... So far, it took about 6 minutes to find a 354-digit counterexample. Going titanic, with more than 1000 decimal digits, seems possible, but rather hard,...
24324
djbroadhurst
Jul 19, 2012 8:32 pm
... Indeed there seems to be little obstacle to manufacturing "Fermat plus Lucas" pseudoprimes that are much larger, given Bernhard's unwise choice of a...
24323
djbroadhurst
Jul 19, 2012 5:52 am
... It takes Pari-GP less than a second to find a 16-digit counterexample and about 7 minutes to find this 25-digit counterexample: g =...
24322
djbroadhurst
Jul 19, 2012 3:07 am
... http://109.90.3.58/devalco/suf_helmes.htm#4 ... I satisfied your conditions, with composite P, as follows: f = 0, A*g^2 = 1 mod P. You assumed...
24321
djbroadhurst
Jul 18, 2012 4:50 pm
... Thanks, Bernhard, for acknowledging my disproof of your over-ambitious claim. I have found a counterexample with composite p > 10^22. You might like to...
24320
bhelmes_1
Jul 18, 2012 4:05 pm
Dear David, the counterexample is right and congratulation to your efforts to find it so fast. I do not know yet where the mistake in the proof comes from. ...
24319
djbroadhurst
Jul 17, 2012 2:33 pm
Bernhard Helmes wrongly claimed that the following conditions are sufficient to prove the primality of a positive non-square integer p = 3 mod 4: 1) there...
24318
djbroadhurst
Jul 16, 2012 7:20 pm
... I did verify that point and all your other points. See the test: g = 21283; a = 2969; p = 29539; {if(p%4 == 3 && !issquare(a) && kronecker(a,p) == 1 && ...
24317
bhelmes_1
Jul 16, 2012 6:54 pm
Dear David, thank you for your friendly help. ... 1. You are right with your counterexample that the certificate is not right. I hope you will enjoy this...
24316
djbroadhurst
Jul 16, 2012 5:28 pm
... Yes: very easily. Here is a counterexample with prime g, prime a, and composite p, for which every part of your 3-selfridge test is applied, including your...
24315
djbroadhurst
Jul 16, 2012 3:20 pm
... Here are some [g, a, p] "certificates" that falsely declare composite p to be prime, using your 1-selfridge test: [1018080, 30271, 1048351] [74061, 13880,...
24314
Maximilian Hasler
maximilian_h...
Jul 16, 2012 9:32 am
It is not clear what you mean by "converges to" in your statement. Definition : A sequence s converges to a point x if for any neighborhood V of x almost all...
24313
bhelmes_1
Jul 16, 2012 8:12 am
Dear David, thanks a lot that you reveal a hole in the proof and in the algorithm. You will surely remark that both counterexample are of the form a=16 and...
24312
kad
yourskadhir
Jul 16, 2012 5:43 am
Let N = pq be any semiprime such that a^2 < N < b^2 & p < q. Let 3l be the multiple of 3 nearer to a^2 and 3m be the multiple of 3 nearer to N and 3n be the...
... (where A^2 = a mod p) You make no reference to the Lucas test in your final "certificate". So all that you seem to require for a final "proof" is that 1) p...
24309
bhelmes_1
Jul 15, 2012 4:47 pm
Dear David ... The jacobi (a,p)=1 is a pretest. Then i make sure that a is a quadratic residuum by calculating a^[(p-1)/2]=1 with p=3 mod 4 That assures that a...
24308
djbroadhurst
Jul 14, 2012 11:37 pm
In primenumbers@yahoogroups.com, ... No. ... Your attempted "proof" wrongly assumes that a positive kronecker implies ... Let P = 15. ... Let A = 2. Then your...
24307
bhelmes_1
Jul 13, 2012 8:45 am
A beautifull day, i have documented the test with the mathematical proof under http://109.90.3.58/devalco/suf_helmes.htm Is the mathematical proof right and...
24306
Mark
marku606
Jul 8, 2012 1:11 am
The message below got stuck in the ether for over a day, and surely the Higgs boson had something to do with it. Anyways, Bernhard has since clarified things...
24305
bhelmes_1
Jul 7, 2012 9:06 pm
Hello Mark ... i distinguish between primes p=x^2+1 and the primes which appear as divisor for the first time p | x^2+1 and p < x^2+1 for x<=2^5 = 32 i get ...
24304
Mark
marku606
Jul 7, 2012 1:02 pm
... Thank you Phil, that was very helpful. Now if it could only level a house made from hypothetical perfect euler bricks! (That happens to be the problem...
24303
Mark
marku606
Jul 7, 2012 7:30 am
Hello Bernhard, Looking at table 4a, it seems to present column c as the number of primes up to 2^n, and column d as the number of primes of the form x^2+1 up...
24302
djbroadhurst
Jul 6, 2012 10:34 pm
... Phil wisely chose to use 24 = 2^3 * 3 for a modular exhaustion. When I first encountered number theory, it took me a while to understand why working mod 8...
24301
Phil Carmody
thefatphil
Jul 6, 2012 9:35 pm
... Most likely attack is mod 24. Squares are {0,1,4,9,16,12}. Prime squares are {1} and the trivial (choice-free) {4,9} So we have some 2s, some 3s, and then...
24300
Mark
marku606
Jul 6, 2012 5:42 pm
True. And those identities from William were pretty neat. I had said, "If the sum of the squares of 5 primes equals a square, two of the primes are 2 and 3" ...
24299
Jack Brennen
jbrennen
Jul 6, 2012 4:12 pm
Yes, the sum of squares of 3 composites can equal a square. But Mark stated that the sum of squares of 3 primes cannot; it's pretty easy to prove. (Choose the...
24298
elevensmooth
Jul 6, 2012 3:53 pm
... 8^2 + 49^2 + 64^2 = 81^2 For any integers p, q, s, and t let d=(p^2+q^2+s^2+t^2)/2 c=(p^2+q^2-s^2-t^2)/2 a=ps+qt b=pt-qs then a^2 + b^2 + c^2 = d^2 ...
24297
Mark
marku606
Jul 6, 2012 3:26 pm
Some tidbits that some may find interesting and perhaps fun to prove: The sum of the squares of 3 primes never equals a square. If the sum of the squares of 5...
24296
bhelmes_1
Jul 4, 2012 9:32 am
A beautiful day, There are some results for primes concerning the polynom f(n)=n^2+1 109.90.3.58/devalco/quadr_Sieb_x^2+1.htm Besides some nice algorithms how...
