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

Let's start from [URL="https://groups.yahoo.com/neo/groups/primenumbers/conversations/topics/20207"]Ken Davis' construction[/URL]. Observe the form that he used (which is similar to the form J.K.Andersen used before him).

Maybe we can find this even cheaper in computrons.
[B]Illustration:[/B] Let's take m=3*2^n, so that m+1 is prime. A tiny example m=3*2^534.
Then we will sieve for two forms:
p = k*m*(m^2 - 1)+ 6*m -1, p[SUB]6[/SUB] = p+6
...and presto, done:
[B]63166*3*2^534*(9*2^1068-1)+18*2^534-1[/B] is prime, quite trivially, and
[B]63166*3*2^534*(9*2^1068-1)+18*2^534+5[/B] (with 3*2^534+1 as a helper)

Now, repeat with m=
3*2^34350
3*2^42294
3*2^42665
3*2^44685
3*2^48150
3*2^55182
3*2^59973

The only part to write is a simple sieve, then sieve, and then do some PRP'ing.

One-up
318725*3*2^2208*(9*2^4416-1)+18*2^2208-1
318725*3*2^2208*(9*2^4416-1)+18*2^2208+5

Next up
363629*3*2^3168*(9*2^6336-1)+18*2^3168-1
363629*3*2^3168*(9*2^6336-1)+18*2^3168+5

And slightly larger
2865046*7*2^6614*(49*2^13228-1)+42*2^6614-1
2865046*7*2^6614*(49*2^13228-1)+42*2^6614+5

I am working on 34350 using -f -o of pfgw for "-1" as feedback to pfgw -f later on an ABC file with +5 & -1 in the header...

If you don't sieve, you will waste enormous amount of time on ineligible k values.
I sieved to 1T.

[QUOTE=Batalov;514175]
Maybe we can find this even cheaper in computrons.
[B]Illustration:[/B] Let's take m=3*2^n, so that m+1 is prime. A tiny example m=3*2^534.
Then we will sieve for two forms:
p = k*m*(m^2 - 1)+ 6*m -1, p[SUB]6[/SUB] = p+6
...
Now, repeat with m=
3*2^34350
3*2^42294
3*2^42665
3*2^44685
3*2^48150
3*2^55182
3*2^59973
[/QUOTE]

You can do it even much denser:
Let r=k*2^n+1 ~ sqrt(N) Proth prime, then search p in the form:

[CODE]
p=c*r*2^n+6*r-5 where c=1,2,3,.. is running.
[/CODE]

Then p-1 is divisible by 2^n, and p+5 is divisible by r, what is a known prime.
Bingo, and the sieve is blazingly fast, because you need only 2^n mod s, where s is prime.

example:
[CODE]
k=165;n=100 for that r=k*2^n+1 is a Proth prime, and turned out that c=2920 is good.
[/CODE]

[QUOTE=paulunderwood;514179]I am working on 34350 using -f -o of pfgw for "-1" as feedback to pfgw -f later on an ABC file with +5 & -1 in the header...[/QUOTE]

I did not get far.

I have now written a sieve in pari-gp (which I will convert to PrimeSieve+GMP) and am testing the world record contender:

p=c*(3*2^34350+1)*2^34350+6*(3*2^34350+1)-5 :smile:

I just can't my sieves to behave :down:

Moreover, I feel that any record breaking sexy pair needs to involve primorials in order to get the required density when searching.

[QUOTE=paulunderwood;514236]I just can't my sieves to behave :down:

Moreover, I feel that any record breaking sexy pair needs to involve primorials in order to get the required density when searching.[/QUOTE]

That is just wrong assumption, if you'd be correct we would search only on the form say k*p#+1 and not Mersenne.

And for sieve why not use my ancient polysieve: [url]https://primes.utm.edu/bios/page.php?id=3934[/url] . That handle this problem also, I'll give how to feed this problem for the code.

