Primenumbers Yahoo Groups Search results for Aurifeuille & factoring Re: Primes again --- In primenumbers@^$1, Robin Garcia wrote: > > "la formule d'Aurifeuille avait échappé à l'attention d'Euler, > > de Lagrange, de Legendre, de Sophie Germain et de Landry"" > > My mathematical knoweledge djbroadhurst Nov 14, 2011 Re: Primes again David wrote : "Historical reminder of the great escape: "la formule d'Aurifeuille avait échappé à l'attention d'Euler, de Lagrange, de Legendre, de Sophie Germain et de Landry"" My mathematical knoweledge is Robin Garcia Nov 14, 2011 Re: Primes again x - 1 x^2 + x - 1 x^4 - x^3 + 2*x^2 + x + 1 x^4 + x^3 + 2*x^2 - x + 1 Historical reminder of the great escape: "la formule d'Aurifeuille avait échappé à l'attention d'Euler, de Lagrange, de Legendre, de Sophie Germain et de Landry" David djbroadhurst Nov 14, 2011 Re: Algebraic factoring --- In primenumbers@^$1, Paul Leyland wrote: > Aurifeuille found only the special case for 2^14+1. According to but let us not persist in the recurrent "canard" that Aurifeuille was without algebraic merit. His case is open. Particularly djbroadhurst Feb 5, 2011 Re: [PrimeNumbers] Re: Algebraic factoring David (ever happy to be stunned) I'll have to think about that one. I'm not hopeful. I have a related historical question. Aurifeuille found only the special case for 2^14+1 and his eponymous factorizations for 2^{4n+2}+1 were due to Lucas, I believe. In any Paul Leyland Feb 5, 2011 Re: Algebraic factoring significantly larger that the other")... Actually, this is more a job for Pollard-Williams-Lenstra that for Léon-François-Antoine Aurifeuille. I must also admit that I'm still stuck on the third game. ;-) JL j_chrtn Feb 1, 2011 Re: Complex Lucas primes lucasV(2+I,2*I,n)/(2+I)) = GM(n)*GQ(n) > GM(n) = norm((1+I)^n-1) > GQ(n) = norm((1+I)^n+1)/5 > > (34): Set g = 1 in my Gauss-Lucas-Aurifeuille factorization: > I*lucasV(P,1,2*k-1) = (1 + I + g*S)*(1 + I - g*S) > with S = (P/2 + 1)*lucasU(P,1,k) - lucasV(P,1,k)/2 > for P Mike Oakes Apr 26, 2009 Re: Complex Lucas primes lucasV(2+I,2*I,n)/(2+I)) = GM(n)*GQ(n) GM(n) = norm((1+I)^n-1) GQ(n) = norm((1+I)^n+1)/5 (34): Set g = 1 in my Gauss-Lucas-Aurifeuille factorization: I*lucasV(P,1,2*k-1) = (1 + I + g*S)*(1 + I - g*S) with S = (P/2 + 1)*lucasU(P,1,k) - lucasV(P,1,k)/2 for P David Broadhurst Apr 26, 2009 Re: Complex Lucas primes In the present work these are:- V() for cases (32) U() for cases (25), (27), (32) [again] and (38). I expect Dr. David "Aurifeuille" B. will have some sharp points to make on this subject at some stage, but meanwhile, here is a pedestrian algebraic demonstration Mike Oakes Apr 24, 2009 Re: Lucas-Lehmer-Aurifeuille de-Henrification --- In primenumbers@^$1, "Mike Oakes" wrote: > what are, in general terms, the properties of P > and/or n that makes such a proof possible. For a start, it is desirable that at least one of n +/- 1 is very smooth: http://primes.utm.edu/top20/page.php?id=47 David Broadhurst Apr 19, 2009 Re: Lucas-Lehmer-Aurifeuille de-Henrification --- In primenumbers@^$1, "David Broadhurst" wrote: > > Happily, I have proven the primality of this gigantic PRP: > > lucasU(14,1,9435)-lucasU(14,1,9434) 10792 Lelio R Paula 04/2009 Many congrats on another difficult and beautiful proof, David. I'm