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 robish Volunteer moderator
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Apart from your own biggest Prime, has anyone got a favourite Prime? Ok I'll go 1st. I like Van Z's gfn20. Not because it's the biggest but because of its symmetry, 919444. Just sits well on the tongue. Hard to forget. Ok maybe I'm mad or OCD but order has an elegance does it not?
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I like that one too. :) |
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robishVolunteer moderator
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I like that one too. :)
:) no you're not allowed, it's your biggest! Apart from your biggest! :)
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Ok, I would call out Tabaluga's 1955556^131072+1. Not so much for symmetry, but all those 5's. |
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robishVolunteer moderator
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Ok, I would call out Tabaluga's 1955556^131072+1. Not so much for symmetry, but all those 5's.
Now you're talking, that's exactly what I mean. I'm with you on that. :)
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To put aside it is (still) my largest prime, but when I found it it was like dream come true.
9*10^1009567-1 or 8 at the start and 9 up to the end, but there is 1009566 nines to the end :)
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92*10^1439761-1 REPDIGIT PRIME :) :) :)
314187728^131072+1 GENERALIZED FERMAT
31*332^367560+1 CRUS PRIME
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robishVolunteer moderator
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To put aside it is (still) my largest prime, but when I found it it was like dream come true.
9*10^1009567-1 or 8 at the start and 9 up to the end, but there is 1009566 nines to the end :)
Awesome!, think your may win with that one hands down! Game over! Ah no, but that is ridiculously cool! :)
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Dave
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Me too in terms of repdigit. Though I am also interested in binary-look primes, especially if they're palindromic... |
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467917 · 2^1993429 - 1
My first T5k prime, and highest ranking prime at the time of finding prime. It was part of the rieselsieve project, and one of my systems found it on Christmas eve. I only found out when I checked my e-mail on Christmas day. Best present ever. |
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robishVolunteer moderator
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Me too in terms of repdigit. Though I am also interested in binary-look primes, especially if they're palindromic...
Any examples? Don't know much about this area.
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robishVolunteer moderator
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467917 · 2^1993429 - 1
My first T5k prime, and highest ranking prime at the time of finding prime. It was part of the rieselsieve project, and one of my systems found it on Christmas eve. I only found out when I checked my e-mail on Christmas day. Best present ever.
Timing counts. How big was it and when?
Speaking of timing, I was hunting for gfn17low on the 4th june. Switched back to gfn18 when I saw one in the pipe and bang 18! But I had been on 18 since march, only switched to 17 low on the 2nd June or so. Almost missed it :)
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Timing counts. How big was it and when?
https://primes.utm.edu/primes/page.php?id=76544
Just over 600k digits in 2005. Today, even a PPS prime would be bigger. I suppose you could put that prime down as the one that got me addicted. |
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robishVolunteer moderator
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Timing counts. How big was it and when?
https://primes.utm.edu/primes/page.php?id=76544
Just over 600k digits in 2005. Today, even a PPS prime would be bigger. I suppose you could put that prime down as the one that got me addicted.
2005? Well impressive for then! Moore's law has had many iterations since then so pc power was comparatively poor back then so fair play to you for getting it.
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robishVolunteer moderator
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How about 288larsons 86884666^16384+1 is pretty cool?
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My lucky numbers 10590941048576+1 and 224584605939537911+81292139*23#*n for n=0..26 |
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Well aside from the FPS i've found I like 2.
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19249*2^13018586+1 . Yes, it made more interested in all these searches. |
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I have found numerous mega primes, an actual SGS prime, a large Fermat divisor, and my favorite prime number is...
2
It is the only even prime number. That is awfully special in my book. :)
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tngSend message
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I have found numerous mega primes, an actual SGS prime, a large Fermat divisor, and my favorite prime number is...
2
It is the only even prime number. That is awfully special in my book. :)
I'll go with 2 as well -- same reason
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Me too in terms of repdigit. Though I am also interested in binary-look primes, especially if they're palindromic...
Any examples? Don't know much about this area.
Mersenne primes are all 1s in binary. Of course they are not searched here at PrimeGrid.
Many primes we search here have trivial binary expansions because they are based on a huge power of two. For example a 321 prime of the minus type is all 1s except for the second-most significant bit which is 0, so:
101111...11
And 321 plus type starts with 11 then has only 0s until it ends in a single 1, that is:
110000...001
Any Proth prime (PPS, EPS, MEGA) is similar; it has a short "arbitrary" word of initial digits (from the k), then a long string of 0s and a single 1 in the end.
/JeppeSN |
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I have many favorites, but the number F4 = 65537 is cool because it is thought to be the largest prime of the form 2^N + 1, and hence the last prime that needs only two 1s in its binary expansion. And in hexadecimal as well.
In decimal, GF(1,10) = 101 is thought to be the last prime with digit sum 2. No multidigt prime in base 10 can have a digit sum of 3. Many primes with decimal digit sum 4 can be found (expect infinitely many, A062339).
In the opposite direction, primes of decimal appearance 899999...9 was already mentioned by Crun-chi.
/JeppeSN |
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In form k*10^n+1 my largest prime is 9×10^100613+1
But now I attacking on two fronts ( 4*10^n+1 and 9*10^n+1), so you can expect one ,bigger, soon :)
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92*10^1439761-1 REPDIGIT PRIME :) :) :)
314187728^131072+1 GENERALIZED FERMAT
31*332^367560+1 CRUS PRIME
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My favorite prime is 22699.
It's one of the remaining SoB k's and the one from the very first SoB task I had from PrimeGrid. I was very interested in the subject and made a lot of my own little research looking into divisors of 22699*2^n+1 and stuff.
But if you need more generic proof, why it's a good prime? Sure. Let's start from 2 being a quite unique prime and 69 being... let's say, a well known number.
Then 269 is prime.
Then 2269 is prime and so is 2699.
Finally, 22699 is prime.
Q. E. D. |
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Michael GoetzVolunteer moderator
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My favorite prime (other than my GFN-19), is 193*2^3329782+1.
It's the only known mega prime that is a Fermat divisor.
I was the one who manually ran the Fermat divisor tests for that prime, so I clearly remember looking at the result on my screen and thinking, "Whoa... does that really say what I think it says???" :)
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My favorite prime (other than my GFN-19), is 193*2^3329782+1.
It's the only known mega prime that is a Fermat divisor.
I was the one who manually ran the Fermat divisor tests for that prime, so I clearly remember looking at the result on my screen and thinking, "Whoa... does that really say what I think it says???" :)
You remind me of 3*2^382449+1!
Transpose mega into 100,000-digit.
I will never forget the "Whoa!!!".
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In decimal, GF(1,10) = 101 is thought to be the last prime with digit sum 2. No multidigt prime in base 10 can have a digit sum of 3.
You made me think! Since the 1st one sounds unproven, then assuming the 2nd one is correct possible solutions must have factors. Am I good enough to work out a proof myself?... so that would be numbers either starting 2 ending 1 with variable number of zeros in between, or starting 1, ending 1, with a 1 in between, each separated by variable numbers of zeros... |
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In decimal, GF(1,10) = 101 is thought to be the last prime with digit sum 2. No multidigt prime in base 10 can have a digit sum of 3.
You made me think! Since the 1st one sounds unproven, then assuming the 2nd one is correct possible solutions must have factors. Am I good enough to work out a proof myself?... so that would be numbers either starting 2 ending 1 with variable number of zeros in between, or starting 1, ending 1, with a 1 in between, each separated by variable numbers of zeros...
think about divisibility by 3 |
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My favorite prime (other than my GFN-19), is 193*2^3329782+1.
It's the only known mega prime that is a Fermat divisor.
I was the one who manually ran the Fermat divisor tests for that prime, so I clearly remember looking at the result on my screen and thinking, "Whoa... does that really say what I think it says???" :)
Good one! I wonder when that record will be beaten (by some PrimeGrid user for sure).
It is quite hard to imagine how huge megaprimes are, but try to think about the Fermat number whose primality status (composite!) is revealed by this number:
F_3329780 = 2^(2^3329780) + 1
/JeppeSN |
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robishVolunteer moderator
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My favorite prime is 22699.
It's one of the remaining SoB k's and the one from the very first SoB task I had from PrimeGrid. I was very interested in the subject and made a lot of my own little research looking into divisors of 22699*2^n+1 and stuff.
But if you need more generic proof, why it's a good prime? Sure. Let's start from 2 being a quite unique prime and 69 being... let's say, a well known number.
Then 269 is prime.
Then 2269 is prime and so is 2699.
Finally, 22699 is prime.
Q. E. D.
Very very cool.
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robishVolunteer moderator
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How about this one, just got overnight,
131400000^16384+1,
Tidy. :)
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My lucky numbers 10590941048576+1 and 224584605939537911+81292139*23#*n for n=0..26 |
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It's the only known mega prime that is a Fermat divisor.
This also made me wonder:
What is the first Fermat number divisible by a megaprime? The first one that has a chance, is F_22.
And what is the last Fermat number divisible by a non-mega-prime (a prime that is too small to be a megaprime)? It must be less than F_3321922, because all factors of F_n are of form k*2^(n+2) + 1 with k>1. So all primes from F_3321922 and up have only megaprimes as prime divisors.
/JeppeSN |
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Michael GoetzVolunteer moderator
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It's the only known mega prime that is a Fermat divisor.
This also made me wonder:
What is the first Fermat number divisible by a megaprime? The first one that has a chance, is F_22.
I had to think about that one for a moment. :)
I don't think any of the searches we're doing, or any of the software, would be useful in answering that question. Certainly, PFGW, when checking for Fermat divisors, finds divisors MUCH larger than the prime being checked. For a prime with a million digits, it's looking at Fermat numbers around F_3300000, not F_22.
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My lucky number is 75898524288+1 |
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So all primes from F_3321922 and up have only megaprimes as prime divisors.
Of course, I meant all Fermat numbers from F_3321922 and up have only megaprimes as prime divisors. /JeppeSN |
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It's the only known mega prime that is a Fermat divisor.
This also made me wonder:
What is the first Fermat number divisible by a megaprime? The first one that has a chance, is F_22.
I had to think about that one for a moment. :)
I don't think any of the searches we're doing, or any of the software, would be useful in answering that question. Certainly, PFGW, when checking for Fermat divisors, finds divisors MUCH larger than the prime being checked. For a prime with a million digits, it's looking at Fermat numbers around F_3300000, not F_22.
I agree! Methods like the one used by David Bessel (Elliptic Curves Method (ECM) with Prime95 software) to find the first factor of F_22, or similar, could be used to find one more small factor, or a couple more, and then the cofactor can be tested, and if we are extremely lucky, that huge cofactor will be prime, a megaprime, and we will have F_22 divisible by a megaprime.
It would be really sensational to have F_22 completely factored like that (no Fermat number greater than F_11 has been completely factored).
I do not think we could find a megaprime factor of a "small" Fermat number like F_22 in any other way than eliminating some small factors and having enough luck that the huge cofactor after that is prime(?).
That megaprime would be expressible only as:
F_22 / [(3853959202444067657533632211*2^24 +1)*(other small factor)*...]
/JeppeSN |
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My favorite number is the 26th Eisenstein-Mersenne Norm prime: 3^2237561+3^1118781+1
My second favorite is the prime partitions number of a number which is square: partitions(14881^2)
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My lucky number is Phi(4, 2^2396029-1)/2.
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6391936^131072+1 found by participant jess is a nice prime too. 6391936 is palindromic. |
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robishVolunteer moderator
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6391936^131072+1 found by participant jess is a nice prime too. 6391936 is palindromic.
Agreed! Like it. Might be best yet :) well for me anyway :)
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My lucky numbers 10590941048576+1 and 224584605939537911+81292139*23#*n for n=0..26 |
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Ä°t is very rare. There are no another example of this property among gfn primes (except gfn14 and below I didnt checked them) May be there are other examples in other subprojects. This is a nice thread I think. Participants can discover interesting properties about their primes and share here. I like and I am interested in prime curios /The numbers which have interesting properties and related to primes. I noticed another nice example ;
9249*2^32767+1 a pps prime with 9868 digits. 32767=2^15-1 :)) 9249*2^(2^15-1)+1 is prime ! wow.
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3909127223745*2^1290000-1 is prime!
126055746^32768+1 is prime!
175894284^16384+1 is prime! (Private gfn server)
127511188^8192+1 is prime! (Private gfn server) |
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Dave
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Are there any golden sequential pandigitals e.g.123456789×2^n? Or k×2^[pandigit]? Another mini-conjecture. |
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e.g.123456789×2^n
123456789*2^1 - 1 is prime :P
Or k×2^[pandigit]?
I would imagine a number of this size would be difficult to check, since it would have at a minimum 37 million digits (123456789 * log10(2) +1 ~ 37,164,197), which is currently 10 million digits longer than the largest prime found. Perhaps in the future, but I wouldn't bank on making that sort of discovery anytime soon. I suppose there would "only" be 362 thousand numbers to check per k - 724 thousand if you wanted +1 and -1 - but it seems like a lot of work.
I think far more interesting would be a prime where the numbers come in order - something like 1 * 2^(345*678) - 9 or something. I haven't found any resources on them, and while I'd like to search for them, I'm unclear on the number theory of the divisors of the power of 2 and primality and that sort of thing. |
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If you go to https://primes.utm.edu/primes/search.php and search for 123456789 in description, and hit the box for all primes then there are a handful containing that sequence. Makes me wonder, how far have people searched 123456789*2^n +/- 1 |
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robishVolunteer moderator
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If you go to https://primes.utm.edu/primes/search.php and search for 123456789 in description, and hit the box for all primes then there are a handful containing that sequence. Makes me wonder, how far have people searched 123456789*2^n +/- 1
Wow there's quite a few including
123456789123456789 · 105513 + 1
New subproject? :)
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robishVolunteer moderator
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I got this one, would have been a beaut, but alas not prime
2244422^524288+1 :(
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Crun-chiVolunteer tester
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If you go to https://primes.utm.edu/primes/search.php and search for 123456789 in description, and hit the box for all primes then there are a handful containing that sequence. Makes me wonder, how far have people searched 123456789*2^n +/- 1
Wow there's quite a few including
123456789123456789 · 105513 + 1
New subproject? :)
Home subproject: even this is Primegrid forum, and we are all members of Primegrid, dont be afraid to start new sieve, make your own project, find your own prime.Even Primegrid give you chance to find primes in many projects it is still and only just grain of sand on whole beach of primes. :)
123456789*2 ^n+/-1 is also interesting
or
11111111111*2^n +/-1
Possibilities are endless
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92*10^1439761-1 REPDIGIT PRIME :) :) :)
314187728^131072+1 GENERALIZED FERMAT
31*332^367560+1 CRUS PRIME
Proud member of team Aggie The Pew. Go Aggie! |
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robishVolunteer moderator
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Got this today 304442222^16384+1
Sweet! :) symmetry.
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My lucky numbers 10590941048576+1 and 224584605939537911+81292139*23#*n for n=0..26 |
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Fav number is 7 and a prime. |
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Dad
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Mine is
555*2^3563328+1
because it's got 555 and it's my first (and only) Mega
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Tonight's lucky numbers are
555*2^3563328+1 (PPS-MEGA)
and
58523466^131072+1 (GFN-17 MEGA) |
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If we talk about the relatively small primes, my favourite from them is 333667. It's probably the largest unique period prime without digits 0 1 9. Also this is the largest prime divisor of the "magic number" 12345679 and of course 111111111. And for its "almost symmetry". |
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If we talk about the relatively small primes, my favourite from them is 333667. It's probably the largest unique period prime without digits 0 1 9. Also this is the largest prime divisor of the "magic number" 12345679 and of course 111111111. And for its "almost symmetry".
https://primes.utm.edu/curios/page.php/333667.html |
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19 -- I've just always thought it was a good number.
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Thanks!!!
What about primes that make numbers after 271129 non-Sierpinski? Really like them.
271169*2^1725+1 and 271463*2^1805+1 (so, between 271129 and 271577 there is nothing missed).
324169*2^15802+1 and 327679*2^24046+1 (the same thing about 322523 and 327737). |
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Timing counts. How big was it and when?
https://primes.utm.edu/primes/page.php?id=76544
Just over 600k digits in 2005. Today, even a PPS prime would be bigger. I suppose you could put that prime down as the one that got me addicted.
2005? Well impressive for then! Moore's law has had many iterations since then so pc power was comparatively poor back then so fair play to you for getting it.
looks over at the Apple //e that shares my desk with the computer I run PG on... (8700K/1080Ti)
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Timing counts. How big was it and when?
https://primes.utm.edu/primes/page.php?id=76544
Just over 600k digits in 2005. Today, even a PPS prime would be bigger. I suppose you could put that prime down as the one that got me addicted.
2005? Well impressive for then! Moore's law has had many iterations since then so pc power was comparatively poor back then so fair play to you for getting it.
looks over at the Apple //e that shares my desk with the computer I run PG on... (8700K/1080Ti)
Apple iPad 2 As Fast As The Cray-2 Super Computer on LINPACK benchmark* (Phoronix, 2012)
* With recursive/parallel LU factorization on the iPad 2
- Cray-2 released 1985
- iPad 2 released 2011 |
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I'd rather have the Cray than any Apple product that wasn't designed by Woz.
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I'd rather have the Cray than any Apple product that wasn't designed by Woz.
Good luck fitting the Cray on your desk beside the 8700K/1080Ti |
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Plenty of space in this room.
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My favourite prime is 1327, because it has quite a large gap of 34 following it.
By the way my favourite number is 1001 because it is sphenic and palindromic. Still looking for the fabled sphenic prime though. |
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Still looking for the fabled sphenic prime though.
Not sure there's a joke i'm missing, but sphenic numbers are composite by definition. |
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Still looking for the fabled sphenic prime though.
Not sure there's a joke i'm missing, but sphenic numbers are composite by definition.
Yeah I have a fondness for oxymorons that sound more plausible on account of the obscurity of the words. "Sceptical omphalist" would be another example. |
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Yeah I have a fondness for oxymorons that sound more plausible on account of the obscurity of the words. "Sceptical omphalist" would be another example.
If you have a fondness for sphenic numbers, the wikipedia page for them needs updating. They have the largest sphenic number known as of Jan of 2018, but there's a new biggest prime since then.
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robishVolunteer moderator
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Another Favourite.
Scott's recent SR5 88444·5^2799269-1 is really cool :)
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My lucky numbers 10590941048576+1 and 224584605939537911+81292139*23#*n for n=0..26 |
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CGBSend message
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I would have thought my first prime find to make the T5K would be my favourite, or possibly the MEGA I found, but for a favourite prime I personally found, it has to be the relatively small yet significant;
87888968^32768+1
It has lots of 8's in it and as I understand, 8 is considered a lucky number in eastern cultures. Also, it's one of only two consecutive GFN-15 primes known to exist. (As is my understanding) https://oeis.org/A118539 |
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Just found this :)
Q What is your favorite prime number and why?
To quote Sheldon Cooper,
"The best number is 73. Why? 73 is the 21st prime number. Its mirror, 37, is the 12th and its mirror, 21, is the product of multiplying 7 and 3... and in binary 73 is a palindrome, 1001001, which backwards is 1001001."
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My lucky numbers 10590941048576+1 and 224584605939537911+81292139*23#*n for n=0..26 |
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robishVolunteer moderator
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Oh and this :)
Belphegor’s prime, which is
1000000000000066600000000000001
This number has 31 digits (13 backwards), 2 sets of 13 zeros separated by 666.
This number is prime as well , and contains many superstitious elements.
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My lucky numbers 10590941048576+1 and 224584605939537911+81292139*23#*n for n=0..26 |
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Dave
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My favourite is the next one I find.
@Rob: https://www.youtube.com/watch?v=zk_Q9y_LNzg |
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robishVolunteer moderator
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My favourite is the next one I find.
