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 strongpseudoprimes) :
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^((N1)/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." *** 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?... 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) *** 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 *** Paul K. Jasper, from Sydney, Australia wrote (20/6/2001):
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:
Mr Perez did it, and Eric himself wrote:
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. ***
___ * 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 