Mike Oakes 19 Apr, 2009 Lucas-Lehmer-Aurifeuille de-Henrification From time to time, I prove the primality of PRPs at Henri Lifchitz's repository of gigantic unprovens: http://www.primenumbers.net/prptop/prptop.php after which Henri diligently removes them from his site, precisely because they are no longer unproven David Broadhurst 18 Apr, 2009 Re: Aurifeuille --- In primenumbers@^$1, "Maximilian Hasler" wrote: > I think Aurifeuille died before he could publish something; > maybe he also didn't make/wouldn't have made an effort to > do this. I seem to recall David Broadhurst 14 Apr, 2009 Re: Aurifeuille > Hugh Williams cites Lucas as saying that Aurifeuille > was already dead by 1878: > > http://tinyurl.com/c2l9pgAurifeuille as an author. I don't think there is any. I think Aurifeuille died before he could publish something; maybe he also Maximilian Hasler 14 Apr, 2009 Re: Aurifeuille 4, (1878), 86, 98. > > Unfortunately Dickson did not list authors in his references. > It may be that none of these papers has Aurifeuille as an author. > > For example, the second paper (1877-8) is by Lucas alone: > > É. Lucas, > Théorèmes d'arithmétique, > Atti David Broadhurst 14 Apr, 2009 Re: Aurifeuille primenumbers@^$1, "Andy Steward" wrote: > A reference to Aurifeuille's papers But we still don't know whether he wrote any papers in his references. It may be that none of these papers has Aurifeuille as an author. For example, the second paper (1877-8) is by David Broadhurst 14 Apr, 2009 Re: Aurifeuille Hi All, A reference to Aurifeuille's papers from Dickson, History of the Theory of Numbers (1991 AMS reprint), Vol 1, p383: "H. LeLasseur and A. Aurifeuille noted Andy Steward 14 Apr, 2009 Re: Aurifeuille paper alluded to Yes, that is indeed the paper referenced by Granville and Pleasants. I have not read [7]. > "Ainsi la formule d'Aurifeuille avait échappé à l'attention > d'Euler, de Lagrange, de Legendre, de Sopie Germain et de Landry So Mike was keeping good historical David Broadhurst 13 Apr, 2009 Aurifeuille > > In an 1878 paper, Lucas explained how Aurifeuille > > proved that there are identities like those above Aurifeuillian because some of these formulas were discovered by Aurifeuille (see page 276 of [7]). where [7] E. Lucas, Théorèmes Maximilian Hasler 13 Apr, 2009 Re: More sequence puzzles 1, "David Broadhurst" wrote: > Léon-François-Antoine Aurifeuille does not appear at > http://www-groups.dcs.st-and.ac.uk AURIFEUILLIAN FACTORIZATION > In an 1878 paper, Lucas explained how Aurifeuille > proved that there are identities like those above > for David Broadhurst 13 Apr, 2009 Re: More sequence puzzles literature: http://tinyurl.com/dezu7j is hard to find. Does anyone know of an on-line copy of this book? Léon-François-Antoine Aurifeuille does not appear at http://www-groups.dcs.st-and.ac.uk/~history/Indexes/A.html David David Broadhurst 13 Apr, 2009 Re: The house that Jack built having more than 250 digits. From this it may be seen that the cyclotomic structure is very rich, thanks to both Lehmer and Aurifeuille. > Here are the odd values of n < 10000 for which V(n) is prime:- > n = 5, 7, 13, 107, 227, 491, 8009 And of course these David Broadhurst 25 Mar, 2009 Re: [PrimeNumbers] Algebraic factor identities that the difference of squares is the same as the sum of > 4th powers. 