Here it is:
[CODE]
$ ./polysieve
Sieve P(s)+a*Q(s)+c for multiple c values, with fixed s=k*b^n+d; P,Q is polynom.
Give k: 1
Give b: 2
Give n: 34350
Give d: 0
Give the degree of the P polynom: 1
Give the 0-th coefficient of P: 0
Give the 1-th coefficient of P: 18
Give the degree of the Q polynom: 2
Give the 0-th coefficient of Q: 0
Give the 1-th coefficient of Q: 1
Give the 2-th coefficient of Q: 3
Give the number of c values for the sieve: 2
0-th c value: 1
1-th c value: 7
Give start and end value for 'a' (in billions)! 0 10
Give the limit for sieving primes (maxp): 1000000000000
Give the name of the file to output the numbers! sexy.txt
Using primes for wheelsieve up to 5
[/CODE]

On line 18 of the c code set:
#define bound_small_primes 5//11 // used up to this bound all primes in wheel sieve (change it, but it is very critical)
to lower the wheelsieve.
The range for 'a' is really what you like (what was c in my previous post), above we test 'a' from 0 to 10 (in billions), and maxp=1000000000000. Note that we needed to use the s=2^n to make it an integer polynom.

ps. use smaller example to test it out, say 534 instead of the large 34350 (you need to change only that line) to handle that case. My code also works for the general case, on every Proth numbers (with some modification on the polynoms).

[QUOTE=R. Gerbicz;514243]That is just wrong assumption, if you'd be correct we would search only on the form say k*p#+1 and not Mersenne.

And for sieve why not use my ancient polysieve: [url]https://primes.utm.edu/bios/page.php?id=3934[/url] . That handle this problem also, I'll give how to feed this problem for the code.[/QUOTE]

When searching for arithmetic progressions we have always used primorials -- I think this sexy types are similar, but I willing to give it another shot without them.

I downloaded and successfully compiled PolySieve.c.

[code deleted due to cross post]

[QUOTE=paulunderwood;514247]When searching for arithmetic progressions we have always used primorials -- I think this sexy types are similar, but I willing to give it another shot without them.[/QUOTE]

In general we use primorials for small numbers, say when you're searching 18 primes in ap, then the sieve bound is also smaller, and it is better to use primorials. In our case with primorials you'd only lost in sieving for these large numbers.

Btw polysieve as you can see is single threaded and there is no save option, so don't stop it, however maybe from Puzzle-Peter there is an updated code with save option somewhere in the forum. Not thought that somebody would run my code for weeks/months(?).

the next world record
 

Behold, the new world record. yay...yawn...

And didn't take "nearly 10 years" to beat, just ~ a day.
[URL="http://factordb.com/index.php?id=1100000001288370666"](153528880*(1369*2^46028-1)+6)*37*2^23014+5[/URL]
[URL="http://factordb.com/index.php?id=1100000001288370665"](153528880*(1369*2^46028-1)+6)*37*2^23014-1[/URL]

I am trying Robert's form for > 30k digits next; I like the form.

[QUOTE=Batalov;514358]Behold, the new world record. yay...yawn...

And didn't take "nearly 10 years" to beat, just ~ a day.
[URL="http://factordb.com/index.php?id=1100000001288370666"](153528880*(1369*2^46028-1)+6)*37*2^23014+5[/URL]
[URL="http://factordb.com/index.php?id=1100000001288370665"](153528880*(1369*2^46028-1)+6)*37*2^23014-1[/URL]

I am trying Robert's form for > 30k digits next; I like the form.[/QUOTE]

Drat, drat and double drat! I am searching for a ~20681 digit pair and was hoping to be the champion.

[QUOTE=R. Gerbicz;514243]That is just wrong assumption, if you'd be correct we would search only on the form say k*p#+1 and not Mersenne.

And for sieve why not use my ancient polysieve: [url]https://primes.utm.edu/bios/page.php?id=3934[/url] . That handle this problem also, I'll give how to feed this problem for the code.[/QUOTE]

Polysieve is pretty amazing. With a little bit of thinking it can do so many things and it is really fast. For some applications you need a lot of RAM though.

