Primes of the Form N!1
PRIMES OF THE FORM N!1
Several years ago we first made the observation that certain the factorial terms N!+1 and N!-1 often have the form of primes such as 3!-1=5, 3!+1=7, 4!-1=23. We want in this note to show why such primes exist and that there probably are an infinite number of these prime sub-group members.
Let us begin by writing out the first fifteen integers and their factorial form and corresponding exponential vectors. We get the following table-
N N! Exponential Vector 1 1 [0 0 0 0 0 0 0 0] 2 2 [1 0 0 0 0 0 0 0] 3 6 [1 1 0 0 0 0 0 0] 4 24 [3 1 0 0 0 0 0 0] 5 120 [3 1 1 0 0 0 0 0] 6 720 [4 2 1 0 0 0 0 0] 7 5040 [4 2 1 1 0 0 0 0] 8 40320 [7 2 1 1 0 0 0 0] 9 362880 [7 4 1 1 0 0 0 0] 10 3628800 [8 4 2 1 0 0 0 0] 11 39916800 [8 4 2 1 1 0 0 0] 12 479001600 [10 5 2 1 1 0 0 0] 13 6227020800 [10 5 2 1 1 1 0 0] 14 87178291200 [11 5 2 2 1 1 0 0] 15 1307674368000 [11 6 3 2 1 1 0 0]
What is noticeable at once from the exponential vector forms shown is that N! is dominated by two taken to a positive power with three following at a lower power and five at a still lower power. We have encountered this type of behavior before when dealing with super-composites S. Such super-composites have the interesting property that their graph of f(S+n) versus n will have a peak when n=0. Here f(n) is the number fraction(NUMFRAC) defined as-
f(x)= [sigma(x)-x-1]/x
, with sigma(x) being the sum of divisors term of number theory. The interesting thing about NUMFRAC is that it vanishes when x is a prime and has values f(x)>>1 when x is a super-composite.
To test out the presence of a peak in an x versus f(x) plot for N! we choose x=11!=39916800. A plot follows-
This clearly shows the super-composite nature of 11!. The peak in the graph stems from the fact that 11! has a total of eleven divisors while its neighbor at 11!+1 is a prime so that f(11!+1)=0. The neighbor at 11!-1 has just six divisors 1,13,17,23,7853. and 39916800. The presence of a prime number in the immediate vicinity of a super-composite was noted in an earlier article and should continue to hold for N!1. We will call these the N! primes. To find them one carries out the search program-
for N from 1 to b do {N,isprime(N!-1),isprime(N!+1)}od;
In the present search we will stop with b=600 since the evaluations beyond this point takes too long for my computer program. Here are the results for the primes found-
For N!-1 Primes: N={3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, --}
For N!+1 Primes: N={1, 2, 3, 11, 27, 37, 41, 73,77, 116, 154, 320, 399, 427,..}
There is no obvious link between these primes other than to note that the number of N!+1 and N!-1primes are about the same, that the evaluation times and the prime spacings increase with N. The primes will become huge with larger N. So, for example, the prime
546!-1= 141302009261418325453762621393526286698658617796079910071353871488907640 504160743565501241236107733762080837422133568434484213467457560621538131 284091550631968816592748754460044234605778793037038634578162615257746896 342308546754776364849349150715175577496877924111556740124186753117294177 162303783405503829723023736326990734833969973121886593758921817486418079 092325492863765948445727040946819291138129804200523981866521636537165676
001267152151758174473660767772438139761519901508251733325317410707450984 627544775891183776230678501637671689726049915871033588824272610663202323 719389931283164345628668801970555064368945390634202638535930076803929062 184736870310291943624402875877497305289749392357778468456405062145336924 250773407066015135454416984550131804906976797622941301057536164271657477 854025104530247400241587703885280095204104624235632766825184645847127845 152855411530623429061372120607469651273815795437429759478958062539197279 629575104299984571584633649528868101236905793259525917436877798780909275 667980803203926354424450148098596386681966959342498581232330994696677186 100809764161773397785695845695608487026255462399999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999.
Using the Stirling approximation we find its value to be about [546!-1] sqrt(2546)546^(546)*exp(-546) -1=0.1412x10^1260. That this 1260 digit long number is a factorial prime is verified by the evaluation of isprime(546!-1). It took 69 seconds for our program to come up with the answer ‘yes’.
A good test of computer power would be to see the time it takes to carry out this last calculation. With more powerful super-computers many additional factorial primes will be found since one suspects that the total number of this subgroup of primes is infinite.
As a final remark, the thirty one factorial primes N!1 we have found have taken us up to a 1260 digit long number at 546!-1. This should be compared to Mersenne Primes where the 19th prime found is 2^(4253)-1 which has about 1281 digits. Hence the prime number density for factorial primes is larger than that for Mersenne Primes.
U.H.Kurzweg October 27, 2019 Gainesville, Florida