20200714, 12:54  #23  
Nov 2003
16444_{8} 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 20200714 at 12:56 

20200714, 13:06  #24 
Feb 2017
Nowhere
2^{3}×569 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 p1.
One answer is, "of order ln(p) at most." I am sure that, given a lower bound for p (say 1000 or 10^{40} 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. 
20200714, 13:33  #25  
"Jeppe"
Jan 2016
Denmark
2^{3}×3×7 Posts 
Quote:
Last fiddled with by Uncwilly on 20200714 at 14:24 Reason: expurgated 

20200714, 14:14  #26  
6809 > 6502
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Aug 2003
101×103 Posts
22561_{8} Posts 
Quote:
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 20200714 at 14:22 

20200714, 14:26  #27 
6809 > 6502
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Aug 2003
101×103 Posts
3^{3}×5×71 Posts 

20200716, 04:36  #28  
May 2004
100111100_{2} Posts 
Quote:
Euler's generalization of Fermat's theorem a further generalization (ISSN #1550 3747 Hawaii international conference on mathematics and statistics2004) 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 conference2006) b)ultimate generalisation of Fermat's theorem(planetmath .org2012) c) modified Fermat's theorem in order to accommodate Gaussian integers as bases(mersenneforum .orgrecent) d)A theorem a la Ramanujan (AMSBENELUX1996) 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 20200716 at 04:40 Reason: A minor correction 

20200716, 07:32  #29  
"Mihai Preda"
Apr 2015
1,361 Posts 
Quote:


20200716, 09:47  #30  
Sep 2002
Database er0rr
2^{2}×919 Posts 
Quote:
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 20200716 at 10:19 

20200716, 13:35  #31 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2·7·367 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 20200716 at 13:44 
20200716, 13:48  #32  
6809 > 6502
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Aug 2003
101×103 Posts
2571_{16} Posts 
Quote:
Please PM RDS with your suggestions for him. 

20200716, 14:36  #33  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2×7×367 Posts 
Quote:
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