Williams' sequence 4*5^n-1 (A046865)

geoff · 2006-07-16, 02:22

The sequence A046865 is mentioned in secion A3 of R. K. Guy's "Unsolved Problems in Number Theory".

The terms of the sequence are the primes of the form 4*5^n-1, a specific case of Williams' sequences of primes of the form (r-1)*r^n-1.

I will start sieving with srsieve. If anyone else is interested in extending this sequence, perhaps we could add it to the Base 5 Sierpinski/Riesel distributed sieve when I catch up (say when sieving reaches p=200e9)?

The sequences in other bases, A003307, A079906, A046866, etc. could also be extended, but would have to be sieved individually.

Citrix · 2006-07-16, 16:45

I am interested in helping out. But what is the open problem related to these numbers? what is the weight of these numbers?

Have you tried to search (b-1)*b^n+1? For fixed n and variable b.
Can b-1 ever be a sierpinki or riesel number for base b?

I am more interested in working on the two questions stated above, if there is no open problems related to these numbers.

geoff · 2006-07-17, 01:24

Warning: this sequence tickles a bug present in srsieve versions 0.3.0 to 0.3.6, upgrade to 0.3.7 or later.

Quote:

Originally Posted by Citrix

I am interested in helping out. But what is the open problem related to these numbers? what is the weight of these numbers?

I have to confess I haven't researched further than the entry in Guy, the 'unsolved problem' seems to be the same as for the Mersenne primes (which are of the form (r-1)*r^n-1 with r=2), what is their distribution, is the sequence infinite etc.

If n > 0 and n=2m then 4*5^n-1 = (2*5^m-1)(2*5^m+1), so only the odd terms have to be sieved. Also all the factors for the odd terms appear to end in either 1 or 9, so the sieve speed can be doubled by filtering out those ending in 3 or 7.

Quote:

Have you tried to search (b-1)*b^n+1? For fixed n and variable b.
Can b-1 ever be a sierpinki or riesel number for base b?

No. I don't know. When I posted I was mainly thinking that if we leave it too long it will be harder to catch up to the Base 5 distributed sieve, but if the factors have special properties then it may be better to sieve the sequence seperately anyway.

I only sieved 4*5^n-1 with 0 < n <= 2e6 (the distributed sieve range) up to p=6e9, I will stop there, but will continue sieving a smaller range, say 0 < n <= 200,000 since that should be enough to find a few more terms to extend the sequence.

If anyone is interested in sieving 5*4^n-1 I have posted the current sieve in NewPGen format and a modified version of srsieve which only sieves primes that are 1 or 9 mod 10 here.

antiroach · 2006-07-17, 13:58

i can sieve these for a while. what program would be best(fastest) to test these numbers for primality?

Citrix · 2006-07-17, 14:30

Quote:

Originally Posted by Citrix

Have you tried to search (b-1)*b^n+1? For fixed n and variable b.
Can b-1 ever be a sierpinki or riesel number for base b?

I will try working on this, to see if I can find the sierpinki or riesel numbers. I leave 4*5^n-1 for you.

Thankyou.

fetofs · 2006-07-17, 15:20

Quote:

Originally Posted by antiroach

i can sieve these for a while. what program would be best(fastest) to test these numbers for primality?

Probably normal 5-base procedure, LLR/PRP for probable primality tests, and OpenPFGW for verifications.

antiroach · 2006-07-17, 15:46

Enter expression followed by carriage return:
4*5^15393+-1
Primality testing 4*5^15393+-1 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 1+sqrt(7)
Running N+1 test using discriminant 7, base 3+sqrt(7)
Calling N+1 BLS with factored part 100.00% and helper 0.11% (300.12% proof)
4*5^15393+-1 is prime! (215.8260s+0.0009s)

Is there someone I can report this to?

antiroach · 2006-07-17, 23:31

I sieved the 0<=n<=2M file upto 12e9. Here's the srsieve formatted file: http://s89744942.onlinehome.us/results.zip

I also started factoring the numbers. Im upto like n = 30k. I plan on going upto like n=50k and then im going to switch over to working on the 6*7^n-1 sequence.

geoff · 2006-07-18, 03:10

Quote:

Originally Posted by antiroach

4*5^15393+-1 is prime! (215.8260s+0.0009s)

Is there someone I can report this to?

Nice one :-) There is a submission page, but I think it has to be checked manually by someone, so it is probably best to wait until you have finished working on the sequence and make just one submission for all the new terms you find.

antiroach · 2006-07-19, 14:08

I prp'd 4*5^n-1 upto n = 50000 without finding any more primes.

geoff · 2006-07-20, 04:14

I will keep a record of what sieving and PRP testing has been done on the 4*5^n-1 sequence in this directory, but it will probably lag a day or two behind this thread.

2006-07-16, 02:22	#1
geoff Mar 2003 New Zealand 10010000101₂ Posts	*Williams' sequence 45^n-1 (A046865)** The sequence A046865 is mentioned in secion A3 of R. K. Guy's "Unsolved Problems in Number Theory". The terms of the sequence are the primes of the form 45^n-1, a specific case of Williams' sequences of primes of the form (r-1)r^n-1. I will start sieving with srsieve. If anyone else is interested in extending this sequence, perhaps we could add it to the Base 5 Sierpinski/Riesel distributed sieve when I catch up (say when sieving reaches p=200e9)? The sequences in other bases, A003307, A079906, A046866, etc. could also be extended, but would have to be sieved individually.

2006-07-16, 16:45	#2
Citrix Jun 2003 3·23² Posts	I am interested in helping out. But what is the open problem related to these numbers? what is the weight of these numbers? Have you tried to search (b-1)b^n+1? For fixed n and variable b. Can b-1 ever be a sierpinki or riesel number for base b? I am more interested in working on the two questions stated above, if there is no open problems related to these numbers. Last fiddled with by Citrix on 2006-07-16 at 16:46*

2006-07-17, 15:46	#7
antiroach Jun 2003 11110100₂ Posts	Sequence has been extended Enter expression followed by carriage return: 45^15393+-1 Primality testing 45^15393+-1 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 2 Running N-1 test using base 3 Running N+1 test using discriminant 7, base 1+sqrt(7) Running N+1 test using discriminant 7, base 3+sqrt(7) Calling N+1 BLS with factored part 100.00% and helper 0.11% (300.12% proof) 45^15393+-1 is prime! (215.8260s+0.0009s) Is there someone I can report this to? Last fiddled with by antiroach on 2006-07-17 at 15:46*

2006-07-17, 23:31	#8
antiroach Jun 2003 2²×61 Posts	Status Update I sieved the 0<=n<=2M file upto 12e9. Here's the srsieve formatted file: http://s89744942.onlinehome.us/results.zip I also started factoring the numbers. Im upto like n = 30k. I plan on going upto like n=50k and then im going to switch over to working on the 6*7^n-1 sequence.

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2006-07-17, 13:58	#4
antiroach Jun 2003 11110100₂ Posts	i can sieve these for a while. what program would be best(fastest) to test these numbers for primality?

2006-07-19, 14:08	#10
antiroach Jun 2003 11110100₂ Posts	I prp'd 4*5^n-1 upto n = 50000 without finding any more primes.

2006-07-20, 04:14	#11
geoff Mar 2003 New Zealand 13·89 Posts	I will keep a record of what sieving and PRP testing has been done on the 4*5^n-1 sequence in this directory, but it will probably lag a day or two behind this thread.