Prime Sums of Exactly Three Distinct Factorials

[Jonathan Post]

Prime Sums of Exactly Three Distinct Factorials

By

Jonathan Vos Post

10-11 Aug 2006



These early results have been checked, but may still have errors. The author
would be grateful if these were pointed out.



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 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 partial 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 after which all higher A
gives the same common prime factor to A! + B! + 1!, and it is not clear
under what circumstances "several" means infinite.



Lemma [trivial proof]:

(a) There are no solutions to "prime = 1! + 4! + n! for n>4."

(b) There are no solutions to "prime = 1! + 5! + n! for n>5."

(c) There are no solutions to "prime = 1! + 6! + n! for n>6."



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, …}.



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, uses the convention that "0" means either that there is no solution at
all for the given A, and/or after the greatest given value in the row there
are no greater solutions.



A! + B! + 1! is prime, A =/= B, A<B



A \ B

-------------------------------------------------------------------

1          0

2          0

3          4, 5, 6, 0

4          0

5          0

6          0

7          8, 12, 16, 23, 27, 33, 37, 42, 53, … [no more through 70]

8          14, 16, 18, 48, … [no more through 60]

9          10, 13, 14, 0

10        0

11        42,… [no more through 70]

12        0

13        21, 26, 29, 44, 45, … [no more through 60]

14        16, 17, 18, 22, 0

15        19, 20, 21, 29, 0  [eventual common factor 59]

16        0

17        46, … [no more through 70]

18        0

19        23, 26, 38, 42, 45, 50, 0 [eventual common factor 73]

20        22, 24, 29, 32, … [no more through 70]

21        24, 32, 36, 39, 0 [eventual common factor 43]

22        0

23        26, 33, 34, 35, 43, 0 [eventual common factor 47]

24        26, … [no more through 60]

25        31, … [no more through 70]

26        29, 36, … [no more through 70]

27        56, 61, … [no more through 80]

28        0

29        33, 37, 50, 62, … [no more through 70]

30        0

31        37, … [no more through 70]

32        44, … [no more through 70]

33        39, 0 [eventual common factor 67]

34        46 , … [no more through 80]

35        76, …

36        0

37        90, …

38        62, …

39        70, …

40        0

41        52, … [no more through 70]

42        0

43        56, …

44        87, …

45        52, …

46        0

47        50, … [no more through 70]

48        62, …

49        76, …

50        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
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://list.seqfan.eu/pipermail/seqfan/attachments/20060811/0b8d91f5/attachment-0001.htm>