Smarandache prime(s?)

[Batalov]

Originally Posted by Neil J. A. Sloane, OEIS Foundation

To Number Theory List,

Consider the sequence with nth term equal to the
concatenation of the decimal numbers 1234...n (https://oeis.org/A007908).
When is the first prime? The comments in A007908 say that there should be infinitely many primes, and that there
are no primes among the first 64000 terms.
If you would like to help with this search, you could leave a comment in A007908 saying that there are no primes among terms X through Y, or, of course, that n = Z gives a (probable) prime, which would be pretty exciting.

Best regards, Neil

There is a PRPNet server now, working on candidates n>200,000. Everyone is welcome to attach their prpclients.

Instructions:
1a. Get PRPclient 5.4.0 for your platform from here. It contains all programs, bundled in.
1b. (optionally) Get PFGW from here if you already have prpclient but not PFGW
2. Edit prpclient.ini : add email, username, machineid, server and pfgwexe :

//============= in prpclient.ini =============
email=youremail@somewhere.com                                                                                                                             
userid=UserId                      
machineid=PhenomII                                                                               
instanceid=1                                                                                     
//...                                                                                                                                                
server=SmarPRP:100:1:smarandache.ddns.net:1200
//...
pfgwexe=pfgw64.exe

The current status is here: http://smarandache.ddns.net:1200/pending_tests.html ( or http://99.121.249.54:1200/ )
The current typical test duration is 16 hours (on a Broadwell), 10-24 hours on any modern CPU.

The credit for the find will go to "A,B,PFGW", where A is the finder, B is me and PFGW is Mark+George and other authors of PFGW.

Happy hunting! And read the whole thread, too (optional but important).
.

[paulunderwood]

Are you going to take a punt at it, Serge

[science_man_88]

well as far as I can tell only prime indexes need to be checked because for example 123456 = 1001 *123 +333 and both 123 and 333 divide 3 and so on for the other primes but this is likely well known already, in fact it sounds familiar ! not much else I can think of that's useful to help though. edit: doh I forgot that it expands the number of digits faster than doubling for doubling n for the most part.

[wombatman]

There are no primes in the first 77000 terms. - Max Alekseyev, Oct 03 2015

Just wanted to this statement out as it's buried at the end of the comments on the OEIS page.

[science_man_88]

I realize now that 1,12,123 .. means each number adds it's value mod 3 to the sum of digits and so the sum of digits is only not divisible by 3 when you are at an index that is 3x+1 and this is only odd if x is even so we get 6y+1 and then when you follow this up you'll see the sequence 1,7,13,19,25 (25 can't work as it ends in 5), mod 30 and then if any of the endings can't work with an even number in front of them I'd say try that to eliminate tests but once again I'm probably randomly ranting again.

[danaj]

sm88, a half-decent primality test will find these cases (divisible by small values) quite easily, even at hundreds of thousands of digits. Filtering out values divisible by the first 10k or more small primes is also quite fast -- a trivial amount of time compared to running a PRP test. I'd be leery of excluding values to test unless one was quite sure of the math.

[science_man_88]

Originally Posted by danaj

sm88, a half-decent primality test will find these cases (divisible by small values) quite easily, even at hundreds of thousands of digits. Filtering out values divisible by the first 10k or more small primes is also quite fast -- a trivial amount of time compared to running a PRP test. I'd be leery of excluding values to test unless one was quite sure of the math.

well of my second post I can be certain because 12 is 0 mod 3 and 123 is 0 mod 3 and the numbers appended have sum digits that are 0 1 and 2 mod 3 0+1+2 =3 if you have a number that is like 123 appending a number that is 1 mod 3 lands you at 1 mod 3 and appending a number that is 2 mod 3 on that will land you at 0 mod 3 and likewise when you append a 0 mod 3 number after that so both 2 and 0 mod 3 are divisible by 3 that leaves only 1 mod 3 or 3x+1 which can only be odd ( the only indexes that can append an odd last digit when x is even leading to 6y+1 and then any of those that have y that are 4 mod 5 lead to a number that will end in 5 so only if y is 0,1,2,3 mod 5 will 6y+1 be a number that doesn't end in 5.

[Batalov]

Originally Posted by wombatman

Just wanted to this statement out as it's buried at the end of the comments on the OEIS page.

Yep, that's why I additionally linked to Mathworld (except for the link, the rest is verbatim from N.J.A.S.' posting to NUBTHRY).

I've previously taken a bite on up-and-down-concatentated primes. And just like for them, I'd like to note that for every 10^k <= n < 10^k+1, there is an explicit summation formula. Additionally, Capt.Obvious reports that only n ≡ 1,3,7,9 (mod 10) need to be considered. And n>=77000.

[LaurV]

Sum of digits of any 3 consecutive numbers is divisible by 3, so only 2/3 of the 1,3,5,7 (mod 10) have to be considered
This is what sm88 tries to tell you hehe, and you don't get it...

edit: pari goes quite fast to few thousands and stays there. Interesting that the "isprime" version is faster than other "probable prime" (modular) tests, because probably the builtin "isprime" filters the numbers which are obviously not primes (like even, etc). It can be seen how the "counter" progresses "in steps", jumping over big chunks of integers.

gp> i=1; n=1; while(!isprime(n),print(i++); n=eval(concat(Str(n),Str(i))))

edit2, this is also faster to generate the sequence than the version on oeis, because it doesn't do any calculus (one should print n instead of i).

[Batalov]

Originally Posted by LaurV

This is what sm88 tries to tell you hehe, and you don't get it...

By G*d, you are right!! I never get what sm88 tries to whatever... because...

[LaurV]

Bad Batalov! Bad! The mods Gods should not put people their sheep on ignore list, they should read everything, that is why they are mods Gods, to protect the other sheep, etc... with the function comes the responsibility


Just re-watched Jackie Brown, in which particular case Michael Keaton's name was the [STRIKE]Birdman[/STRIKE] Lord, as everyone most probably knows. It is a so much more mellow film :rolleyes: