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2020-07-14, 12:54   #23
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

750810 Posts

Quote:

One might also want to ask:

When does x^q = a mod p have a single root, when does it have multiple roots,
and when does it split completely for given a, p, q???

Welcome to the wonderful world of Galois groups.

Note that this question also arises during study of the Special Number Field Sieve.

Last fiddled with by R.D. Silverman on 2020-07-14 at 12:56

 2020-07-14, 13:06 #24 Dr Sardonicus     Feb 2017 Nowhere 3×31×67 Posts Reading the last posts, it occurred to me to wonder, given a prime p, how large can the smallest q be (in terms of p), that does not divide p-1. One answer is, "of order ln(p) at most." I am sure that, given a lower bound for p (say 1000 or 1040 or something), a constant C near 1 could be given for which q is at most C*ln(p). This is a consequence of PNT, though it might be possible to get by with less, e.g. some of Chebyshev's estimates which predate proofs of PNT.
2020-07-14, 13:33   #25
JeppeSN

"Jeppe"
Jan 2016
Denmark

18510 Posts

Quote:
 Originally Posted by R.D. Silverman Hey clueless. You can't even do simple arithmetic. Nor do you bother to study any math. The result makes you look stupid every time you open your mouth. [...] Finally STFU until you can be bothered studying at least some of this subject. If you want to post mindless numerology go to the misc.math sub-forum or open your own sub-forum in the blogorrhea.
Silverman, in my opinion many of your posts are unnecessarily offensive. It is nice that you spend time to help people who do not have the mathematical insight you have, but you do it in a way that is rude and much too condescending. If you are unable to write in a polite and friendly manner, no matter how stupid other participants may seem to you, I think you should s*** t** f*** u*. /JeppeSN

Last fiddled with by Uncwilly on 2020-07-14 at 14:24 Reason: expurgated

2020-07-14, 14:14   #26
Uncwilly
6809 > 6502

"""""""""""""""""""
Aug 2003
101×103 Posts

1089610 Posts

Quote:
 Originally Posted by R.D. Silverman Finally STFU until you can be bothered studying at least some of this subject.
That sort of language does not belong in the Number Theory forum.

If you can't be civil, refrain from posting. There is no reason you have to post. If you do choose to get involved, I would suggest that you give an OP 2 rounds of comments. If they are hopeless after that, state so politely, then no longer engage.

Last fiddled with by Uncwilly on 2020-07-14 at 14:22

2020-07-14, 14:26   #27
Uncwilly
6809 > 6502

"""""""""""""""""""
Aug 2003
101×103 Posts

2A9016 Posts

Quote:
 Originally Posted by JeppeSN I think you should s*** t** f*** u*. /JeppeSN
That sort of language does not belong in the Number Theory forum. You will be joining R.D. Silverman in time away from posting.

2020-07-16, 04:36   #28

May 2004

4748 Posts

Quote:
 Originally Posted by R.D. Silverman One might also want to ask: When does x^q = a mod p have a single root, when does it have multiple roots, and when does it split completely for given a, p, q??? Welcome to the wonderful world of Galois groups. Note that this question also arises during study of the Special Number Field Sieve.
Ok so I have been hasty.here is a summary of my contributions to number theory:
Euler's generalization of Fermat's theorem- a further generalization
(ISSN #1550 3747- Hawaii international conference on mathematics and statistics-2004)
The theorem: let f(x) = a^x + c
where a belongs N and is fixed, c belongs to Z and is fixed and x belongs to N. Then a^(x +k*f(x)) + c is congruent to 0 (mod f(x)).
Here k belongs to N.
Proof is based on Taylor's theorem.
Applications: 1) finding some factors of very large rational integers when expressed in an exponential form 2)finding impossible prime factors of exponential functions ( see A 123239 of OEIS)
Other contributions to number theory: a) Universal exponent generalization of Fermat's theorem(Hawaii international conference-2006) b)ultimate generalisation of Fermat's theorem(planetmath .org-2012)
c) modified Fermat's theorem in order to accommodate Gaussian integers as bases(mersenneforum .org-recent)
d)A theorem a la Ramanujan (AMS-BENELUX-1996)
Also search for "akdevaraj" on youtube.
e) a property of Carmichael numbers conjectured in '89 and proved by Carl Pomerance (generalised conjecture proved by Maxal- see A 104016 and A 104017 on OEIS)