2x^2 = a y^2 = b So 2ab is a perfect square, and we have an Aurifeuillean factorization you noticed. If 3ab, 5ab, 6ab, 7ab, 10ab, etc. is a perfect square, we'll be able to use the corresponding line in Andrey Kulsha 17 May, 2006 Algebraic factor identities Re: A surprising algebraic factorization > 4 x^4 + y^4 = ( 2 x^2 - 2 x y + y^2) ( 2 x^2 + 2 x y + y^2) http://xyyxf.at.tut.by/aurifeuillean.pdf Which contains many identities such as A^2 + B^2 = {1, 1}^2 2AB Thanks. I figure out the notation thusly. (A + B)^2 = {1 Kermit Rose 17 May, 2006 Re: [PrimeNumbers] A surprising algebraic factorization > 4 x^4 + y^4 = ( 2 x^2 - 2 x y + y^2) ( 2 x^2 + 2 x y + y^2) http://xyyxf.at.tut.by/aurifeuillean.pdf :) [Non-text portions of this message have been removed] Andrey Kulsha 16 May, 2006 Re: [PrimeNumbers] RE: Estimating log (B^pi(B)) David Broadhurst " wrote: > > Woh! I can't see Dickman in that at all. > > What am I missing? > > Let me backtrack a bit, to Aurifeuille. ... > (more or less) T1 and T2 digits then Dickman suggests > (don't eek me, I only said sugests) that we will > get them twice Phil Carmody 31 Jan, 2003 RE: Estimating log (B^pi(B)) > Woh! I can't see Dickman in that at all. > What am I missing? Let me backtrack a bit, to Aurifeuille. Some time ago, Bouk and I were trying to develop a theory of Generalized Lucas Aurifeuillian parts: the generalization of http David Broadhurst <d.broadhurst@open.ac.u 31 Jan, 2003 RE: [PrimeNumbers] Re: decoding factors.gz Woodall, GCW and > factorial\pm. It's what I use to keep my tables up to date. I happen to prefer the Pxyz/Cxyz > convention and Aurifeuillean factorizations but it's straightforward to keep all factors > explicitly (indeed, it's easier to do so). Once I'd taken google Phil Carmody 24 Dec, 2002 RE: [PrimeNumbers] Re: decoding factors.gz Woodall, GCW and factorial\pm. It's what I use to keep my tables up to date. I happen to prefer the Pxyz/Cxyz convention and Aurifeuillean factorizations but it's straightforward to keep all factors explicitly (indeed, it's easier to do so). To get things started Paul Leyland 24 Dec, 2002 Re: Aurifeuillian Bells All the Aurifeuillian algebraic factorizations for the Cunningham project are provided in situ: pmain1102. If you want to master Aurifeuille for bases b>12 then as, I told you, here is where best to start reading: > Math.Comp. 65 (96) 383 > which Andy and I long since David Broadhurst <d.broadhurst@open.ac.u 23 Dec, 2002 Re: [PrimeNumbers] Re: decoding factors.gz --- "David Broadhurst " wrote: > Phil: You forgot Aurifeuille: > > 339 (1,113) L.M > L (3) 16273.11179590817.1052915894677891021.612796856501249407843 > M 29833.1783853257.506559378229.P29 Phil Carmody 23 Dec, 2002 Re: decoding factors.gz Phil: You forgot Aurifeuille: 339 (1,113) L.M L (3) 16273.11179590817.1052915894677891021.612796856501249407843 M 29833.1783853257.506559378229.P29 So Richard meant: I have given you all the factors you need if you also use your brain :-) David David Broadhurst <d.broadhurst@open.ac.u 23 Dec, 2002 Re: Sierpinski Problem with y=2^s*x b) find a covering set for n != 2 mod 4 c) if desired, make x=0 mod 5, so that p=5 does not cover what you did with Aurifeuille in (a) Question: What is the smallest k=x^4 for which this can be done, with and without the grace note (c)? Back in a few weeks djbroadhurst 31 Aug, 2002 Re:(2^37933+1)^4-2 is PRP... No dice. Cyclotomy only pushes you up from 25% to 26% and I was wrong to suggest that Aurifeuille might give additional help. This is one for Henri, I fear. David djbroadhurst 16 Jun, 2002 RE: [PrimeNumbers] n^n+1, n^n-1 ago, Paul Jobling had an idea for a paper on > factorisations of (n^n-1)/(n-1). I've done some work on n<=512. > Cyclotomy and Aurifeuille make it interesting. Yes, it was a *long* while ago. For the interested, it was looking to expand the work performed by Sam Wagstaff Paul Jobling 12 Oct, 2001 Re: [PrimeNumbers] n^n+1, n^n-1 while ago, Paul Jobling had an idea for a paper on factorisations of (n^n-1)/(n-1). I've done some work on n<=512. Cyclotomy and Aurifeuille make it interesting. Here's an extract from the draft paper: Currently, repunits have been completely factorized for 161 values Andy Steward 12 Oct, 2001 Re: Online factorization applet is now faster! few pluses: 1) The ECM curves are not random, but on restart your cookies remember what you have already done. 2) Cyclotomy and Aurifeuille are fully implemented for both Cunningham and Fibonacci numbers. Since Dario's ecm.java file is downloadable, anyone interested d.broadhurst@open.ac.uk 21 Sep, 2001 Re: base 10 factors database. formula and then a > CPU millisecond to merely *prove* it, > with computer algebra. > > but discovery is an art. Else how would > Aurifeuille have become so celebrated? > > David Looking at the "Algebraic Factors" appendix in Riesel, the aurifeuillian factorization for ajw01@uow.edu.au 18 Sep, 2001 Re: base 10 factors database. 1) 3: lhs:=(10^(10*h)+1)/(10^(2*h)+1); 8*h 6*h 4*h 2*h lhs := 10 - 10 + 10 - 10 + 1 4: rhs:=(A-B)*(A+B); 8*h 6*h 4*h 2*h rhs := 10 - 10 + 10 - 10 + 1 ===== but discovery is an art. Else how would Aurifeuille have become so celebrated? David d.broadhurst@open.ac.uk 18 Sep, 2001 Re: [PrimeNumbers] Re: base 10 factors database. x^2 + 1 > > 2) Now set x=10 and discover that > > Phi(4,10)=101 > Phi(12,10)=9901 > > are already prime! > > 3) Now use Aurifeuille for > > Phi(20,10)=L*M; L=3541; M=27961 > > which also give primes! > > 4)So the only *work* was extracting the 61 in Phil Carmody 18 Sep, 2001 Re: base 10 factors database. L and M does, is it splits the expression into > 2 large factors L and M such that L.M = N. .. > L=A-B > M=A+B However, to use Aurifeuille when N is already a cyclotomic factor (as it should be, for efficiency's sake) it's best to use the gcd's L=gcd(N,A-B) M=gcd(N d.broadhurst@open.ac.uk 18 Sep, 2001 Re: base 10 factors database. x^10 - x^8 - x^6 + x^2 + 1 2) Now set x=10 and discover that Phi(4,10)=101 Phi(12,10)=9901 are already prime! 3) Now use Aurifeuille for Phi(20,10)=L*M; L=3541; M=27961 which also give primes! 4)So the only *work* was extracting the 61 in Phi(60,10)=L*M d.broadhurst@open.ac.uk 17 Sep, 2001 Re: Williams-Lenstra Cyclotomy n) is NOT truly primitive, for any odd n; use 2+sqrt(3)=(sqrt(1/2)+sqrt(3/2))^2 to further factorize it 2) primV(4,1,n) admits Aurifeuille for n=3 mod 6 3) N=primV(4,1,2^a*p) with prime p has N^2-1 factorizable by cyclotomy + (1) + (2) 4) N=primV(4,1,2^a*3^b*p) has d.broadhurst@open.ac.uk 02 Sep, 2001