[CODE]./pfgw64 -t -q"5298*(1*2^8922-0)+709263602*(883*(1*2^8922-0)^2+1*2^8922-0)+1"
PFGW Version 3.7.10.64BIT.20150809.x86_Dev [GWNUM 28.7]

Primality testing 5298*(1*2^8922-0)+709263602*(883*(1*2^8922-0)^2+1*2^8922-0)+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 49.90%
5298*(1*2^8922-0)+709263602*(883*(1*2^8922-0)^2+1*2^8922-0)+1 is prime! (0.3782s+0.0001s)
[/CODE]

[CODE]./pfgw64 -t -hsexy.helper -q"5298*(1*2^8922-0)+709263602*(883*(1*2^8922-0)^2+1*2^8922-0)+7"
PFGW Version 3.7.10.64BIT.20150809.x86_Dev [GWNUM 28.7]

Primality testing 5298*(1*2^8922-0)+709263602*(883*(1*2^8922-0)^2+1*2^8922-0)+7 [N-1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file sexy.helper
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 49.94%
5298*(1*2^8922-0)+709263602*(883*(1*2^8922-0)^2+1*2^8922-0)+7 is prime! (0.3656s+0.0006s)[/CODE]

My first sexy pair, based on a the 1993 Harvey Dubner prime [URL="https://primes.utm.edu/primes/page.php?id=30139"]883*2^8922+1[/URL].

Note the huge variable :devil:

[QUOTE=Puzzle-Peter;514420]Polysieve is pretty amazing. With a little bit of thinking it can do so many things and it is really fast. For some applications you need a lot of RAM though.[/QUOTE]

Off-topic: Indeed! Congrats for [URL="https://primes.utm.edu/primes/page.php?id=126414"]425521077* 2^3321910 - 1[/URL] (1,000,004 digits) :toot:

And for the new world record for a triplet: [url]https://primes.utm.edu/primes/page.php?id=126416[/url] (20,008 digits) :toot: :toot: :toot:

[QUOTE=paulunderwood;514496]...And for the new world record for a triplet: [url]https://primes.utm.edu/primes/page.php?id=126416[/url] (20,008 digits) :toot: :toot: :toot:[/QUOTE]
Which also beats the 19,503-digit 'sexy' from primepairs.com , but [B]not [/B]the current WR. :rolleyes:

And now, the world record sexy pair with the size more appropriate for 2019:

[URL="http://factordb.com/index.php?id=1100000001289544195"](187983281*2^51478+4)*(5*2^51478-1)+5[/URL]
[URL="http://factordb.com/index.php?id=1100000001289544353"](187983281*2^51478+4)*(5*2^51478-1)-1 [/URL]
(31,002 decimal digits)

[QUOTE=Batalov;514552]And now, the world record sexy pair with the size more appropriate for 2019:

[URL="http://factordb.com/index.php?id=1100000001289544195"](187983281*2^51478+4)*(5*2^51478-1)+5[/URL]
[URL="http://factordb.com/index.php?id=1100000001289544353"](187983281*2^51478+4)*(5*2^51478-1)-1 [/URL]
(31,002 decimal digits)[/QUOTE]

Congrats, Serge :toot:

I am searching for a [URL="https://en.wikipedia.org/wiki/Sexy_prime#Sexy_prime_triplets"]sexy prime triplet[/URL]. Do you care to outdo my efforts?

[QUOTE=Batalov;514552]And now, the world record sexy pair with the size more appropriate for 2019:

[URL="http://factordb.com/index.php?id=1100000001289544195"](187983281*2^51478+4)*(5*2^51478-1)+5[/URL]
[URL="http://factordb.com/index.php?id=1100000001289544353"](187983281*2^51478+4)*(5*2^51478-1)-1 [/URL]
(31,002 decimal digits)[/QUOTE]

Question: if one of the primes is PRP [url]http://factordb.com/index.php?id=1100000001289544353[/url] Why is it considered a record?

I suppose it can be proven definitely prime with Primo if someone with the resources tries.