Last fiddled with by devarajkandadai on 2020-07-16 at 04:40 Reason: A minor correction

2020-07-16, 07:32   #29
preda

"Mihai Preda"
Apr 2015

101101001012 Posts

Quote:
 Originally Posted by Uncwilly That sort of language does not belong in the Number Theory forum. If you can't be civil, refrain from posting. There is no reason you have to post. If you do choose to get involved, I would suggest that you give an OP 2 rounds of comments. If they are hopeless after that, state so politely, then no longer engage.
Let's cut him some slack. If he's right, knowledgeable and informative, I'm willing to let him choose the manner of expressing himself just to hear his ideas.

2020-07-16, 09:47   #30
paulunderwood

Sep 2002
Database er0rr

24×281 Posts

Quote:
 Originally Posted by devarajkandadai Ok so I have been hasty.here is a summary The theorem: let f(x) = a^x + c where a belongs N and is fixed, c belongs to Z and is fixed and x belongs to N. Then a^(x +k*f(x)) + c is congruent to 0 (mod f(x)). Here k belongs to N.
Let a=2, x=4, c=3 and k=1.

Then f(2) = 2^4 + 3 = 19

2^(4+1*19)+3 = 2^23 + 3 = 16 mod 19 ???

However if you are saying: Let f(x)=a^x+c. For all a in N and for all c in Z then there exists a k such that f(x+k*f(x))=0 mod f(x) for all x in N; that may be a different matter.

Last fiddled with by paulunderwood on 2020-07-16 at 10:19

 2020-07-16, 13:35 #31 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 53·59 Posts I suggest that a moderator edit out the deliberate nastiness from the thread. Something along the lines of "(redacted abusive content)" would appear, twice in https://www.mersenneforum.org/showpo...7&postcount=22 and also in such quoted or original content in other posts as in 23 and 25. That sort of deliberately abusive language does not belong anywhere in the mersenne forum. To originate it, as RDS did, seems to me a greater issue, than to object to it as JeppeSN did, mimicking RDS to give RDS back a little taste of his own vitriol. I suggest RDS spend his time off reading Dale Carnegie's "How to Win Friends and Influence People" and https://www.mersenneforum.org/showpo...00&postcount=1 and employ them upon return. Last fiddled with by kriesel on 2020-07-16 at 13:44
2020-07-16, 13:48   #32
Uncwilly
6809 > 6502

"""""""""""""""""""
Aug 2003
101×103 Posts

24·3·227 Posts

Quote:
 Originally Posted by kriesel To originate it, as RDS did, seems to me a greater issue, than to object to it as JeppeSN did, mimicking RDS to give RDS back a little taste of his own vitriol.
RDS posted it via an initialism. Jeppe did it in plain language that was expurgated by a moderator. Answering vitriol with vitriol just covers everyone with vitriolic acid.

2020-07-16, 14:36   #33
kriesel

"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

53×59 Posts

Quote:
 Originally Posted by Uncwilly RDS posted it via an initialism. Jeppe did it in plain language
The meaning is the same. Except for those rare readers of such a tender age as not to have encountered it fully spelled out before.

Quote:
 that was expurgated by a moderator.
Missed that detail on first read.
Quote:
 Answering vitriol with vitriol just covers everyone with vitriolic acid.
I'm all in for less vitriol. None would be a good level. I stand by the claim that to originate it is more serious than to reflect it, just as throwing the first punch defines who is at fault.

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