Seq. of largest QR mod the primes

Adorján Ferenc fadorjan at
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:
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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