mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Miscellaneous Math

Reply
 
Thread Tools
Old 2022-01-31, 15:32   #34
paulunderwood
 
paulunderwood's Avatar
 
Sep 2002
Database er0rr

109C16 Posts
Red face Generalization

Based on a Lucas PRP test over x^2-b^r*x+b, I have come up with a general test:

Code:
{tst(n,b,r)=local(t=lift(Mod(b,n)^(2*r-1)));
kronecker(b,n)==-1&&
kronecker(t-4,n)==1&&
gcd(t-3,n)==1&&
gcd(t-2,n)==1&&
gcd(t-1,n)==1&&
gcd(r-1,n-1)==1&&
Mod(b,n)^((n-1)/2)==-1&&
Mod(Mod(z,n),z^2-(t-2)*z+1)^((n+1)/2)==-1;}
Testing is slow with 3 nested loops. Can you find a counterexample?

Edit: I just noticed this test does not work for primes 3, 5, 7, 17, 23 and 31. Bigger primes have plenty of scope given by the parameters.

I made a programming error in Pari/GP, taking a gcd of a fraction when coding the above in a different way. So the above script easily gives counterexamples. Oh well!

Last fiddled with by paulunderwood on 2022-02-01 at 02:40
paulunderwood is offline   Reply With Quote
Old 2022-02-05, 06:26   #35
paulunderwood
 
paulunderwood's Avatar
 
Sep 2002
Database er0rr

22·1,063 Posts
Question Generalization II

I have now devised a general test of the polynomial x^2-b^r*x+b with an Euler PRP test for the discriminant b^(2*r)-4*b:

Code:
{tst(n,b,r)=local(t=lift(Mod(b,n)^(2*r-1)),k=kronecker(b,n));
gcd(b,n)==1&&
gcd(t-1,n)==1&&
gcd(t-2,n)==1&&
gcd(t-3,n)==1&&
kronecker(t-4,n)==-k&&
Mod(b,n)^((n-1)/2)==k&&
Mod(t-4,n)^((n-1)/2)==-k&&
Mod(Mod(z,n),z^2-(t-2)*z+1)^((n+1)/2)==k;}
(I think that is coded up properly).

Can you fool this 1+1+2 Selfridge "restricted domain" PRP test?



Edit: I found some frauds for n=287051.

Because 2 appears as the denominator for the solution to x I have added a Fermat base 2 PRP test and am trying to find a pseudoprime to this new test.

Edit 2: A counterexample is tst(79786523,206932265,1)

Last fiddled with by paulunderwood on 2022-03-10 at 00:20
paulunderwood is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Lucas and Fibonacci primes Batalov And now for something completely different 9 2017-06-28 16:56
Lucas Table R.D. Silverman Factoring 19 2012-09-07 17:24
Need help with Lucas Sequences... WraithX Programming 11 2010-09-23 23:22
Lucas-Lehmer Test storm5510 Math 22 2009-09-24 22:32
Lucas-Lehmer Dougal Information & Answers 9 2009-02-06 10:25

All times are UTC. The time now is 18:50.


Sat Aug 13 18:50:46 UTC 2022 up 37 days, 13:38, 2 users, load averages: 0.98, 1.12, 1.09

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2022, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.

≠ ± ∓ ÷ × · − √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °
∠ ∟ ° ≅ ~ ‖ ⟂ ⫛
≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘ ∏ ∐ ∑ ∧ ∨ ∩ ∪ ⨀ ⊕ ⊗ 𝖕 𝖖 𝖗 ⊲ ⊳
∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟
¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣ … ⋯ ⋮ ⋰ ⋱
∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ
𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎𝜍 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