Proth primes
While working on GFNs (N=60006150, k=50,000,0002,500,00,000), Markus Tervoonen gathered a huge list of Proth primes (about 40 millions).
If you are interested please leave a message... Luigi 
Starting Proth prime test of 123173*2^333333+1
Using allcomplex FMA3 FFT length 30K, Pass1=640, Pass2=48, clm=2, a = 3 [B]123173*2^333333+1 is prime![/B] (100349 decimal digits) Time : 46.491 sec. Starting Proth prime test of 182931*2^333333+1 Using allcomplex FMA3 FFT length 30K, Pass1=640, Pass2=48, clm=2, a = 5 [B]182931*2^333333+1 is prime![/B] (100349 decimal digits) Time : 46.459 sec. Starting Proth prime test of 1460231*2^333333+1 Using zeropadded FMA3 FFT length 32K, Pass1=512, Pass2=64, clm=2, a = 3 [B]1460231*2^333333+1 is prime![/B] (100350 decimal digits) Time : 49.041 sec. Starting Proth prime test of 1569345*2^333333+1 Using zeropadded FMA3 FFT length 32K, Pass1=512, Pass2=64, clm=2, a = 11 [B]1569345*2^333333+1 is prime![/B] (100350 decimal digits) Time : 49.245 sec. Starting Proth prime test of 1714923*2^333333+1 Using zeropadded FMA3 FFT length 32K, Pass1=512, Pass2=64, clm=2, a = 5 [B]1714923*2^333333+1 is prime![/B] (100350 decimal digits) Time : 48.322 sec. Starting Proth prime test of 1751013*2^333333+1 Using zeropadded FMA3 FFT length 32K, Pass1=512, Pass2=64, clm=2, a = 5 [B]1751013*2^333333+1 is prime![/B] (100350 decimal digits) Time : 49.114 sec. Starting Proth prime test of 1852761*2^333333+1 Using zeropadded FMA3 FFT length 32K, Pass1=512, Pass2=64, clm=2, a = 5 [B]1852761*2^333333+1 is prime![/B] (100350 decimal digits) Time : 49.546 sec. 
I'm interested. Besides, today I've found one: 305147*2^1030527+1 (310226 digits long) from [url]http://irvinemclean.com/maths/sierpin3.htm[/url] . How can I check it for being a Fermat, GF, xGF divisor?

[QUOTE=Chara34122;522691]I'm interested. Besides, today I've found one: 305147*2^1030527+1 (310226 digits long) from [url]http://irvinemclean.com/maths/sierpin3.htm[/url] . How can I check it for being a Fermat, GF, xGF divisor?[/QUOTE]
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 [CODE]pfgw64 gxo q”305147*2^1030527+1”[/CODE] 
If anyone is interested : one more 285473*2^530921+1 is prime! (159829 decimal digits).

A new prime has been discovered (pending verification) and has 5,269,954 digits.
It is 7 ·6[SUP] 6772401[/SUP] + 1 . Although I am doubtful it was checked as a Proth Prime, it might be represented as such [STRIKE][7*3**6772401* 2**6772401 +1 .[/STRIKE] [URL="https://primes.utm.edu/primes/page.php?id=129914"]https://primes.utm.edu/primes/page.php?id=129914[/URL] 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>2[SUP]n[/SUP] N=[I]k[/I]2[SUP]n[/SUP]+1 so I ammend my previous statement. 
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. 
He mentioned a few years ago that he had access to a cluster: [url]https://mersenneforum.org/showthread.php?t=17690[/url]
I imagine he still does. 
[QUOTE=rudy235;525594]...A Proth number is restricted to k k>2[SUP]n[/SUP] N=[I]k[/I]2[SUP]n[/SUP]+1 so I ammend my previous statement.[/QUOTE]
Would the simple form of this not be k*2^n+1? 
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