Problems & Puzzles: Puzzles

Puzzle 10.- Primes associated to Primorials and Factorials

Jaime Ayala and myself - as probably many prime hunters world wide - have been looking for a formula - or set of formulas - that always offer the possibility of finding a prime. In this case we were looking for "big" primes (titanic) and with the added condition of providing enough known factors enable to make the rigorous primality test.

The most obvious candidates were the "primorials" just because they provide in themselves the needed factors. But is known that simple primorials aren’t a safe way to provide a prime.

After testing several variations around the primorials we arrived to the following couple of conjectures :

I.- Primes & Primorials :

"For every primorial pk# there is at least one pj from the primorial (1<=j<=k), such that at least one of the following expressions give us a prime :

N = pk# x pj +1

N = pk# x pj - 1

N = pk# / pj +1

N = pk# / pj - 1"

(using the Caldwell’s nomenclature, pk#=p1 x p2 x p3 x … x pk, p1=2, p2=3, p3=5,…)

After verifying the above conjecture, very naturally we asked ourselves if this conjecture remained for the factorials ; surprisingly the numbers answer was positive. This is the second conjecture :

II.- Primes & Factorials :

"For every factorial n! there is at least one nj from the factorial (1<=j<=k), such that at least one of the following expressions give us a prime :

N = n! x nj +1

N = n! x nj - 1

N = n! / nj +1

N = n! / nj - 1"

 The first conjecture has been exhaustively verified and confirmed from K=2 up to K=360, while the second one from K=2 to K=450

After the mentioned verification we jumped the K values of these conjectures to get the largest primes affordable with Ubasic 8.74 (2033 digits). This are the results :

N= p641#*p75 -1…….…….digits = 2033

N= p640#*p315 -1………….digits = 2030

N= p641#/p320 +1………….digits = 2027

N= p639#*p12 +1…………..digits = 2025

N= p640#/p136 -1 ………….digits = 2024

N= 819 !/39 +1 ………..….digits = 2031

N= 818 !*21 +1 ………..….digits = 2031

N= 817 !*165 -1 ………..….digits = 2029

N= 818 !/102 -1 ………..….digits = 2028

Maybe you would like :

  1. disapprove our conjectures
  2. get bigger primes (out of Ubasic) using our conjectures.

As said, every puzzle has his solver. This puzzle found to Phillip Poplin almost 3 years after being posted. Phillip has shown not only that our 2nd conjecture fails but he proposes a new conjecture in return.

According to Phillip the 2nd conjecture fails for n=2308. He used as tools "...the wonderful programs pfgw (primeform) by Chris Nash, and apsieve by M. Bell".

In return he proposes the following alternative conjecture:

"There exists a k such that N=k*n!+1 is prime."( Phillip's Conjecture)

And adds "It would be nice to put some bound on the maximum k-value required to try. Maybe someone with more math knowledge in number theory could help. For smaller (n=10000), I think a good bound is k<=4*n. I have been searching for these numbers for a few months, and have verified this to be true for n<=2650."


To tell the truth, I suppose that also our 1st Conjecture is false reaching to the proper primorial. Now that Phillip has shown the way to crack this nuts maybe someone else may try the 1st Conjecture of this puzzle.

Needles to say that the very Phillip's conjecture is now on the show.


False! The Phillip's conjecture is not a conjecture; it's just another way of state an old theorem... as was seen by J. K. Andersen and Luis Rodríguez, who wrote:

Phillips Conjecture doesn't exist, because is exactly equal to Dirichlet Theorem:
"In an arithmetical progression, a + rk, with (a,r)=1; k = 1,2,3,.. there are infinitely many primes." In this case Phillips took : a = 1 ; r = n!


David Kokales wrote (Set,. 2004): "I have tested conjecture #1 through the 1000th prime, namely 7919, and found that it holds true so far."


Bill McEachen wrote (June 07):

The desired solution stems from my June 2007 conjecture; this gives the following solution

Q=pk#(pj#)...(pz#)  +q   where Q is a "new" prime, q is a smaller (known) prime, and the product terms defined to match OEIS A129912 entries

thus a new prime is found by summing the product of multiple unique primorials with a smaller prime.

The primorial products must be from OEIS A129912 following from the conjecture

For example, the prime 189239 is found from  P3#*P13# + 9059
or  = 6 * 30030 + 9059


Steven Harvey reported, on Set. 2011, that the Conjecture #1 is false and that the smallest counter example is on p(5843)=53819, that is to say:

[Contrary to the Conjecture #1 statement]...for all the primes p(k), where k is <= 5483, the four equations produce composite values:


Here is his complete report:

I am writing to report that a counterexample has been found to the conjecture #1 in Puzzle ten of the Prime Puzzles and Problems Connection,

Mark Rodenkirch contributed cycles, custom sieves, and pfgw scripts, and deserves much of the credit. Other contributors include Anderson/Robinson(p21), Gary Barnes,Peter Benson(p67), D. Heuer(p16), T. Rajala, H. Rosenthal(p10); T. Sorbara, and Stephano D'Urso.

Details at

Steven Harvey,

I asekd him:

Can you provide other info: Total time spent in the search? Total CPU-time spent, etc.? When the counterexample was released? When has been confirmed by an indepenedent search?

His answer:

I have searched off and on for years, I am not sure when I started, By April of 2007, I was up to p(2700). After then, up to the present, I've had a core running on the problem; constantly since a coordinated search started on the Mersenne forums in November of 2010. I found it this Thursday, and ran a double-check, then realized I hadn't logged the double check so a third is currently running and logging.





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