[QUOTE=rudy235;514559]Question: if one of the primes is PRP [url]http://factordb.com/index.php?id=1100000001289544353[/url] Why is it considered a record?

I suppose it can be proven definitely prime with Primo if someone with the resources tries.[/QUOTE]

~50% factorisation of N+1 is given by [url]https://primes.utm.edu/primes/page.php?id=11814[/url] and so leads to a quick BLS proof:

[code]/pfgw64 -tp -hsexy.helper -q"(187983281*2^51478+4)*(5*2^51478-1)-1"
PFGW Version 3.7.10.64BIT.20150809.x86_Dev [GWNUM 28.7]

Primality testing (187983281*2^51478+4)*(5*2^51478-1)-1 [N+1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file sexy.helper
Running N+1 test using discriminant 2, base 1+sqrt(2)
Calling Brillhart-Lehmer-Selfridge with factored part 49.99%
(187983281*2^51478+4)*(5*2^51478-1)-1 is prime! (63.8154s+0.0122s) [/code]

[QUOTE=rudy235;514559]Question: if one of the primes is PRP [url]http://factordb.com/index.php?id=1100000001289544353[/url] Why is it considered a record?

I suppose it can be proven definitely prime with Primo if someone with the resources tries.[/QUOTE]
No need for Primo. Have a look at the N+1 for that number and note that 5*2^51478-1 is a prime.

[QUOTE=Batalov;514561]No need for Primo. Have a look at the N+1 for that number and note that 5*2^51478-1 is a prime.[/QUOTE]

Sure. But when I posted it appeared as PRP . Now it has just been changed to P (prime)

[QUOTE=paulunderwood;514553]Congrats, Serge :toot:

I am searching for a [URL="https://en.wikipedia.org/wiki/Sexy_prime#Sexy_prime_triplets"]sexy prime triplet[/URL]. Do you care to outdo my efforts?[/QUOTE]

What size are you trying to find? I might be tempted to go for it

[QUOTE=paulunderwood;514496]Off-topic: Indeed! Congrats for [URL="https://primes.utm.edu/primes/page.php?id=126414"]425521077* 2^3321910 - 1[/URL] (1,000,004 digits) :toot:

And for the new world record for a triplet: [url]https://primes.utm.edu/primes/page.php?id=126416[/url] (20,008 digits) :toot: :toot: :toot:[/QUOTE]

Thanks! I am amazed how quickly you guys learn about such news. Are you constantly monitoring the Top5000?

[QUOTE=Puzzle-Peter;514660]What size are you trying to find? I might be tempted to go for it[/QUOTE]

It is based on this puny number: [url]https://www.mersenneforum.org/showpost.php?p=514447&postcount=15[/url]

Please go for it!

[QUOTE=Puzzle-Peter;514663]Thanks! I am amazed how quickly you guys learn about such news. Are you constantly monitoring the Top5000?[/QUOTE]

You guess right. The "status page" is bookmarked. :smile:

Triplets, -- sky is the limit
 

The form that I adapted from Robert's sketch allows for a separate, modified search of a triplet of uber-Primo size.

[U]Miniscule proof-of-concept triplet:[/U]
(47172023*2^3052-4)*(5*2^2548-1)+1
(47172023*2^3052-4)*(5*2^2548-1)-3
(47172023*2^3052-4)*(5*2^2548-1)-5

Now this can be repeated for a 25K size, with a bit of compute expense. No primo proof will be needed.

Not practical to apply this construction to a quad or higher. (Quads are still of adolescent size, so proving two 10K-digit primes with Primo is less of a problem than sieve for special form and PRP for this form is slower than for a k,b,n,c form.)

8704161957*(2^20000)-1
8704161957*(2^20000)+5
8704161957*(2^20000)+11


+5 and +11 proven with PRIMO

[QUOTE=Puzzle-Peter;516145]8704161957*(2^20000)-1
8704161957*(2^20000)+5
8704161957*(2^20000)+11


+5 and +11 proven with PRIMO[/QUOTE]

You should update [url]https://en.wikipedia.org/wiki/Sexy_prime#Sexy_prime_triplets[/url]

10409207693*(2^20000)-1
10409207693*(2^20000)+5

10409207693*(2^20000)+11


Not searching this particular form any further, so now I'll update wikipedia. Not sure about trying something a little bit bigger yet...

According to the [URL="https://en.wikipedia.org/wiki/Sexy_prime#Sexy_prime_quadruplets"]wiki page[/URL] we now have a record quadruplet:

"n May 2019 Marek Hubal announced a 1138-digit quadruplet with p = 1567237911*2677# + 3301 + 6*n"

Just for those that may not realise it:
Sexy primes are pairs that differ by six (sex is Latin for six).
[url]https://en.wikipedia.org/wiki/Sexy_prime[/url]

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

New big triplet
 

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

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:
[CODE]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)
[/CODE]

Congrats! Nice sexy pair, 50539 digits!

Shorter written as:
[URL="http://factordb.com/index.php?id=1100000001367019108"](520461*2^55931+1)*(98569639289*(520461*2^55931-1)^2-3)-1[/URL]
[URL="http://factordb.com/index.php?id=1100000001367019163"](520461*2^55931+1)*(98569639289*(520461*2^55931-1)^2-3)+5[/URL]

[QUOTE=Batalov;527198]Congrats! Nice sexy pair, 50539 digits!

Shorter written as:
[URL="http://factordb.com/index.php?id=1100000001367019108"](520461*2^55931+1)*(98569639289*(520461*2^55931-1)^2-3)-1[/URL]
[URL="http://factordb.com/index.php?id=1100000001367019163"](520461*2^55931+1)*(98569639289*(520461*2^55931-1)^2-3)+5[/URL][/QUOTE]

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

[QUOTE=paulunderwood;527200][url]https://en.wikipedia.org/wiki/Sexy_prime[/url] needs updating with Peter's new pair.[/QUOTE]

Done :smile:

record quadruplet
 

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

Norman Luhn:

[QUOTE]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
[/QUOTE]

Another record triplet
 

Wiki:

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

p = 22582235875×2[sup]22224[/sup]+1.[/QUOTE]

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

Primo certificates for the latter two are uploaded to factordb

[QUOTE=Puzzle-Peter;535761]18416522281203*2^33222-1
18416522281203*2^33222+5
18416522281203*2^33222+11

Primo certificates for the latter two are uploaded to factordb[/QUOTE]

Congratulations. I have updated the Wiki page.

Unfortunately, Jens K. Andersen don't make an update on his prime page....


Gerd and I found on 2019/12/24 this sexy prime triplet at 10602 digits. This was the 1st case of a gigantic triplet.


[B]2683143625525 · 2^35176 + 1,7,13[/B]


Certificates was uploaded to factordb.com


[URL]https://primes.utm.edu/top20/page.php?id=13[/URL]


It set also an CPAP-3 record.

Paul, can you make also an wiki entry,please ?



best
Norman

I have edited the [URL="https://en.wikipedia.org/wiki/Sexy_prime"]Wiki page[/URL]. Peter's recent "record" no longer shows.

Thank you !


BTW, the 1st report was here:


[url]https://matheplanet.de/matheplanet/nuke/html/viewtopic.php?topic=244976&start=0&lps=1784598#v1784598[/url]

Congrats, Serge, for the new sexy triplet (and CPAP) record (10,753 digits):

[url]https://primes.utm.edu/primes/page.php?id=130856[/url]

I trust you will update the [URL="https://en.wikipedia.org/wiki/Sexy_prime#Sexy_prime_triplets"]wiki page[/URL].

As this is the thread for the more unpopular constellations, how about a pair of cousin primes?


(520461*2^55931+1)*(43439253939*(520461*2^55931-1)^2-3)+1
(520461*2^55931+1)*(43439253939*(520461*2^55931-1)^2-3)+5



To prove, use the -tc switch in PFGW and have a helper file that contains
520461*2^55931+1
520461*2^55931-1
2584831267