Is 1! + 11! + 300! prime?
Jonathan Post
jvospost3 at gmail.com
Tue Aug 15 00:43:42 CEST 2006
Prime Sums of Exactly Three Distinct Factorials
By
Jonathan Vos Post
10-11 Aug 2006, revisions of 12-14 Aug 2006
Acknowledgment:
The author would like to acknowledge corrections and suggestions by Dr.
George Hockney, Andrew Carmichael Post, Joshua Zucker, Gerald McGarvey,
Franklin T. Adams-Watters, Hugo Pfoertner, Ralf Stefan, Stefan
Steinerberger, and Joseph Biberstine.
For the purpose of this note, we do NOT consider 0!=1 and 1!=1 to be
distinct factorials.
The only prime factorial, of course, is 2!=2. The only way that the sum of
two distinct factorials can be prime is if the sum is of the form n! + 1,
the so-called "factorial primes"
A002981 Numbers n such that n! + 1 is prime { 0, 1, 2, 3, 11, 27, 37, 41,
73, 77, 116, …}
A088332 Primes of the form n!+1 { 2, 3, 7, 39916801,
10888869450418352160768000001, …}. A! + B! cannot be prime for A>1, B>1,
A=/=B because there would be at least one prime factor in common between A
and B.
It is well-known that there can be prime sums of three or more
factorials. These
are the prime subset of A059590 Sum of distinct factorials (0! and 1! not
treated as distinct)
{ 0, 1, 2, 3, 6, 7, 8, 9, 24, 25, 26, 27, 30, 31, 32, 33, 120, …}, namely:
A089359 Primes which can be partitioned into distinct factorials. 0! and 1!
are not considered distinct { 2, 3, 7, 31, 127, 151, 727, 751, 5167, …}. We
are interested in a specific subset of that.
We now tabularize and partially characterize Prime Sums of Exactly Three
Distinct Factorials.
Lemma [trivial proof]:
The prime sums of three distinct factorials must be of the form A! + B! + 1.
Lemma :
There are no solutions to "prime = 1! + 2! + n! for n>2."
[trivial proof]: For n>2, we have 3 | 1! + 2! + n! = 3 + n! and 1! + 2! + n!
> 3.
The smallest solutions are:
4! + 3! + 1! = 31 is prime.
5! + 3! + 1! = 127 is prime.
6! + 3! + 1! = 727 is prime.
Lemma :
There are no solutions to "prime = 1! + 3! + n! for n>6."
[trivial proof]: For n>6, we have 7 | 1! + 3! + n! = 7 + n! and 7 + n! > 7.
It is the case in general, for fixed A distinct from B, that A! + B! + 1! =
prime has either no solutions, or several solutions with the upper bound for
B being the least prime factor of
A! + 1 = A051301(A). [proof left to reader]. The number of such solutions
for A = 1, 2, 3, 4, … is: 0, 0, 0, 3, 0, 0, 0, 9, 4, 3, 0, …
Examples:
(a) There are no solutions to "prime = 1! + 4! + n! for n>4."
The upper bound for B is LPF(1 + 4!) = least prime factor (25) = 5, so we
need only check that 1! + 4! + 5! = 145 = 5 x 29 is divisible by 5, as is
the case with all higher B.
(b) There are no solutions to "prime = 1! + 5! + n! for n>5."
The upper bound for B is LPF(1 + 5!) = LPF(121 = 11^2) = 11. We check for
5<B<11 and find that 1 + 5! + 6! = 841 = 29^2; that 1+ 5! + 7! = 5161 = 13 x
397; that 1+ 5! + 8! = 40441 = 37 x 1093; that 1+ 5! + 9! = 363001 = 17 x
131 x 163; that 1+ 5! + 10! = 3628921 = 67 x 54163; and we know that 1+ 5! +
11! = 39916921 = 11 x 3 628811 and all higher values of B give multiples of
11.
(c) There are no solutions to "prime = 1! + 6! + n! for n>6."
