20130706, 18:43  #1 
Jun 2003
2×3^{3}×29 Posts 
Prime generating series
Question: Are there finite terms in these sequence?
Sierpinski:1, 1, 2, 1, 5, 1, 1, 29, 3, 37, 31, 227, 835, 115 Riesel: 2, 1, 2, 1, 1, 7, 15, 35, 619, 2191, 1267, 187 Checked up to 5000 bits. How to generate the sequence: (for Sierpinski side) Start with k=1 Then find lowest n such that k*2^n+1 is prime n is the first term Then k=k*2^n+1 Then find lowest n such that k*2^n+1 is prime n is the second term and so on.... (Similar for the riesel side). Any thoughts on what the weights of the k generated will be and if they will ever end in a sierpinski or riesel number. What is the most efficient way to generate this series. PFGW's script seems to fail after the first few sequences. 
20130708, 04:53  #2 
Romulan Interpreter
Jun 2011
Thailand
5·11·157 Posts 
7615 is the next. Small pari script, quite fast. Have no idea about the theoretical question, tho.

20130708, 10:02  #3 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT)
2^{2}·1,423 Posts 
I can add 3755 to the Riesel list.
I can also add 6071 after LaurV's addition on the Sierpinski list. I have tested upto n=12500 and proved all primes. I have been using a pfgw script. It runs nice and fast assuming you use f to turn on trial factoring. You could do with sieving to go much further than I have done though. My script will do sequences of the form k*b^n+c. b and c can be changed as can the minimum n to test and the starting k. Last fiddled with by henryzz on 20130708 at 10:04 
20130708, 10:16  #4 
Romulan Interpreter
Jun 2011
Thailand
5·11·157 Posts 
Confirm for 6071. I got it too, but being busy here and forgot to post.

20130708, 10:23  #5 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT)
13074_{8} Posts 
Other bases don't really work as you end up reaching a k with a trivial factor quickly.

20130708, 10:37  #6 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT)
2^{2}·1,423 Posts 
I don't know how to prove that it never stops at a riesel k for starting k=1 but it is trivial to prove that some prime starting ks terminate.
For k=127301 n=2 is prime and this gives k=509203 for the next iteration. 509203 is a riesel k. 
20130708, 15:48  #7 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT)
2^{2}×1,423 Posts 
Discovered I left riesel k=7 running by accident.
The current status of all sequences is: k=1, 1: 2, 1, 2, 1, 1, 7, 15, 35, 619, 2191, 1267, 187, 3755 >12500 k=1, +1: 1, 1, 2, 1, 5, 1, 1, 29, 3, 37, 31, 227, 835, 115, 7615, 6071 >12500 k=5, +1: 1, 1, 1, 583, 79, 371, 1439, 27, 1067 >6000 k=7, 1: 1, 3, 3, 11, 35, 3, 19, 23, 11, 59, 315, 2707, 223, 2807, 1627, 2739, 5171, 2243 >26500 
20130708, 16:55  #8 
Jun 2003
2×3^{3}×29 Posts 
Could you share your PFGW script with me. I was getting errors with PFGW when I programmed it. I would like to extend the series higher.
What bases give trivial factors for k=1? 
20130708, 18:29  #9  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT)
2^{2}·1,423 Posts 
Quote:
Code:
SCRIPT DIM base, 2 DIM k, 5 DIM min_n, 1 DIM n, min_n DIM type, 1 DIMS type_str,+1 DIMS test_str OPENFILEAPP prp_file,primeseriesprps.txt LABEL next_k SET n, min_n LABEL next_n SETS test_str,%d*%d^%d%s;k;base;n;type_str PRP k * base ^ n + type, test_str IF (ISPRP) THEN GOTO prp_found SET n,n+1 GOTO next_n LABEL prp_found PRINT test_str SET k, k * base ^ n + type WRITE prp_file,test_str GOTO next_k 

20130708, 19:36  #10 
Jun 2003
2×3^{3}×29 Posts 
Thanks!
I tried your script for k=1 and base=30, tested upto 40,000 bits but it has not turned into trivial candidates yet. For base=2 it crashed on me. I am trying it again. Can we develop some trivial solutions like: 1) If the base is odd... it will turn into trivial solution. 
20130708, 22:40  #11 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT)
5692_{10} Posts 
Well done in finding another that works. I have had very limitted success in finding more. All were multiples of 30 although I am not sure that is a requirement.

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