The reason k must be odd for base 2 is that if k is even, you can always divide it by 2 until you get an odd k, and increase n accordingly. Ex. 10*2^2+1=5*2^3+1. This simply eliminates testing the same number multiple times, and provides for a common format for these numbers. As far as n=0, I really think this should not be included for the following two reasons. First, it is not included in the original Sierpinski numbers, which we are trying to represent in base 5. Second, including n=0 eliminates all information on the base. For example, 4*2^0+1=4*5^0+1=4*45569^0+1. This defeats the purpose by reducing the expression k*b^n+1 to the much more general form k+1, or basically k.

Update:
Primality testing 123910*5^136268+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 11 123910*5^136268+1 is prime! (1597.4078s+0.0268s) Yes I know that Robert believes that we don't need to test this number, but as I had sieved fairly deeply and had already done so much PRP testing, I decided to continue knowing that if I found a prime it would be near 100000 digits. It is, at 95253 digits. It will be stored as 24782*5^136269+1 in Chris Caldwell's database. 109208 done to 188000 (still reserved) 71492 done to 55000 (still reserved) 
Wow
Rogue  many congrats are in order for finding such a large prime  all power to you.
Actually though, and I am being pedantic, it is the second prime for this value of k, the first being n=0, as is shown on Geoff's list 24782 when n=1 removed that number from checking. But how nice to find a juicy big prime !! Regards Robert Smith 
Delighted
Hello all,
After a delightfull stay in Disney Resort Paris, I came home to a computer stating me a delightful find: PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8] Primality testing 37718*5^104499+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 11 Calling BrillhartLehmerSelfridge with factored part 99.99% 37718*5^104499+1 is prime! (1165.4437s+0.0267s) Number was ranked 1737th largest prime ever yesterday evening Cheers, Micha 
Congratulations rogue and michaf for the big primes.
If n=0 is allowed in the definition of base5 Sierpinski number, then k=7528 has a prime 7528*5^0+1, but we will still need to eliminate k=5*7528=37640, so a prime 7528*5^n+1 for n >=1 must be found either way. This is the only exceptional case in Roberts list above, I have added an asterisk beside the other candidates in status.txt that don't need to be tested if n=0 is allowed. My own results: 83032*5^39408+1 is prime. 33448*5^n+1 is not prime for n <= 100,000 and I am releasing it. 
One more down.
67282*5^45336+1 is prime. Lars 
And the next result:
68294*5^33723+1 is prime! I keep the rest of my ranges reserved. Lars 
six down for me now
Hya's
found myself my sixth prime: PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8] Primality testing 46922*5^37483+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 2 46922*5^37483+1 is prime! (155.2515s+0.0060s) Cheers, Micha 
k=34094
34094*5^27305+1 is prime!!! [19090 digits]
Resevering k=26798 and 27676 
Hey, 51460*5^50468+1 is prime. (35281 digits)
Primality testing 51460*5^50468+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 2 Calling BrillhartLehmerSelfridge with factored part 99.99% 51460*5^50468+1 is prime! (917.7062s+0.0104s) 
Hi,
my next update. 68416*5^44578+1 is prime. All my ranges are tested to at least n=59000. I will keep them reserved. Lars 
All times are UTC. The time now is 01:08. 
Powered by vBulletin® Version 3.8.11
Copyright ©2000  2021, Jelsoft Enterprises Ltd.