Seq. of largest QR mod the primes
Adorján Ferenc
fadorjan at freemail.hu
Sat Nov 27 11:04:00 CET 2004
Dear/Cher Sequence fans,
I joined recently this group and now I wish to turn your attention to a
set of sequences (A088190-88201) which are related to the sequence
of largest quadratic residues (also of non-residues) modulo the
subsequent primes. I submitted them in Sept. 2003, and also
elaborated a quite detailed draft describing several surpising findings:
http://web.axelero.hu/fadorjan/qrp.pdf
Most of the findings remained unproven, I was only able to prove a
few. However, all of them are verified up to the index of 10^5.
The most remarkable things are as follows:
- the sequence of larges QR modulo the primes is increasing "almost"
monotonically, but from time to time there occur some "glitches", where
a(i)<=a(i-1). It is quite surprising that the p(i)-s (the i-th prime) have
special features where such things occur: they are always congruent to
7 (mod 8) but never 2 (mod 5), however not all primes with these
features produce the glitch.
- it could be proven that the distance between the primes and the
related largest QR is either 1 or some prime number, and also the
condition of being this difference as 1 is given.
Altogether, there are 20 specific observations given in the paper, but
most of them remained unporven conjectures supported by remarkable
numerical evidence.
I am sure that there are several people in this list who are much more
experienced and also more smart than me, therefore I hope that some
of you could provide important contributions or new ideas to the matter.
Best regards
Ferenc (Francois) Adorjan
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