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 arent 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 :
(using the Caldwells 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 :
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
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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, et.al. 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:
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):
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:
Here is his complete report:
I asekd him: