Problems & Puzzles: Puzzles
Puzzle 256. Jack Brennen old records
Jack Brennen sent to me by email in May 1999, the following records obtained by him:
I used to have these records in one page of "Records" that was changed some two years ago. As far as Jack and me know, these records haven't been broken since then. Maybe it's time to do it.
Q1. Can you improve the Jack Brennen's records (get larger runs of consecutive primes of the forms 4k+1 and 4k+3?
In the page 13 of the well known R.K.Guy's Book (UPiNT) we may read:
Q2. Can you explain the Turán's claim?
Jim Fougeron broke both Jack Brennen's records.
Now the Fougeron's records are a run of 33 consecutive primes 4k+1 and 32 consecutive primes 4k+3) .
BTW Fougeron found a larger run (32 consecutive primes, 12 digits each ) than the Brennen's record (24 consecutive primes, 13 digits each) for the primes 4k+3 in a region of smaller numbers than the reported by Jack. See below the Fougeron's range in this color. Does this means that Jack did not make an exhaustive ordered search for his records?
Jim Fougeron wrote (Feb. 2004):