@Rob: https://www.youtube.com/watch?v=zk_Q9y_LNzg
TY cool video. Cheers Dave!! ;)
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My lucky numbers 10590941048576+1 and 224584605939537911+81292139*23#*n for n=0..26 |
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I have synaesthesia, and 19 has a special quality that feels really good. 1 is an awful number, but the 9 completely tames it so 19 is really ool.
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My lucky number is 3504206+2 |
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BurVolunteer tester
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Relatively small with 7 digits:
1281979
But my birthday... :) |
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Michael GoetzVolunteer moderator
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I have decided my favorite is 8675309, since it's probably the largest prime number that literally millions of people actually remember. Even if they don't know it's prime. :)
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My lucky number is 75898524288+1 |
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I have decided my favorite is 8675309, since it's probably the largest prime number that literally millions of people actually remember. Even if they don't know it's prime. :)
For those that are younger than a certain amount of time... Jenny's phone number
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I have decided my favorite is 8675309, since it's probably the largest prime number that literally millions of people actually remember. Even if they don't know it's prime. :)
For those that are younger than a certain amount of time... Jenny's phone number
I actually didnt know that!
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SHSID Electronics Group
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GFN-14: 50103906^16384+1
Proth "SoB": 44243*2^440969+1
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I have decided my favorite is 8675309, since it's probably the largest prime number that literally millions of people actually remember. Even if they don't know it's prime. :)
For those that are younger than a certain amount of time... Jenny's phone number
I actually didnt know that!
It's also the code ALOT of people use at stores if they don't want it to have their own phone number, ie type in the (local are code) 867-5309 and see if you get the discount for being a local on the stuff you buy. You'd be surprised how often it works!!! I often ask the checkout person what the local area code is and then type the numbers in and they say''what did you do?" when it works, most laugh while a few have no clue. |
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Michael GoetzVolunteer moderator
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Supposedly Jenny Craig (the weight loss company) tried really hard to acquire the toll free phone number 1-800-867-5309 but the owner wouldn't sell. :)
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My lucky number is 75898524288+1 |
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Scott BrownVolunteer moderator
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I believe that, when the song was in the top 40 charts originally, there was a woman named Jenny living at that number with one of the New York area codes...poor her, I think she was forced to change her number.
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Here are some interesting things from a Wikipedia article about the number.
Popularity and litigation
The song, released in late 1981, initially gained popularity on the American West Coast in January 1982; many who had the number soon abandoned it because of unwanted calls.
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BurVolunteer tester
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Playing around a bit with 12/8/1979 and these are prime
1281979
1281979 * 2^2 + 1
1281979 * 2^3 - 1
1281979 * 6^6 - 1
There are three twins close by: 1282007 & 1282009 and 1282031 & 1282033 (consecutive twins!) and 1282079 & 1282081 but it's not a twin prime itself unfortunately. |
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Playing around a bit with 12/8/1979 and these are prime
1281979
1281979 * 2^2 + 1
1281979 * 2^3 - 1
1281979 * 6^6 - 1
There are three twins close by: 1282007 & 1282009 and 1282031 & 1282033 (consecutive twins!) and 1282079 & 1282081 but it's not a twin prime itself unfortunately.
Got a few more for you if you want them :)
These are all of the exponents of 2 up to 10000 that result in a prime of the form 1281979*2n+1:
2, 142, 202, 242, 370, 578, 614, 754, 6430, 7438, 7894
The last one of those has 2383 digits.
These are all of the exponents of 2 up to 10000 that result in a prime of the form 1281979*2n-1:
3, 7, 43, 79, 107, 157, 269, 307, 373, 397, 1005, 1013, 1765, 1987, 2269, 6623, 7083, 7365
And then, just for fun, I also checked 1281979n+2 and 1281979n-2. Nothing for the +2 form up to n=1000, but 12819794-2 is also prime :)
Enjoy!
Edit: Tested the 1281979*2n+1 form up to n=65536 and found the following n values resulting in primes: 10474, 11542, 45022, 46802. Last one has 14095 digits. Primes of the form 1281979*2n+1 are now officially called Bur Primes in my books ;) |
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1+2+8+1+9+7+9 = 37 and 128*1979+1281979 are also prime :) |
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Lol that was really impressive!
12*81979+1281979=2265727 is also prime :)
I'll try 432007, it itself is prime. Dunno other variants :)
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SHSID Electronics Group
SHSIDElectronicsGroup@outlook.com
GFN-14: 50103906^16384+1
Proth "SoB": 44243*2^440969+1
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Relatively small with 7 digits:
1281979
But my birthday... :)
I guess that is 12 Aug 1979; not Dec 8, 1979; or Jan 28, 1979.
19790812^4 + 1 is prime.
/JeppeSN |
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Relatively small with 7 digits:
1281979
But my birthday... :)
I guess that is 12 Aug 1979; not Dec 8, 1979; or Jan 28, 1979.
19790812^4 + 1 is prime.
/JeppeSN
I think it's december 8th (sorry for reading dates the american way)
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SHSID Electronics Group
SHSIDElectronicsGroup@outlook.com
GFN-14: 50103906^16384+1
Proth "SoB": 44243*2^440969+1
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BurVolunteer tester
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Thanks, that's some very interesting results. To be honest I used this online tool to check for primality so I didn't get too far... Also I didn't think about sum/product of individual digits.
1281979^n takes ages using the website. 1281979*6^666 - 1 is still running since a couple of days. I guess I should finally download LLR and try some combinations. Maybe in some base there's even a Cullen or Woodall prime? Unlikely, I know.
It's August, 12th. European here. It feels a bit egocentric now, so maybe someone elses birthday might also produce interesting results? :D
edit: I downloaded LLR and 1281979*6^666-1 took 35 ms. The website uses ECM, so apparently that's really inefficient for these numbers compared to LLR.
Is there a readme for ABC input file format? Specifically how to easily test ranges like 123*2^n+-1 with x<n<y. |
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Thanks, that's some very interesting results. To be honest I used this online tool to check for primality so I didn't get too far... Also I didn't think about sum/product of individual digits.
1281979^n takes ages using the website. 1281979*6^666 - 1 is still running since a couple of days. I guess I should finally download LLR and try some combinations. Maybe in some base there's even a Cullen or Woodall prime? Unlikely, I know.
I think C/W primes are already sieved thru that range. Also the "k" of the prime could easily go unnoticed if not specifically searching for C/W primes, especially when the k has lots of factors.
It's August, 12th. European here. It feels a bit egocentric now, so maybe someone elses birthday might also produce interesting results? :D
:D
I'll use that online tool too :P Or my calculator (which factors!) for small numbers
edit: I downloaded LLR and 1281979*6^666-1 took 35 ms. The website uses ECM, so apparently that's really inefficient for these numbers compared to LLR.
Is there a readme for ABC input file format? Specifically how to easily test ranges like 123*2^n+-1 with x<n<y.
If you downloaded LLR, there should be a readme file that teaches you how to do that.
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SHSID Electronics Group
SHSIDElectronicsGroup@outlook.com
GFN-14: 50103906^16384+1
Proth "SoB": 44243*2^440969+1
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Playing around a bit with 12/8/1979 and these are prime
1281979
1281979 * 2^2 + 1
1281979 * 2^3 - 1
1281979 * 6^6 - 1
There are three twins close by: 1282007 & 1282009 and 1282031 & 1282033 (consecutive twins!) and 1282079 & 1282081 but it's not a twin prime itself unfortunately.
Playing around the same way with Apr. 3rd, 2007 and:
No twin primes in the range 431907 to 432107 (I hand-tested them lol)
432007
432007*2^1-1 (I think this is a riesel prime, ie. it eliminates k=432007 in TRP)
432007*2^6+1
......
432007*6^1+1
432007*6^9-1
......
432007*10^3+1
432007*10^x-1 always has 3 as an algebraic factor
......
432007^n+2 always has 3 as an algebraic factor
432007^2-2
:)
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SHSID Electronics Group
SHSIDElectronicsGroup@outlook.com
GFN-14: 50103906^16384+1
Proth "SoB": 44243*2^440969+1
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Thanks, that's some very interesting results. To be honest I used this online tool to check for primality so I didn't get too far... Also I didn't think about sum/product of individual digits.
1281979^n takes ages using the website. 1281979*6^666 - 1 is still running since a couple of days. I guess I should finally download LLR and try some combinations. Maybe in some base there's even a Cullen or Woodall prime? Unlikely, I know.
It's August, 12th. European here. It feels a bit egocentric now, so maybe someone elses birthday might also produce interesting results? :D
edit: I downloaded LLR and 1281979*6^666-1 took 35 ms. The website uses ECM, so apparently that's really inefficient for these numbers compared to LLR.
Is there a readme for ABC input file format? Specifically how to easily test ranges like 123*2^n+-1 with x<n<y.
Hi Bur! Glad to see you digging into these a bit more! It can be pretty addictive :)
That Alpertron calculator is amazing for finding factors, but not the best for prime testing as you noticed. If you ever need (or want) to find factors of large numbers it is a good choice. You have definitely reached a much better solution for prime testing with LLR.
As far as the ABC file format goes, the LLR readme file contains a description in Section 8, but there may be a different solution that will get you up to bigger numbers faster ;)
You can use NewPGen to sieve ranges to test with LLR and that makes it much, much faster. It will find all of the candidates with small factors so that LLR can focus on things that have a higher chance of being prime. You can download it here https://primes.utm.edu/programs/NewPGen/
I've sieved 1281979*2^n+1 from n=65536 to n=4000000 up to p=118453653886 (meaning that every candidate was divided by every prime less than ~118 billion to see if it was composite) and ran the file for a few minutes with LLR to find that n=70382 and n=74938 also result in primes. You can download that candidates file here: https://drive.google.com/file/d/1Ympv4_4SU7cHUW_m7KGZ1r9OMLR60qQ5/view?usp=sharing
You can either load that file into NewPGen and keep sieving deeper to remove more candidates, or load that straight into LLR and start the prime hunt with it. And you can break it into smaller ranges manually too, so that you can run multiple copies of LLR (1 per CPU core if you want), just make sure to leave the file header in there (same goes for ABC file format; every file needs a header of some sort). For a lot of searches I do, I'll break the files down so that they span a range of n values of 200,000 if n<1M, 100,000 if 1M<n<1.5M, 50,000 if 1.5M<n<2M, etc, but you can find the amount that works best for you.
Good luck! |
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Bur, remember that determining whether a number is prime or not is easy, but factoring the number is hard!
A number like 1281979*6^666 - 1 has 525 digits when written in the normal decimal way. 525-digit numbers are too huge to be factored, usually. Of course, if you are lucky, you can "sieve" away some small factors (and in that case you know the number is composite), but if there are no small factors, you are doomed.
But to make a probabilistic primality test of a 525-digit number takes only a fraction of a second with any tool. If the test returns "no", that is proof the number is composite. If the test returns "yes", the number is very, very likely prime. What I am talking about here is general numbers like 1281979*6^666 + 163.
When a number like 1281979*6^666 + 163 comes out "probable prime" (PRP), with 525 digits, it is still quite fast to make a deterministic test of it. But we are talking many seconds, not a few milliseconds like the probabilistic test.
The original number you had, N = 1281979*6^666 - 1, had the special property that it is very easy to see how N+1 factors. In such case, where either N-1 or N+1 is easy to factor, something magically happens: There exist deterministic primality tests that are just as fast as the probabilistic (PRP) tests! So 525-digit number can be proved prime in a few milliseconds, instead of many seconds. That is why all the primes we find at PrimeGrid are of N-1 or N+1 type.
You could try PARI/GP for general investigations. Use
factor(...)
ispseudoprime(...)
isprime(...)
where "..." is your number. You can try 1281979*6^666 - 1 as a 525-digit number that is composite. Try 1281979*6^666 + 163 as a 525-digit number which is prime but for which N-1 and N+1 methods do not apply. Or look at 1281979*6^1608-1 which is a 1258-digit prime, and the N+1 technique works (LLR does it better than PARI).
You can also run PARI/GP in your browser.
/JeppeSN |
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In fact, LLR does some quite complicated things with the base-6 number 1281979*6^1608-1. On my very slow laptop, with Windows, with .\cllr64.exe -d -q"1281979*6^1608-1" it says:
Base factorized as : 2*3
Base prime factor(s) taken : 3
Starting N+1 prime test of 1281979*6^1608-1
Using FFT length 448, a = 3
1281979*6^1608-1 may be prime. Starting Lucas sequence...
Using FFT length 448, P = 3
1281979*6^1608-1 may be prime, trying to compute gcd's
1281979*6^1608-1 may be prime, but N divides U((N+1)/3), P = 3
Restarting Lucas sequence with P = 7
Using FFT length 448, P = 7
1281979*6^1608-1 may be prime, trying to compute gcd's
1281979*6^1608-1 may be prime, but N divides U((N+1)/3), P = 7
Restarting Lucas sequence with P = 8
Using FFT length 448, P = 8
1281979*6^1608-1 may be prime, trying to compute gcd's
1281979*6^1608-1 may be prime, but N divides U((N+1)/3), P = 8
Restarting Lucas sequence with P = 9
Using FFT length 448, P = 9
1281979*6^1608-1 may be prime, trying to compute gcd's
U((N+1)/3) is coprime to N!
1281979*6^1608-1 is prime! (1258 decimal digits, P = 9) Time : 524.296 ms.
In this case, old OpenPFGW is faster. I do .\pfgw64.exe -tp -q"1281979*6^1608-1" and see:
PFGW Version 3.8.3.64BIT.20161203.Win_Dev [GWNUM 28.6]
Primality testing 1281979*6^1608-1 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
1281979*6^1608-1 is prime! (0.1375s+0.0051s)
/JeppeSN |
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BurVolunteer tester
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432007*2^1-1 (I think this is a riesel prime, ie. it eliminates k=432007 in TRP) That's a nice one.
Actually, I read the readme and couldn't figure out the ABC format. For example:
Fixed k and c : ABC%d*$a^$b+%d or ABC%d*$a^$b-%d
Apparently %d is used for constants and %x for variables? But where do I define their values?
Hi Bur! Glad to see you digging into these a bit more! It can be pretty addictive :) That's true. Unfortunately, while I'm really interested in math, it is hard for me. I feel like someone with paralyzed arms loving archery. Sometimes one of my colleagues who has a natural grasp for math explains some proof and it's fascinating. But I can't do it myself.
One curio I was able to find: 12819*79-2 and 12819*79+2 are both prime. The 2 is a bit arbitrary though.
525-digit numbers are too huge to be factored, usually. If you ever find an efficient algorithm for factorization, you can choose between becoming very famous or very rich (maybe both). ;)
You can also run PARI/GP in your browser. Quite a handy website, thanks.
I've sieved 1281979*2^n+1 from n=65536 to n=4000000 up to p=118453653886 (meaning that every candidate was divided by every prime less than ~118 billion to see if it was composite) and ran the file for a few minutes with LLR to find that n=70382 and n=74938 also result in primes. Thanks for sharing the candidates file! n > 3300000 produces mega primes, that would be a really nice find. Now I have to find a computer to run it on... |
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That same website is where you can download PARI/GP installers for different platforms.
