 2 * 401 * 5113

58^4 + 1 = 11316497 = 2393 * 4729

59^4 + 1 = 12117362 = 2 * 17 * 593 * 601

60^4 + 1 = 12960001 = 17 * 281 * 2713

and so on.

Best,

Jonathan Vos Post

Jonathan said:

> What are the general guidelines for referencing a paper in the
> hardcopy or online literature that uses an OEIS sequence but fails to
> notice or cite that?

Please use the web page to submit them.

The format is as follows:

For a literature reference:

%D A007067 C. Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.

(and there are a few other examples in the OEIS already)

For a web page:

%H A117045 K. Shankar, <a href="http://www.math.ou.edu/~shankar/papers.html">Square roots and continued fractions</a>.

(ditto)

Thanks

Neil

jvp> From seqfan-owner at ext.jussieu.fr  Tue Jun 10 19:01:01 2008
jvp> Date: Tue, 10 Jun 2008 09:59:48 -0700
jvp> From: "Jonathan Post" <jvospost3 at gmail.com>
jvp> To: "Sequence Fans" <seqfan at ext.jussieu.fr>
jvp> Subject: A005097 (Odd primes - 1)/2 implicitly referenced in new arXiv paper
jvp>
jvp> A005097 (Odd primes - 1)/2 is implicitly referenced in the new arXiv
jvp> paper, Theorem 2.1, p.5
jvp>
jvp> http://arxiv.org/pdf/0806.0251
jvp> (replaced)
jvp>     Title: On Hamilton Decompositions
jvp>     Authors: Dhananjay P. Mehendale
jvp> tournament on 9 vertices is added
jvp>     Subjects: General Mathematics (math.GM)
jvp>
jvp>
jvp> What are the general guidelines for referencing a paper in the
jvp> hardcopy or online literature that uses an OEIS sequence but fails to
jvp> notice or cite that?
jvp> ...

If the paper is contributing to the OEIS such that it becomes one of
the references (%H) or links (%Y) in the OEIS, you can send the
typical snapshot of the associated OEIS sequence to the paper's author.
This would be a straight way to promote use/awareness/acknowledgment of the
OEIS in the long term. This is in particular useful if the reference list
in the paper is of sub-standard quality (short in the sense that it indicates
the author does not have access to some library with mathematical journals
and would be willing to cite the OEIS--an open-access database).

Richard

```