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Let $p,q$ be primes with $p \leq q$. The product $2\cdot3\cdot\dots\cdot p$ is denoted with $p\#$, the product $2\cdot3\cdot\dots\cdot q$ is denoted with $q\#$ (primorials).

Now $z(p,q)$ is defined by $z(p,q) = p\#+q\#/p\#$

For example $z(11,17) = 2\cdot3\cdot5\cdot7\cdot11 + 13\cdot17$

What can be said about the prime factors of $z(p,q)$ besides the simple fact that they must be greater than $q$?

Stahl
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Peter
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    Probably nothing can be said. Do you have some reason for thinking there is something to say? – Gerry Myerson Jun 23 '13 at 07:58
  • A hard number of this form seems to be z(31,311) with 114 digits which I still have not factored. Perhaps someone can run msieve on it ... – Peter Jun 23 '13 at 18:51
  • 31#+311#/31# = 107480167189516405372388671076196134826345717005513453425763190407*9052304252819204826879243465754849774511655928277 (about 10 hours without any special tuning) – Peter Košinár Jun 28 '13 at 06:55
  • Thanks again! I wonder why my programs are so slow. Is it my machine (2,7Ghz) or must the programs be compiled? I use executable files, but I doubt if they are precompiled. Did you use a compiled or a executable version ? – Peter Jun 28 '13 at 09:48
  • z(307,331) and z(5,311) are the next two holes, the latter number having a small factor. Perhaps, someone joins in the factorizations ... – Peter Jul 14 '13 at 17:11
  • Incidentally, a misplaced "=" leads to the discovery that $z(\cancel{307}\ \mathit{293}, 331)$ is prime, which factors in surprisingly little time when a composite is expected. The other prime for $z(p, 331)$ occurs when $p = 199$. – user130144 Mar 07 '14 at 02:58
  • I notice $z(p, q)$ starts getting difficult to factor around $q = {73, 79}$, e.g. $z(2, 73)$, $z(11, 79)$. How quickly have those been factored, and by what methods? – user130144 Mar 07 '14 at 04:31

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The number $z(p,q)$ is coprime to any of these primes.

It is more likely to be prime, especially for small values, but not necessarily so. For example, $z(7,11) = 13*17$ is the smallest composite example, but one fairly easily finds composites (like $z(11,13)$, $z(5,19)$, $z(3,11)$, $z(13,19)$, $(13,23)$, and $z(13,37)$ to $z(13,59)$ inclusive. For $z(17,p)$, it is composite for all $p$ between $23$ and $113$ inclusive, only $19$ and $127$ yielded primes.

There is nothing exciting, like proper powers, for valeues of $q<120$.

There does not seem to be any particular pattern to the primes. Some are small, and some are fairly large.

wendy.krieger
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