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SHSID Electronics Group
SHSIDElectronicsGroup@outlook.com
GFN-14: 50103906^16384+1
Proth "SoB": 44243*2^440969+1
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Speaking of favorite numbers, there is a new tag #MegaFavNumbers on YouTube where people explain their "mega" favorite number. Note that "mega" there means at least 7 digits (when we say megaprime here at PrimeGrid, it means at least 1'000'000 digits). Many of the videos are going to interest the readers of this forum! /JeppeSN |
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robishVolunteer moderator
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Having been born in 1968, a few months back I decided to try find a prime with 1968 or 68 in it. It was a bit of a disaster (but that's another story) but earlier Kellen while looking through GFP-5's found the following:-Â
686868688968989880Â
And to quote Kellen ;)
"Best part is that where there isn't a 68, you have 89, which is just a rotated 68 or a 98, which is just a flipped 68, with the exception of the final two digits."Â
This is now a favourite of mine and 5 times over! Really cool find.Â
I can't ever hope to find more 68s in a prime 🙂😎
Has to be a hard one to beat, in my book anyway ðŸ˜
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My lucky numbers 10590941048576+1 and 224584605939537911+81292139*23#*n for n=0..26 |
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I can't ever hope to find more 68s in a prime 🙂😎
Has to be a hard one to beat, in my book anyway ðŸ˜
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686869 is prime.
;)
Edit: Got a bigger one:
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
6868686868686868686868686868686868686868686868686868686868686868686868
68686868686868686868686868686868686868686869
2074 digits. 1036 "68"s followed by a 69. Enjoy! :D |
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robishVolunteer moderator
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🤣🤣🤣 Awesome! Speechless ðŸ˜ðŸ˜Ž
I can't ever hope to find more 68s in a prime 🙂😎
How wrong can you be? a new record? :)
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My lucky numbers 10590941048576+1 and 224584605939537911+81292139*23#*n for n=0..26 |
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GellyVolunteer tester
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(10^(2*4132)-1)*68/99 - 1 has 4131 68's in a row, followed by 67. It's about 8k digits and hasn't been proven, but I'm waiting until pfgw finishes the range I gave it before I throw primo at the biggest one. |
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robishVolunteer moderator
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(10^(2*4132)-1)*68/99 - 1 has 4131 68's in a row, followed by 67. It's about 8k digits and hasn't been proven, but I'm waiting until pfgw finishes the range I gave it before I throw primo at the biggest one.
Nice! Cheers :)
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My lucky numbers 10590941048576+1 and 224584605939537911+81292139*23#*n for n=0..26 |
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BurVolunteer tester
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Edit: Got a bigger one: [...] 2074 digits. 1036 "68"s followed by a 69. Enjoy! :D So few primes compared to N, but still enough to come up with all sorts of weird patterns. Infinities are beyond the human mind. :D
There's probably a prime that begins with the first 10^100^100 digits of Pi.
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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 & 12^8-1979 & 1281979 + 4 (cousin prime) |
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Edit: Got a bigger one: [...] 2074 digits. 1036 "68"s followed by a 69. Enjoy! :D So few primes compared to N, but still enough to come up with all sorts of weird patterns. Infinities are beyond the human mind. :D
There's probably a prime that begins with the first 10^100^100 digits of Pi.
Oh of course!
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GFN-14: 50103906^16384+1
Proth "SoB": 44243*2^440969+1
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Edit: Got a bigger one: [...] 2074 digits. 1036 "68"s followed by a 69. Enjoy! :D So few primes compared to N, but still enough to come up with all sorts of weird patterns. Infinities are beyond the human mind. :D
There's probably a prime that begins with the first 10^100^100 digits of Pi.
There is: Proof that there are infinitely many prime numbers starting with a given digit string
Easier to prove that there are primes that contain those digits as a substring: Just append a 1 (if necessary) and use Dirichlet's theorem on primes in arithmetic progression. For example, to prove that there are infinitely many primes containing the string 314, consider 10000*x + 3141.
Of course, almost all primes contain this digit sequence (first 10^(10^200) digits of π) as a substring (sum of reciprocal of all primes diverges; but sum of integers missing a given digit string converges, I guess (cf. Kempner series)).
/JeppeSN |
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GellyVolunteer tester
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https://factordb.com/index.php?id=1100000001571209570 points to the current entry (and eventual proof) that 68686868686868....{4131 68's}67 is a prime number. Took a 7 hour bite out of my current Primo project, which is relatively nothing at all. |
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robishVolunteer moderator
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https://factordb.com/index.php?id=1100000001571209570 points to the current entry (and eventual proof) that 68686868686868....{4131 68's}67 is a prime number. Took a 7 hour bite out of my current Primo project, which is relatively nothing at all.
ðŸ‘👊 nice one 😉
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My lucky numbers 10590941048576+1 and 224584605939537911+81292139*23#*n for n=0..26 |
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