New test

[allasc]

Originally Posted by CRGreathouse

Here's some quick Pari code that shows that there are no other terms (beside 2^29-1) of your sequence in A054723, at least as far as you have patience to check.

у последовательности A054723 и исключений A175625
есть определенная связь :)

расматривая второе известное исключение
2*i+7=46912496118443 => i+3=23456248059221=(2^23-1)*2796203=(2^46-1)/3
предположим что 46+1=47 имеет место быть в последовательности A054723

почемубы не проверить остальные числа из последовательности A054723 по этому принципу :-)
в итоге получил новые исключения для последовательности A175625, по извесным элементам A054723

первый столбец элемент последовательности A054723
второй столбец исключение 2*i+7 для последовательности A175625 и прицип получения по i+3

A054723		2*i+7 exception(A175625)
-----------------------------------------------------
47		46912496118443 (Known exception)
New find exception for A175625
59		192153584101141163 => i+3=(2^58-1)/3
83		3223802185639011132549803 => i+3=(2^82-1)/3
179		255415923477648143059724504525051530603123187030600363 => i+3=(2^178-1)/3
227		71893191112401706119112040232052348463032385126774859949609627331243 => i+3=(2^226-1)/3	
263		4940462474125491004739028693704017401739519345733997399016856917670960197970603 => i+3=(2^262-1)/3

Excuse. On English I can not issue.

machine translation

At sequence A054723 and exceptions A175625
There is a certain communication :)

Considering the second known exception
2*i+7=46912496118443 => i+3=23456248059221 = (2^23-1 *2796203 = (2^46-1)/3
Let's assume that 46+1=47 takes place to be in sequence A054723

Why not to check up other numbers from sequence A054723 by this principle :-)
As a result has received new exceptions for sequence A175625, on извесным to elements A054723

The first column an element of sequence A054723
The second column an exception 2*i+7 for sequence A175625 and a reception principle on i+3

Batalov буду благодарен за перевод... если конечно в этом есть какойто смысл :) (в смысле если это все не бред)

[CRGreathouse]

I didn't find any other direct examples so far -- none to exponent ~100,000. I'll try to look at factors of Mersenne numbers and multiples of Mersenne numbers tomorrow.

[maxal]

Counterexamples are given, in particular, by numbers of the form for equal:

47, 59, 83, 107, 179, 227, 263, 359, 383, 467, 479, 503, 563, 587, 683, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2543, 2579, 2819, 2879, 2903, 2963, 2999, 3023, 3119, 3167, 3203, 3467, 3623, 3779, 3803, 3863, 3947, 4007, 4079, 4127, 4139, 4259, 4283, 4547, 4679, 4703, 4787, 4799, 4919, 4931, 5087, 5099, 5387, 5399, 5483, 5507, 5639, 5879, 5927, 5939, 6047, 6599, 6659, 6719, 6779, 6827, 6899, 6983, 7079, 7187, 7247, 7523, 7559, 7607, 7643, 7703, 7727, 7823, 8039, 8147, 8423, 8543, 8699, 8747, 8783, 8819, 8963, 9467, 9587, 9719, 9743, 9839, 9887, ...

[CRGreathouse]

Originally Posted by maxal

Counterexamples are given, in particular, by numbers of the form for equal:

47, 59, 83, 107, 179, 227, 263, 359, 383, 467, 479, 503, 563, 587, 683, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2543, 2579, 2819, 2879, 2903, 2963, 2999, 3023, 3119, 3167, 3203, 3467, 3623, 3779, 3803, 3863, 3947, 4007, 4079, 4127, 4139, 4259, 4283, 4547, 4679, 4703, 4787, 4799, 4919, 4931, 5087, 5099, 5387, 5399, 5483, 5507, 5639, 5879, 5927, 5939, 6047, 6599, 6659, 6719, 6779, 6827, 6899, 6983, 7079, 7187, 7247, 7523, 7559, 7607, 7643, 7703, 7727, 7823, 8039, 8147, 8423, 8543, 8699, 8747, 8783, 8819, 8963, 9467, 9587, 9719, 9743, 9839, 9887, ...

Are these the only composites in the sequence, or just an accessible subsequence?

