For the last 11 years (on and off) a search has been going on to find the most prime series of the form k*2^n-/+1, k fixed. Now the record has been smashed and a k has been found with at least 202 primes.
The least prime series are provided where k is a Sierpinski number, where there are zero primes. There is no equivalent conjecture/ theory for the most prime series, although it can be mooted that an extension of Cunningham Chain logic suggests that any number might be achievable given a sufficiently large k, but this is likely to be non computable given today's approaches.
Ray Ballinger and Wilfred Keller appears to have provided the initial impetus - see
They had discovered a k with 73 primes. The record was extended by them to 90 primes. Jack Brennen took the record to 113 primes. I became involved with the search working with Phil Carmody, following an observation by Payam Samidoost that it was possible to construct k, using CRM, that eliminate factors that have a small multiplicative base 2 and are not of the form p such that the multiplicative order base 2 is p-1. The subsequent k are referred to as Payam numbers.
The record quickly advanced to 162 primes through Phil Carmody's Payam number generator.
The project fell into abeyance until Payam number generator/ prime checker software produced by Anand Nair allowed a revival. The software was improved on by Robert Gerbicz, and this software is currently used for the initial search. Because of the size of the k, PFGW is used for checking the primality higher than= 10000 (n+1 test).
Over 24,000 k's have been found with 100 primes or more before n=10,000 (these are known as very prime series), with approximately 200 billion Payam numbers checked using the Gerbicz software, primarily by myself (- side) and Thomas Ritschel (+ side)
Thomas extended the record to 181 primes before a recent find on the - side that has eclipsed all records to date. The k is a Payam number of the E(130) classification, namely
As of this morning this candidate has 202 primes at n=496187 with small gaps in the n=229000-250000 range. I found the primes smaller than n=250000 an Thomas those between 250000 and 500000.
The historical search for the most prime series has been concentrated at:
And the ongoing search for primes for the E(130) candidate, with the help of RPS will be on