I have long had an interest in prime numbers and utilizing my CPUs for searching for primes. Over the years, I have found 9 primes which have been on Chris Caldwell’s Prime Pages. My results are here.

I am embarking on a new project to locate both Riesel and Proth Primes. Riesel primes are numbers in the form of k*2^n-1 and Proths are those of the form k*2^n+1. The k values that I will be searching are those of the Hypotenuses of Primitive Pythagorean Triples (OEIS:A195502).

The following work has been done so far:

Riesel Primes

k=5: Searched for n’s up to 3,066,000 & 3,499,000 – 3,646,000 with 43 primes found. To help complete the gap, see the post at the Mersenne Forum.

k=397: Searched for n’s up to 1,200,000 with 18 primes found.

k=533: Searched for n’s up to 1,000,000 with 31 primes found.

k=9161: Searched for n’s up to 10,000 with 15 primes found.

The remaining 14 k’s have not been searched.

Proth Primes

k=5: Searched for n’s through 5,660,000 with 25 primes found.

k=397: Searched for n’s through 920,000 with 37 primes found.

k=533: Searched for n’s through 920,000 with 18 primes found.

k=9161: Searched for n’s through 200,000 with 12 primes found.

There is no evidence of any of the other k’s being searched for Proth primes.

Initial thoughts and strategy

• Start with the Riesel primes to search all n’s through 1,000,000.
• Primes large enough to make the Prime Pages occur once n>666000.
• Use sieving tools from Geoffrey Reynolds (srsieve, sr1sieve, sr2sieve).
• Worry about the Proth’s later.
• Ultimate goal is to search all n’s < 3,000,000 for both Riesel and Proth primes.

Tags: Proth Primes, Riesel Primes