Harder question: Can you give a simple characterization of the sequence, including the primes?

[maxal]

Originally Posted by CRGreathouse

Are these the only composites in the sequence, or just an accessible subsequence?

These are the composites in the sequence that fail the proposed test.

[allasc]

Originally Posted by maxal

Counterexamples are given, in particular, by numbers of the form for equal:

47, 59, 83, 107, 179, 227, 263, 359, 383, 467, 479, 503, 563, 587, 683, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2543, 2579, 2819, 2879, 2903, 2963, 2999, 3023, 3119, 3167, 3203, 3467, 3623, 3779, 3803, 3863, 3947, 4007, 4079, 4127, 4139, 4259, 4283, 4547, 4679, 4703, 4787, 4799, 4919, 4931, 5087, 5099, 5387, 5399, 5483, 5507, 5639, 5879, 5927, 5939, 6047, 6599, 6659, 6719, 6779, 6827, 6899, 6983, 7079, 7187, 7247, 7523, 7559, 7607, 7643, 7703, 7727, 7823, 8039, 8147, 8423, 8543, 8699, 8747, 8783, 8819, 8963, 9467, 9587, 9719, 9743, 9839, 9887, ...

спасибо за анализ!
а Вы целеноправленно рассматривали только простые числа, или проверяли все числа подряд ? :)
а то прям так и напрашивается - новый тест на простоту :) любое k дающее контрпример для последовательности A175625

Machine translation

Thanks for the analysis
And you considered only primary numbers, or checked all numbers successively?:)
And that it is direct and arises - the new test for primary:) any K giving a counterexample for sequence A175625

[allasc]

Originally Posted by CRGreathouse

I edited the sequence.

%I A175625
%S A175625 7,11,23,31,47,59,83,107,167,179,227,263,347,359,383,467,479,503,563,
%T A175625 587,683,719,839,863,887,983,1019,1123,1187,1283,1291,1307,1319,1367,
%U A175625 1439,1487,1523,1619,1823,1907,2027,2039,2063,2099,2207,2447,2459,2543
%N A175625 Numbers n such that gcd(n, 6) = 1, 2^(n-1) = 1 (mod n), and 2^(n-3) = 1 (mod (n-1)/2).
%C A175625 All composites in this sequence are 2-pseudoprimes, A001567. That subsequence begins 536870911, 46912496118443, ...; these were found by 'venco' from the dxdy.ru forums.
%C A175625 Intended as a pseudoprimality test, but note that most primes are also missing as a result of the third test.
%H A175625 Alzhekeyev Ascar M, <a href="b175625.txt">Table of n, a(n) for n = 2..11941610</a>
%o A175625 (PARI) isA175625(n)=gcd(n,6)==1&Mod(2,n)^(n-1)==1&Mod(2,n\2)^(n-3)==1
%K A175625 nonn
%O A175625 1,1
%A A175625 Alzhekeyev Ascar M (allasc(AT)mail.ru), Jul 28 2010, Jul 30 2010

updated, looks correct!
thanks!

[CRGreathouse]

It looks like 2^161039-1 is another composite member of this sequence. (I've been testing Mersenne numbers to see if any others would be in.)

[allasc]

пока для всех найденных составных чисел верно утверждение

(n-1)/2 имеет делитель вида 2^x-1

интересно найти составное число (n) не удовлетворяющее этому правилу :)

----------
While for all found composite member of this sequence truly statement

(n-1)/2 has a kind divider 2^x-1

It is interesting to find composite member of this sequence (n) not satisfying to this rule

[CRGreathouse]

Of course we've all been doing special searches for numbers of that form, so it's not surprising we'd find exceptions of that type.

Unfortunately the individual tests take so long that it seems hard to test all numbers up to a reasonable level.

[J.F.]

Charles asked me for the database of (base 2 Fermat) pseudoprimes I've been working on.
We've been sitting on this one for more than a half year, simply because I got sidetracked/lost focus. I am truly somewhat ashamed of myself. This seems to be a good time to get back to it and finish the project.

Anyway, Will Galway has been so kind to provide a better hosting location than my little website which only goes upto 10^17. You can find the full database here:

http://www.cecm.sfu.ca/Pseudoprimes/index-2-to-64.html

HTH.