New Fermat factors

[Batalov]

We would be amiss to not notice this new find that has just shown on radars...

13 · 2^5523860 + 1 divides Fermat F(5523858)

Congratulations to Scott and the numerous PrimeGrid miners!

[paulunderwood]

Originally Posted by Batalov

We would be amiss to not notice this new find that has just shown on radars...

13 · 2^5523860 + 1 divides Fermat F(5523858)

Congratulations to Scott and the numerous PrimeGrid miners!

There is now no need to run a PRP/N-1 prime check on F(5523858)

[LaurV]

Originally Posted by paulunderwood

There is now no need to run a PRP/N-1 prime check on F(5523858)

haha, that was a good one.

Congrats to the finder(s) !

[JeppeSN]

Good old F(5523858): I always thought it would be composite, but I did not expect such a tiny prime divisor! /JeppeSN

[Dr Sardonicus]

Once upon a time, long long ago, I noted that if n > 2 and k < 2ⁿ⁺² + 2, the fact that N = k*2ⁿ⁺² + 1 divides F_n in and of itself proves N is prime. In the case at hand (n = 5523858), k = 13 satisfies this condition.

[ET_]

Any more infos about the discovery? like the surname of Scott, or the setup he used for the search, or the time he devoted to it?

Luigi
---

P.S. Dr. James Scott Brown.

[mathwiz]

Originally Posted by ET_

Any more infos about the discovery? like the surname of Scott, or the setup he used for the search, or the time he devoted to it?

Luigi
---

https://primes.utm.edu/bios/page.php?id=1298 may at least answer some of your questions.

[ATH]

Originally Posted by ET_

Any more infos about the discovery? like the surname of Scott, or the setup he used for the search, or the time he devoted to it?

His name is Scott Brown it seems:

http://primegrid.com/forum_thread.php?id=8778

https://www.primegrid.com/workunit.php?wuid=638323572

https://www.primegrid.com/show_user.php?userid=1178

[JeppeSN]

It is the same Scott Brown who found another Fermat factor, 9*2^2543551+1, back in 2011, in a similar way.

In PrimeGrid, the participants detect the primality (two persons do it concurrently, the one finishing first is declared the finder). Whether the new Proth prime divides any Fermat number (and/or generalized Fermat numbers with bases at most 12) is detected by PrimeGrid's server, not the participant's computer.

The link posted by ATH shows the timing (primality was reported "22 Jan 2020 | 15:00:58 UTC") and some hardware ("Intel(R) Core(TM) i7-4790 CPU @ 3.60GHz [Family 6 Model 60 Stepping 3]").

The other participant was Stefan Larsson (returned "22 Jan 2020 | 15:11:39 UTC").

At some point PrimeGrid will publish an official announcement (PDF).

This was PrimeGrid's first Fermat divisor in five years. They recently introduced a new subproject that focuses on Proth primes with low "k" (the odd multiplier) because that gives higher probability of Fermat divisors. This approach was recommended by Ravi Fernando and others.

/JeppeSN

[Dr Sardonicus]

Just for fun, I looked for small prime factors of the numbers k*2^5523860 + 1, k = 1 to 12. For all but three of them, the smallest factor can be found mentally. For two of the remaining three, the smallest factor is still quite small.

k = 1 p = 17
k = 2 p = 3
k = 3 p = 14270779
k = 4 p = 5
k = 5 p = 3
k = 7 p = 2625617
k = 8 p = 3
k = 9 p = 5
k = 10 p = 11
k = 11 p = 3
k = 12 p = 7

For the remaining value, 6*2^5523860 + 1, I didn't look far enough to find any factors, but I didn't look all that far. Has (as I suspect) someone already found a factor by trial division, or otherwise shown it to be composite?

[JeppeSN]

PrimeGrid's official announcement is here and can be found in this thread of theirs. /JeppeSN