The upper bound for B is LPF(1+ 6!) = LPF(721 = 7 x 103) = 7, so immediately
we are preclused from solution as this and all higher values of B give a sum
divisible by 7.
The next smallest solutions to "prime = A! + B! + 1! for distinct A, B > 6"
are:
8! + 7! + 1! = 45361 is prime.
12! + 7! + 1! = 479006641 is prime.
Or, more compactly:
7! + n! + 1! is prime for n = {8, 12, 16, 23, 27, 33, 37, 42, 53}. We need
check no further than 53<B<LPF(1 + 7! = 5041 =71^2) = 71, and we find no
solutions in that range. Hence there are exactly those 9 solutions to B for
"prime = 7! + B! + 1! for A<B."
I have used the "Alpertron" by Dario Alejandro Alpern to perform factorization
using the Elliptic Curve Method for "prime = A! + B! + 1! for distinct A, B,
where 0 < A<= 50."
In that range, the hardest minimal solution is for A = 50, namely 50! + 111!
+ 1!, a prime of 181 digits. Coincidently, 181 is prime.
The following table of prime A! + B! + 1! Where, for convenience, we order A
< B. We also show, in the 2nd column, the number of solutions if known, and
in the 3rd column the least prime factor of A!+1 (i.e. the upper bound on
B). For each nontrivial A, solutions have been searched for through B = 100
A! + B! + 1! is prime, A =/= B, A<B
A #solns Bmax B values in solutions
-------------------------------------------------------------------
1 0 2
2 0 3
3 3 7 4, 5, 6
4 0 5
5 0 11
6 0 7
7 9 71 8, 12, 16, 23, 27, 33, 37, 42, 53
8 4 61 14, 16, 18, 48
9 3 19 10, 13, 14
10 0 11
11 ? 39916801 46, 77, 85 [no more through B = 500]
12 0 13
13 5 83 21, 26, 29, 44, 45
14 4 23 16, 17, 18, 22
15 4 59 19, 20, 21, 29
16 0 17
17 3+ 661 46, 183, 560
18 0 19
19 6 73 23, 26, 38, 42, 45, 50
20 5+ 20639383 22, 24, 29, 32, 130, …
21 4 43 24, 32, 36, 39
22 0 23
23 5 47 26, 33, 34, 35, 43
24 2+ 811 26, 90, …
25 2+ 401 31, 83, …
26 3+ 1697 29, 36, 89, … [no more through 90]
27 2+ 10888869450418352160768000001 56, 61, …
28 0 29
29 5+ 14557 33, 37, 50, 62, 99, …
30 0 31
31 2+ 257 37, 87, …
32 2+ 2281 44, 97, …
33 1 67 39
34 1+ 67411 46 , …
35 1+ 137 76, …
36 0 37
37 1+ 13763753091226345046315979581580902400000001 90,
…
38 1+ 14029308060317546154181 62, …
39 1 79 70
40 0 41
41 1+ 33452526613163807108170062053440751665152000000001
52, …
42 0 43
43 1 59 56
44 1+ 1694763 87, …
45 1+ 293 52, …
46 0 47
47 2+ 6007711 50, 86, …
48 2+ 12893 62, 68, …
49 2+ 1021 76, 83, …
50 1+ 149 111, …
-------------------------------------------------------------------
We note the sequence of record values of smallest B as a function of
increasing A:
{4, 8, 14, 42, 46, 56, 76, 87, 111.
Meaning:
A=3, min B = 4;
A=7, min B = 8;
A=8, min B = 14;
A=11, min B = 42;
A=17, min B = 46;
A=27, min B = 56;
A=35, min B = 76;
A=44, min B = 87;
A=50, min B = 111.
There are other interesting integer sequences implicit in the major table.
Further investigations will be detailed in the future.
