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Extended generalized Fermat prime?
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BurVolunteer tester
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Joined: 25 Feb 20 Posts: 332 ID: 1241833 Credit: 22,611,276 RAC: 4,081

I read in Riesel's 1994 book "Prime Numbers and Computer Methods for Factorization" that he tried finding primes of that form but to no avail. Even today there is no such catergory at Caldwell's prime list, so I assume none are known?
Are there any conjectures there might not be any at all?
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Primes: 1281979 & 12+8+1979 & 1+2+8+1+9+7+9 & 1^2+2^2+8^2+1^2+9^2+7^2+9^2 & 12*8+19*79 & 12^81979 & 1281979 + 4 (cousin prime)  

Ravi FernandoProject administrator Volunteer tester Project scientist Send message
Joined: 21 Mar 19 Posts: 144 ID: 1108183 Credit: 8,505,009 RAC: 158

Not sure where you're getting the idea that none of them are known. There are plenty of small onessee e.g. the table with "a" and "b" columns a little ways below here. But there are no proven xGFN primes that are large enough for Caldwell's list, simply because it's usually not feasible to factor p+1 or p1. For example, the two PRPs of the form a^2^16 + b^2^16 listed here cannot be proved prime with current methods, even though they're pretty small by T5K standards.  


Also try the PRP Top search a^x+b^x. /JeppeSN  

BurVolunteer tester
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Joined: 25 Feb 20 Posts: 332 ID: 1241833 Credit: 22,611,276 RAC: 4,081

Thanks, that's a bit embarassing. I didn't get the division by gcd(a+b,2) in the wikipedia table. But apparently that just means: if a+b is even, the sum will be divided by 2, same as the "half generalized Fermats"?
Regarding Caldwell's, I thought he included special types of primes irregardless of their size, as long as they are the top 20 largest known ones. For example Wagstaff primes. The smallest have only about 1000 digits. That should be achievable for xGFs.
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Primes: 1281979 & 12+8+1979 & 1+2+8+1+9+7+9 & 1^2+2^2+8^2+1^2+9^2+7^2+9^2 & 12*8+19*79 & 12^81979 & 1281979 + 4 (cousin prime)  


Yeah, if a and b are both odd, divide by 2. PRP Top search: http://www.primenumbers.net/prptop/searchform.php?form=%28a%5Ex%2Bb%5Ex%29%2F2 (Kellen and me). /JeppeSN  

BurVolunteer tester
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Joined: 25 Feb 20 Posts: 332 ID: 1241833 Credit: 22,611,276 RAC: 4,081

So these would need to be proven via ECPP or APRCL? How long does a proof approximately take, compared to LLR? Since it apparently wasn't undertaken by you, I guess: very long?
Is it even possible with current software? Primo doesn't seem to support candidates larger than 50 000 digits.
On the other hand this comparison from 2014 makes these methods look relatively fast. Maybe a few days for 100 000 digits. Or do things change dramatically when the number of digits increases further?
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Primes: 1281979 & 12+8+1979 & 1+2+8+1+9+7+9 & 1^2+2^2+8^2+1^2+9^2+7^2+9^2 & 12*8+19*79 & 12^81979 & 1281979 + 4 (cousin prime)  

GellyVolunteer tester
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Joined: 13 Nov 16 Posts: 40 ID: 468732 Credit: 522,960,451 RAC: 95,438

ECPP is achingly slow. There is a reason the official version of primo only goes up to 50k decimal digits. The current record, 40k digits, was set by Paul Underwood on 48 thread hardware that took more than a year and was nothing to sneeze at. My 3970x threadripper, a 32 core, 64 thread beast, takes two weeks to months to prove candidates in the 20k digit range.
As a rule of thumb, when you double the amount of digits in the number to be tested, the time it takes goes up 16 to 32 times, depending on how lucky you are. It will likely be years before Paul proves the next big step in ECPP (the smallest unproven repunit, 49k digits), and I'm sure he either got a 3990x (64 cores!) or maybe even doshed out on EPYC.
100k digits, on current hardware with current methods, would take decades.  

BurVolunteer tester
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Joined: 25 Feb 20 Posts: 332 ID: 1241833 Credit: 22,611,276 RAC: 4,081

Ok, thanks. One can loose the feeling for how large these 20k+ digits numbers actually are, because their primality can be proven so easily.
Projects like WW help a bit since we're suddenly dealing with a mere 18 digits and already things take a while.
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Primes: 1281979 & 12+8+1979 & 1+2+8+1+9+7+9 & 1^2+2^2+8^2+1^2+9^2+7^2+9^2 & 12*8+19*79 & 12^81979 & 1281979 + 4 (cousin prime)  


Projects like WW help a bit since we're suddenly dealing with a mere 18 digits and already things take a while.
However, in WW you start with a range of one trillion (10^12) numbers. Then you find out which of those numbers are primes and which are not; there are going to billions of primes. Then for each of those billions of primes, you check if they have the Wieferich property or the WallSunSun property.
So this is in no way comparable to testing one specific 20'000digit candidate number for primality.
/JeppeSN  

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General discussion :
Extended generalized Fermat prime? 