Let's find some large sexy prime pair (and, perhaps, a triplet)

[Puzzle-Peter]

Had my machine fool around with polysieve and pfgw when I was on vacation. This is what I came back to:


3039197559912*2^5000+17
3039197559912*2^5000+11
3039197559912*2^5000+5
3039197559912*2^5000-1


12995721764330*2^4900+131
12995721764330*2^4900+137
12995721764330*2^4900+143
12995721764330*2^4900+149


9838746720240*2^4900+131
9838746720240*2^4900+137
9838746720240*2^4900+143
9838746720240*2^4900+149


19299420002127*2^5050+17233
19299420002127*2^5050+17239
19299420002127*2^5050+17245
19299420002127*2^5050+17251


15803817109527*2^5050+17233
15803817109527*2^5050+17239
15803817109527*2^5050+17245
15803817109527*2^5050+17251


8728595737567*2^5050+17233
8728595737567*2^5050+17239
8728595737567*2^5050+17245
8728595737567*2^5050+17251


7315722716197*2^5050+17233
7315722716197*2^5050+17239
7315722716197*2^5050+17245
7315722716197*2^5050+17251


15703644477605*2^4900+131
15703644477605*2^4900+137
15703644477605*2^4900+143
15703644477605*2^4900+149

[paulunderwood]

Ken Davis further improved the record with a 6,180 digit Brillhart-Lehmer-Selfridge provable triplet in Oct 2019:

p = (72865897*809857*4801#*(809857*4801#+1)+210)*(809857*4801#-1)/35+1

[Puzzle-Peter]

I had a bit of fun with (x+2)*(ax²-3) -1 / +5
= -3x - 6 +a(x³+2x²) -1 / +5

helpers are x for N+1 proof of the +5 candidate and x+2 for N+1 proof of the -1 candidate, so x and x+2 form a twin prime pair. Using Polysieve and PFGW I got a sexy pair for

x=520461*2^55931-1 and

a=98569639289

Here's the PFGW output:

Primality testing -3*(520461*2^55931-1)-6+98569639289*((520461*2^55931-1)^3+2*(520461*2^55931-1)^2)-1 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 2, base 2+sqrt(2)
Calling Brillhart-Lehmer-Selfridge with factored part 33.34%
-3*(520461*2^55931-1)-6+98569639289*((520461*2^55931-1)^3+2*(520461*2^55931-1)^2)-1 is prime! (174.8309s+0.0070s)


Primality testing -3*(520461*2^55931-1)-6+98569639289*((520461*2^55931-1)^3+2*(520461*2^55931-1)^2)+5 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 2, base 1+sqrt(2)
Calling Brillhart-Lehmer-Selfridge with factored part 33.34%
-3*(520461*2^55931-1)-6+98569639289*((520461*2^55931-1)^3+2*(520461*2^55931-1)^2)+5 is prime! (260.7965s+0.0059s)

[Batalov]

Congrats! Nice sexy pair, 50539 digits!

Shorter written as:
(520461*2^55931+1)*(98569639289*(520461*2^55931-1)^2-3)-1
(520461*2^55931+1)*(98569639289*(520461*2^55931-1)^2-3)+5

[paulunderwood]

Originally Posted by Batalov

Congrats! Nice sexy pair, 50539 digits!

Shorter written as:
(520461*2^55931+1)*(98569639289*(520461*2^55931-1)^2-3)-1
(520461*2^55931+1)*(98569639289*(520461*2^55931-1)^2-3)+5

https://en.wikipedia.org/wiki/Sexy_prime needs updating with Peter's new pair.

[paulunderwood]

Originally Posted by paulunderwood

https://en.wikipedia.org/wiki/Sexy_prime needs updating with Peter's new pair.

Done

[paulunderwood]

Ken Davis:

1901870849*(269504*1601#*(269504*1601#+1)*(269504*1601#-1)/385)+6*(269504*1601#-1)-5

is a is a 2053 digit sexy prime quadruplet for n=0-3

-5 proof courtesy of Primo

+1,+7,+11 proofs via pfgw

[paulunderwood]

Norman Luhn:

Gerd Lamprecht and I found a new sexy prime quadruplet at 3025 digits [10049 bit].
This set also a record for CPAP-4.

The lucky numbers are 121152729080*7019#/1729+1+6n, n=0...3.

All numbers are proven primes by PRIMO.

Project time was only near 2 days.

best wishes

Norman & Gerd

[paulunderwood]

Wiki:

Norman Luhn & Gerd Lamprecht improved the record to 6,701 digits in Oct 2019:

p = 22582235875×2²²²²⁴+1.

[Puzzle-Peter]

18416522281203*2^33222-1
18416522281203*2^33222+5
18416522281203*2^33222+11

Primo certificates for the latter two are uploaded to factordb

[paulunderwood]

Originally Posted by Puzzle-Peter

18416522281203*2^33222-1
18416522281203*2^33222+5
18416522281203*2^33222+11

Primo certificates for the latter two are uploaded to factordb

Congratulations. I have updated the Wiki page.