Sum3Factorials.doc
On 8/14/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
>
> Sequence (1) is problematic, in that each row is finite. If you fill
> with zeros, the table winds up being mostly zeros. Better, I think, to
> just put the solutions for a given n into the same row. Then you can
> advertise sequence (4) as the row lengths. You'll have to put the
> later values into an extension, since you don't know all the primes for
> some relatively small n.
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: Jonathan Post <jvospost3 at gmail.com>
>
> I agree, Hugo. I was thinking of 4 sequences:
> (1) The array of n and k such that n! + k! + 1 = a prime, n<k, read by
> antidiagonals;
> (2) a(n) = the least k such that n! + k! + 1 = a prime, or 0 if none;
> (3) a(n) = the greatest k such that n! + k! + 1 = a prime, or 0 if none;
> (4) a(n) the number of solutions, i.e. the number of k such that n! +
> k! + 1 = a prime, from 0 to upper bound A051301(n).
>
> Or is there a way to compress this into fewer sequences?
>
> I'm not saying that it's inherently interesting, but that it builds on
> existing OEIS seqs such as A002981, A088332, A089359, is elementary, is
> new, is not trivial, allows some number theory to be taught to
> non-science-major Math students such as I had for many semesters.
>
> I know quite a few 1 + A! + B! + C! primes but those don't compress
> into a few sequences, 3-D arrays not as easy as 2-D.
>
>
>
> On 8/14/06, Hugo Pfoertner <all at abouthugo.de> wrote:Joshua Zucker wrote:
>
> [...]
>
> > So someone
> > should still check 693 and 845 ...
>
> PFGW Version 1.2.0 for Windows [FFT v23.8]
> Primality testing 693!+11!+1 [N-1, Brillhart-Lehmer-Selfridge]
> Running N-1 test using base 13
> 693!+11!+1 is PRP! (69.7657s+0.0193s)
> Done.
>
> PFGW Version 1.2.0 for Windows [FFT v23.8]
> Primality testing 845!+11!+1 [N-1, Brillhart-Lehmer-Selfridge]
> Running N-1 test using base 13
> Running N-1 test using base 29
> 845!+11!+1 is PRP! (237.4787s+0.0485s)
> Done.
>
> The given times are on a 800 Mhz Athlon, running another job. I am
> currently preparing tests for the next round of Al Zimmermann's
> programming contest, which will be on self-avoiding zigzag paths
> ("matchstick snakes")
>
> >
> [...]
> >
> > For 1! + 17! + n!, my system produced the following candidate primes:
> > 11 14 46 183 560
>
> Confirmed with pfgw:
> 11!+1+17! is 3-PRP! (0.1049s+0.9518s )
> 14!+1+17! is 3-PRP! (0.0001s+0.0538s)
> 46!+1+17! is 3-PRP! (0.0003s+0.0415s)
> Switching to Exponentiating using Woltman FFT's
> 183!+1+17! is 3-PRP! (0.9605s+0.9951s)
> 560!+1+17! is 3-PRP! (30.1122s+6.2870s)
>
> checked:
> PFGW Version 1.2.0 for Windows [FFT v23.8]
> Primality testing 560!+17!+1 [N-1, Brillhart-Lehmer-Selfridge]
> Running N-1 test using base 19
> Running N-1 test using base 23
> 560!+17!+1 is PRP! (75.0765s+0.0597s )
> Done.
> Primality testing 183!+17!+1 [N-1, Brillhart-Lehmer-Selfridge]
> Running N-1 test using base 19
> Running N-1 test using base 23
> Running N-1 test using base 37
> 183!+17!+1 is PRP! (9.1043s+0.0516s)
> Done.
> >
> [...]
> > --Joshua Zucker
>
> To JVP:
> Please don't conclude from my contribution that I find this problem very
> interesting; I just wanted to confirm Joshua's results.
>
> What I definitely don't support is a series of new sequences:
>
> Numbers n such that n!+k!+1 is prime, for k=1...100.
>
> Hugo Pfoertner
>
>
>
>
>
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