20230124, 01:24  #1  
Dec 2022
1B8_{16} Posts 
Math subforum (I am not a crank)
Just recently I made a post titled 'LL pseudoprimes' about a new class of pseudoprimes I investigated, which I had been thinking of for some time, then wrote a program to compute. My posts were moved to the crackpot forum, although, as few new posts are allowed to remain in Math, I am not sure if that is an implication of crankery. I admit my initial post was poorly written due to my state of mind and uncharacteristic haste, and I would rather replace it with a better version I have now composed, incorporating my program's results.
I do not see where the belief that I am a mathematical crank could come from; though my post (even in its revised version) may not being understood by many here, it has none of the crackpot characteristics, nor does my posting history. As it wasn't meant to contain a proof of anything, no real rigor was attempted. Here is the revised version, that I would like to see in the Math forum: Quote:


20230124, 03:14  #2 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2^{2}·5·19·29 Posts 
The forum that it got moved to: "Other Mathematical Topics" is not the crackpot forum.
Miscellaneous Math serves that function. Maybe you should reread your posts and then post them. Or compose them offline, read them, then post them. 
20230124, 03:26  #3 
Dec 2022
110111000_{2} Posts 
I'm pretty sure I saw it in Miscellaneous Math at the time I wrote that, but OK, what _is_ allowed to stay in the Math forum? I'd rather get it right the first time.
And I do usually compose offline, as I did for this improved version. 
20230124, 06:53  #4 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2·3·1,697 Posts 
mersenneforum.org > Great Internet Mersenne Prime Search > Math
That "Math" is a subforum of "Great Internet Mersenne Prime Searchrelated Math"  discussion of how GIMPS functions. _______ Generally speaking, one can take any specialized primality test and deliberately use it outside of its use domain, for example BerrizbeitiaIskra test, and then express a genuine surprise that "it doesn't work! It produces pseudoprimes!" Well, of course it does. Not really surprising. Lucas test is no different than Fermat test  except when used in correct context. Talk to Paul Underwood and other folks about combining several tests together to produce gold out of two or three pieces of led. 
20230124, 08:42  #5 
Sep 2002
Database er0rr
123D_{16} Posts 
Here BPSW is king. I have come up with several test over the years for example:
( The last 3 require a strong discriminant with jacobi symbol = 1.) All these test, like BPSW, are believed to produce pseudoprimes eventually. Last fiddled with by paulunderwood on 20230210 at 19:32 Reason: The second test needed some extra conditions, 
20230125, 01:18  #6 
Dec 2022
440_{10} Posts 
That makes sense re: the Math forum.
Yes, I thought there would be pseudoprimes but I needed to search to be sure, and I correctly realised a computer program would be required, and as a benefit I discovered how to efficiently compute Lucas sequences  which I know was known, but not to me. It's a good bet that any test will have pseudoprimes unless proved not to, and infinitely many of them. This includes, yes, all those Paul Underwood mentions. 
20230125, 15:14  #7 
Feb 2017
Nowhere
2^{4}·3·7·19 Posts 
1) The usage "LL pseudoprime" is gratuitously confusing. "LL" meaning "LucasLehmer" is universally understood to mean the conclusive primality test for Mersenne numbers. The terms "Lucas test" and "Lucas pseudoprime" are standard, and apply in particular to what you are describing, as well as to any number of other similar tests.
2) Let L_{n} = trace(norm(Mod(2 + x, x^2  3)^n)); L_{0} = 2, L_{1} = 4, L_{n+2} = 4*L_{n+1}  L_{n}. It is true (and not difficult to prove) that if p is prime, and p == 7 (mod 24), then L_{(p+1)/4} == 0 (mod p). 3) Curiously, the proof that L_{(p+1)/4} == 0 (mod p) [at least, the one that imitates the usual proof that this holds for Mersenne primes] depends on the fact that, for p prime, p == 7 (mod 8) we have 2^{(p1)/2} == 1 (mod p). Using a clumsy, inefficient, totally mindless PariGP script, I checked the composite numbers n == 7 (mod 24), n < 50000000 for which L_{(n+1)/4} == 0 (mod n). And the congruence 2^{(n1)/2} (mod n) does not hold for any of them. Code:
? u=Mod(x+2,x^23);forstep(n=7,50000000,24,r=lift(trace((Mod(1,n)*u)^((n+1)/4)));if(r==0&&!isprime(n),print(n" "factor(n)" "lift(Mod(2,n)^((n1)/2))))) 1037623 [337, 1; 3079, 1] 246074 2211631 [271, 1; 8161, 1] 1898884 4196191 [31, 1; 223, 1; 607, 1] 1556449 7076623 [271, 1; 26113, 1] 3020878 9100783 [1231, 1; 7393, 1] 8262536 11418991 [2287, 1; 4993, 1] 8616593 15219559 [919, 1; 16561, 1] 13719061 21148399 [1327, 1; 15937, 1] 19931655 29486239 [31, 1; 607, 1; 1567, 1] 745287 32060503 [2311, 1; 13873, 1] 29795787 36035383 [5479, 1; 6577, 1] 21243887 ? 5) This is the information age. Before posting, try doing a Forum search. If that is unfruitful, try feeding likely search parameters, e.g. "lucas pseudoprime" into your favorite Internet search engine. But be prepared to jump back. The hit parade will be a long one. It will take some practice to learn how to sort the wheat from the chaff. It is a skill worth learning. Last fiddled with by Dr Sardonicus on 20230125 at 15:19 Reason: Remove extraneous paren 
20230125, 15:54  #8  
Sep 2002
Database er0rr
7·23·29 Posts 
Quote:
The "smart money" on there being inifinitely many pseudoprimes may never see the light of day! Last fiddled with by paulunderwood on 20230125 at 15:57 

20230126, 01:01  #9 
Dec 2022
2^{3}·5·11 Posts 
The list Dr. Sardonicus shows is the initial part of the pseudoprimes I computed. It doesn't surprise me that they all fail to be 2SPRP, and it may be a matter of faith that some tests or combined tests will have pseudoprimes because of the practical limits of how far we can compute.
The algorithm I came up with requires twice as many mulmods as a Fermat test, and this may well be optimal. Note that I computed, as I discussed, the bisection V(14,1), not V(4,1)  it saves time and the odd terms are never used anyway. The term 'Lucas pseudoprime' was avoided because of its vagueness (it can refer to several different tests for each of infinitely many suitable Lucas sequences) so that it would not have adequately described what specifically I was looking at. 
20230126, 01:11  #10 
If I May
"Chris Halsall"
Sep 2002
Barbados
2·5,647 Posts 
If I may please share. Reflect, etc et al...
I spent a delightful day today with three friends. We talked to each other. We heard each other. We supported each other. We got things done. Thanks for lunch, BTW... 8^) When I read your language, Andrew, all I do is sigh... And ask myself how can you possibly have so much time to waste? That is serious ask. Have a nice day. 
20230126, 02:38  #11 
Dec 2022
2^{3}·5·11 Posts 
Even if that were a sincere question it would not be your business, and also out of place in a thread about actual mathematics  which no one interested would call a waste of time.

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