Problems & Puzzles: Puzzles

Puzzle 8.- Primes by Listing

A "number by listing" is simple a string composed by a list of certain numbers. For example :

N=2468101214161820 is the number composed by all the even numbers from 2 to 20.

I have produced the following numbers and probed that they are primes (or at least strong-pseudo-primes) : 

Would you like to find larger primes of this "primes by listing" ? Please send them to add to this list.

Solution

Yves Gallot discovered (11/6/98) that the number 235711.....11927 - a list of prime numbers from 2 to 11927 - is prime. Here is his communication : 

"The number is 5719 digits long (you can perhaps verify this length, I am not absolutely sure). Time for one test it is very short, just about 3 minutes on my PII/266. a^((N-1)/2) = (a/N) = -1 (mod N) has been verified for more than 20 different values of a, then there is absolutely no doubt that 235711.....11927 is prime". 

Last news (20/6/98): 

Yves Gallot stop his search for a prime obtained listing natural numbers (*) at 8000. Here is an excerpt of his mail: 

"All the lists of natural numbers, from 1 to 8000 are composite. The number '12345...79998000' is more than 30,000 digits long. Then if a prime number is a list of natural numbers, it should have more than 30,000 digits."  

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Patrick De Geest proposes (14/07/98) this variation of the puzzle "Prime by listing", concerned with "natural numbers" : Instead of starting the natural number series with the number "1" start now with an arbitrary "k" number. 

Does anybody wants to keep looking?... 
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Felice Russo take this last challenge and here are his results (28/06/99):

(ki, kf, digits), (2, 27, 44), (3, 19, 27), (5, 17, 21), (5, 71, 129), (5, 99, 185),(5, 123, 257), (7, 127, 267), (8, 149, 332), (9, 187, 445), (11, 309, 808), (14, 47, 68), (16, 43, 56), (17, 39, 46), (23, 41, 38), (25, 49, 50), (26, 147, 292), (28, 73, 92), (102, 109, 24), (102, 139, 114), (103, 127, 75), (108, 127, 60)
He also listed palprimes:
(2, 11411, 76), (2, 37273, 276), (7, 383, 30), (7, 10501, 58), (7, 15451, 118),> (11, 10301, 52), (101, 1123211, 636)  (The three terms are  palindromes ), (131, 797, 36), (151, 383, 21), (151, 75557, 354), (181, 93239, 426)

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Patrick  De Geest has verified (30/9/2000) with a code in Ubasic by his own, that the prime found by Yves Gallot, 235711.....11927 is exactly composed by 5719 digits 

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Paul K. Jasper, from Sydney, Australia wrote (20/6/2001):

1) 8281807978...121110987654321 (all the decreasing integers from 82, 81, 80, 79, ..., 3,2,1)...This beautiful prime of this shape has been discovered by Ralf Stephan from Germany and confirmed by Eric W. Weisstein from USA. It has 155 digits. This is the only discovered prime so far from the Smarandache reverse sequence...  Ralf Stephan posted the paper on internet (it is still there) in 1996-1997, and was paper-published in Spring 1998; see below the paper reference.  You may also contact Ralf himself - see him on the internet.

Stephan, R. W., "Factors and Primes in Two Smarandache Sequences", <Smarandache Notions Journal>, Vol. 9, No. 1-2, second edition, 1998, 5-11.

2) A second record discovered by Eric Weisstein in puzzle 8:

There are infinitely many primes in the Smarandache concatenated prime sequence 2, 23, 235, 2357, 235711, 23571113, 2357111317, ... . (Not proved yet). Weisstein checked the first 1,624 terms and found the terms 1, 2, 4, 128, 174, 342, 435, 1429 primes.

Therefore the largest such Smarandache prime found by Weisstein is: 23571113171923...p(1429)

Regarding the Ralf Stephan's prime no doubt that he got it before me, in the middle of a more wide search for factors & primes of Smarandache concatenated numbers.

As the readers can see the second number (p(1429) = 11927) is the same discovered and reported by Gallot (11/6/98) and should be considered until now a probable prime and not a rigorous prime (I will ask to Eric when he got this number).

Jean Marie Charrier, from Lyon, France, wrote about this last issue:

"Reading the paper by Stephan, R. W., "Factors and Primes in Two Smarandache Sequences", <Smarandache Notions Journal>, Vol. 9, No. 1-2, second edition, Spring 1998, 5-11, in the footnotes of this paper it is specified that Mr. Weisstein has found a record of the largest Smarandache prime from the Smarandache concatenated sequence of primes, i.e. the number 23571113171923...p(1429), where p(1429) is the 1429-th prime. Thus Mr. Weisstein published first in the Spring 1998 his record, while Mr. Gallot sent you his result (according to your nice web site) on 6 November 1998. Mr. Perez, editor of "Smarandache Notions Journal", might confirm this too."

Mr Perez did it, and Eric himself wrote:

"I unfortunately was not date-stamping my searches at the time I found 2.3.5.7...p(1429) so I don't know the exact date, but I do know for a fact I found it before 1998-07-27, which is when I did start date-stamping, and by then I had reached a search limit of p(1955)."

So, about this number there is no doubt that it was discovered independently by Eric and by Yves the same year, 1998, but while Eric got it the spring Yves got it at the end of the year.

Regarding the number itself, being -as it is - a probable prime it's a good candidate to be tested as a rigorous prime by the Titanix in the near future.

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Farideh Firoozbakht wrote (May 2003):

If we set 2000 between each two elements of the set {82,81,80,...,3,2,1} and obtain a prime number with 479 digits

(m=822000812000802000...32000220001 is prime).

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* Nota Bene: Dr. M.L. Pérez (5/2/99) asks me to call these concatenation of natural numbers as "Smarandache Concatenated Numbers".

Please see http://www.gallup.unm.edu/~smarandache/ for the status of the factorization of these numbers