Proth primes

[ET_]

While working on GFNs (N=6000-6150, k=50,000,000-2,500,00,000), Markus Tervoonen gathered a huge list of Proth primes (about 40 millions).

If you are interested please leave a message...

Luigi

[ATH]

Starting Proth prime test of 123173*2^333333+1
Using all-complex FMA3 FFT length 30K, Pass1=640, Pass2=48, clm=2, a = 3
123173*2^333333+1 is prime! (100349 decimal digits) Time : 46.491 sec.

Starting Proth prime test of 182931*2^333333+1
Using all-complex FMA3 FFT length 30K, Pass1=640, Pass2=48, clm=2, a = 5
182931*2^333333+1 is prime! (100349 decimal digits) Time : 46.459 sec.

Starting Proth prime test of 1460231*2^333333+1
Using zero-padded FMA3 FFT length 32K, Pass1=512, Pass2=64, clm=2, a = 3
1460231*2^333333+1 is prime! (100350 decimal digits) Time : 49.041 sec.

Starting Proth prime test of 1569345*2^333333+1
Using zero-padded FMA3 FFT length 32K, Pass1=512, Pass2=64, clm=2, a = 11
1569345*2^333333+1 is prime! (100350 decimal digits) Time : 49.245 sec.

Starting Proth prime test of 1714923*2^333333+1
Using zero-padded FMA3 FFT length 32K, Pass1=512, Pass2=64, clm=2, a = 5
1714923*2^333333+1 is prime! (100350 decimal digits) Time : 48.322 sec.

Starting Proth prime test of 1751013*2^333333+1
Using zero-padded FMA3 FFT length 32K, Pass1=512, Pass2=64, clm=2, a = 5
1751013*2^333333+1 is prime! (100350 decimal digits) Time : 49.114 sec.

Starting Proth prime test of 1852761*2^333333+1
Using zero-padded FMA3 FFT length 32K, Pass1=512, Pass2=64, clm=2, a = 5
1852761*2^333333+1 is prime! (100350 decimal digits) Time : 49.546 sec.

[Chara34122]

I'm interested. Besides, today I've found one: 305147*2^1030527+1 (310226 digits long) from http://irvinemclean.com/maths/sierpin3.htm . How can I check it for being a Fermat, GF, xGF divisor?

[Dylan14]

Originally Posted by Chara34122

I'm interested. Besides, today I've found one: 305147*2^1030527+1 (310226 digits long) from http://irvinemclean.com/maths/sierpin3.htm . How can I check it for being a Fermat, GF, xGF divisor?

You could use pfgw to test the number for Fermat divisibility. You’ll want to use the -gxo flag (which does the tests without testing to see if the number is prime). An appropriate command line for Windows would be

pfgw64 -gxo -q”305147*2^1030527+1”

[Chara34122]

If anyone is interested : one more 285473*2^530921+1 is prime! (159829 decimal digits).

[rudy235]

A new prime has been discovered (pending verification) and has 5,269,954 digits.

It is 7 ·6^6772401 + 1 .

Although I am doubtful it was checked as a Proth Prime, it might be represented as such

[7*3**6772401* 2**6772401 +1 . https://primes.utm.edu/primes/page.php?id=129914

It is by far the largest Prime of 2019 and if verified it will rank as #18 in the list of Largest Primes kept by CC

Congratulations to Ryan Propper.

Edit: A Proth number is restricted to k k>2ⁿ N=k2ⁿ+1 so I ammend my previous statement.

[pepi37]

Congratulations to Ryan Propper.
He really earn this prime: but I cannot even imagine what resources he has.
If we make initial sieve , then assume sieve depth, from last prime we have at least 100000 candidates from 2.8M digits and above :)
And he process 100000 those candidates in 34 days. Some supercomputer must be behind scene.

[ixfd64]

He mentioned a few years ago that he had access to a cluster: https://mersenneforum.org/showthread.php?t=17690

I imagine he still does.

[storm5510]

Originally Posted by rudy235

...A Proth number is restricted to k k>2ⁿ N=k2ⁿ+1 so I ammend my previous statement.

Would the simple form of this not be k*2^n+1?

[paulunderwood]

Congrats to Ryan for the 17th largest known prime 7*2^18233956 + 1 with 5,488,969 decimal digits,