Quote:
Originally Posted by Flatlander
Sorry I missed your post. Yes, those three reservations are correct.
Status:
Base 72 primes for:
116*72^13887-1
291*72^26322-1
79*72^28009-1
k=4 tested to >41000. (I assume it doesn't have falgebraic actors?  )
Base 93 tested to >46000, no primes.
Base 100 prime for:
74*100^44709-1 
k=653 tested to >55,000
I'll take base 100 to 60k, bases 93 and 72 to 50k. (edit: at least) |
4*72^n-1 is very low weight but eventually should yield a prime. Any Riesel k that is a perfect square will always be lower weight than average because the even-n have algebraic factors.
For this one, many of the odd-n's are also knocked out by factors of 7 and 17 as follows:
n==(1 mod 3) have a factor of 7
n==(3 mod 4) have a factor of 17
This leaves only: n==(5, 9, 17, 21) mod 24 that don't have algebraic factors nor a factor of 7 or 17.
I'm seeing plenty of random factors in k's with those mods so there should be a prime at some point.
Here is an interesting side note: To show how low weight this form is, there are very few n's that are yielding just 2 prime factors. I only found 3 up to n=100:
4*72^1-1 = 7*41
4*72^15-1 = 17*1,704,505,929,743,291,922,743,496,463
4*72^57-1 = 35,406,097*(99-digit prime)
n=57 is the only n<=100 that contains only 2 prime factors that don't include a 7 or 17.
I earlier did a test on all bases <= 1024 for primes at k=2. The lowest base I found so far without a prime for k=2 was b=170 on the Riesel side and b=101 on the Sierp side (tested to n=25K). I wonder if Riesel base 72 is the lowest base that does not have a prime for k=4? I may check that eventually. Of course we don't count the bases that have a proven full covering set nor a trivial factor for k=4.
Oh, BTW, if there are any even n's left after you sieve, you can manually remove them. If the smallest factor for them is > than your sieve limit, sr(x)sieve will not remove them.
Gary