Gary Chaffey wrote:
> Congratulations to Jens K. Andersen who has just broken the existing
> records for generalised bitwins of lengths 3-6 and has found the
> first generalised bitwin of length 7.
> The bitwin tables are maintained by Henri Lifchitz and can be found
> at http://ourworld.compuserve.com/homepages/hlifchitz/
A generalized BiTwin with k links (length k+1) is k+1 twin primes on the form
n*b^i+/-1 for k+1 consecutive positive values of i, and b>2.
My new records follow. Each took less than a GHz week.
for i = 1,2,3 (1006,1203,1401 digits)
for i = 1,2,3,4 (261,360,459,558 digits)
Four links: (1250561*151#+423642234015)*4^i+/-1
for i = 1,2,3,4,5 (67,67,68,68,69 digits)
Five links: (526583*83#+27663009*19#)/4*4^i+/-1
for i = 1,2,3,4,5,6 (39,39,40,40,41,42 digits)
Six links: 1394855870347655081*13#*4^i+/-1
for i = 2,3,4,5,6,7,8 (24,25,26,26,27,27,28 digits)
The GMP library and PrimeForm/GW were used for prp testing.
PrimeForm/GW proved the primes above 100 digits.
Marcel Martin's Primo proved the remaining primes.
My usual tuplet sieve was used for 4, 5 and 6 links.
Here b was fixed at 4 for convenience to my existing code.
2 and 3 links were found with an algorithm designed for generalized BiTwins:
Choose large numbers A and C with lots of divisors.
Compute a pool of twins on the form a*A*C+/-1, where a divides A.
Compute a second pool of twins on the form c*A^2*C+/-1, where c divides C.
Every twin combination a*A*C+/-1 and c*A^2*C+/-1 is now a generalized BiTwin:
n*b^1+/-1 and n*b^2+/-1, where n = a^2*C/c and b = c*A/a.
For each twin combination, check whether it extends.
If n*b^3+/-1 happens to be a twin then there are 2 links.
3 links requires n*b^4+/-1 to be a twin as well.
For 2 links, I chose A = 2^10*3^6*5^3*7^2*13#*19#*43#*400#, C = 31#^6*1723#.
For 3 links: A = 2^6*3^4*5^3*7^2*13#*23#*197#, C = A with 317# instead of 197#.
A and C must be relatively large to have enough divisors a and c for the two
I estimate the algorithm could be used on 4 links with some work, but I'm not
planning to do it.
5 links appears unrealistic at a reasonable size.
Henri Lifchitz' record page was getting long (60 screenfuls on my system) and
still growing. I suggested shortening it. It seems he now only lists BiTwins
with k links when they are above the record for k+1 links. This currently means
only I am listed for generalized BiTwins above 1 link. If anybody is curious
about the old page, it is here for a while:
Jens Kruse Andersen