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## Generalised factorial primes??

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• Using the ordinary equipment on my old home computer, I found that (13! + 2)/2 is prime. How many primes of the form (a! + n)/n [or (a! - n)/n] are actually
Message 1 of 3 , Dec 5, 2003
Using the ordinary equipment on my old home computer, I found that (13!
+ 2)/2 is prime.

How many primes of the form (a! + n)/n [or (a! - n)/n] are actually
known? Do you think there might by many to be found??
• Hi all, I just joined the group. I found it searching for a solution to a prime problem I had and am trying to prove. This caught my interest so i though i
Message 2 of 3 , Dec 5, 2003
Hi all,
I just joined the group. I found it searching for a solution to a prime
problem I had and am
trying to prove. This caught my interest so i though i would chime in.

>From: "julienbenney" <jpbenney@...>
>To: primenumbers@yahoogroups.com
>Subject: [PrimeNumbers] Generalised factorial primes??
>Date: Fri, 05 Dec 2003 11:32:54 -0000
>
>Using the ordinary equipment on my old home computer, I found that (13!
>+ 2)/2 is prime.
>
>How many primes of the form (a! + n)/n [or (a! - n)/n] are actually
>known? Do you think there might by many to be found??
Yes Many possiblt infinite. But watch out

From the Pari script below.
Here is a list n =1,2..50 and n=1,2 for sums

a n (a! + n)/n
1,1,2,
2,1,3,
2,2,2,
3,1,7,
4,2,13,
5,2,61,
7,2,2521,
8,2,20161,
11,1,39916801,
13,2,3113510401,
16,2,10461394944001,
27,1,10888869450418352160768000001,
30,2,132626429906095529318154240000001,
37,1,13763753091226345046315979581580902400000001,
41,1,33452526613163807108170062053440751665152000000001,
43,2,30207631531686917818677566034256998753632256000000001,
49,2,304140932017133780436126081660647688443776415689605120000000001,

Another list for n = 1,2..50 for diff
a n (a! - n)/n
3,1,5,
3,2,2,
4,1,23,
4,2,11,
5,2,59,
6,1,719,
6,2,359,
7,1,5039,
9,2,181439,
12,1,479001599,
14,1,87178291199,
30,1,265252859812191058636308479999999,
31,2,4111419327088961408862781439999999,
32,1,263130836933693530167218012159999999,
33,1,8683317618811886495518194401279999999,
38,1,523022617466601111760007224100074291199999999,
41,2,16726263306581903554085031026720375832575999999999,

Pari Script.
nfactp2d2(n,m) =
{
for(x=1,n,
for(k=1,m,
y=floor((x!+ k)/k);
if(isprime(y),print(x","k","y","))
)
)
}
nfactm2d2(n,m) =
{
for(x=1,n,
for(k=1,m,
y=floor((x!- k)/k);
if(isprime(y),print(x",",k","y","))
)
)
}
If you dont have Pari I recommend it over all the expensive programs
Maple,Mathematica etc
it is free and available at

http://pari.math.u-bordeaux.fr/

Also, it has prime proving capability isprime() in addition to the much
faster ispseudoprime.
The script language is c-like but much better than c in terms of use and of
course number
theory capability.

I don't know what your system is but if it is windows, I recommend you
download the binary
executable Pari.exe that will build all the files and folders for you. You
will need to modify the
environment path to include c:\program files\pari; This will enable you to
call gp.exe from other
folders.

I submitted some sequences to sloane's and referenced this email and your
question.

Have fun in Primelandia
Cino

Behind some primes are other primes
with other primes behind um.
And behind these primes
are still more primes
and so ad infinitum.

_________________________________________________________________
Our best dial-up offer is back. Get MSN Dial-up Internet Service for 6
months @ \$9.95/month now! http://join.msn.com/?page=dept/dialup
• ... for a from 2 to 30 and n from 2 to 30 I get the following primes of the form (a!+n)/n. There is apparently no shortage. a,n,(a!+n)/n 2, 2, 2 3, 3, 3 3, 6,
Message 3 of 3 , Dec 5, 2003
On Fri, 5 Dec 2003, julienbenney wrote:

> Using the ordinary equipment on my old home computer, I found that (13!
> + 2)/2 is prime.
>
> How many primes of the form (a! + n)/n [or (a! - n)/n] are actually
> known? Do you think there might by many to be found??
>

for a from 2 to 30 and n from 2 to 30 I get the following primes of the
form (a!+n)/n. There is apparently no shortage.

a,n,(a!+n)/n

2, 2, 2
3, 3, 3
3, 6, 2
4, 2, 13
4, 4, 7
4, 6, 5
4, 12, 3
4, 24, 2
5, 2, 61
5, 3, 41
5, 4, 31
5, 10, 13
5, 12, 11
5, 20, 7
5, 30, 5
6, 3, 241
6, 4, 181
6, 10, 73
6, 12, 61
6, 18, 41
6, 20, 37
6, 24, 31
7, 2, 2521
7, 5, 1009
7, 8, 631
7, 12, 421
7, 15, 337
7, 18, 281
7, 21, 241
7, 24, 211
7, 28, 181
8, 2, 20161
8, 3, 13441
8, 9, 4481
8, 12, 3361
8, 15, 2689
8, 16, 2521
8, 20, 2017
9, 5, 72577
9, 8, 45361
9, 12, 30241
9, 18, 20161
9, 24, 15121
9, 27, 13441
9, 30, 12097
10, 6, 604801
10, 8, 453601
10, 14, 259201
10, 15, 241921
10, 21, 172801
10, 24, 151201
10, 27, 134401
11, 3, 13305601
11, 5, 7983361
11, 6, 6652801
11, 7, 5702401
11, 10, 3991681
11, 20, 1995841
11, 28, 1425601
12, 8, 59875201
12, 10, 47900161
12, 12, 39916801
12, 16, 29937601
12, 21, 22809601
13, 2, 3113510401
13, 4, 1556755201
13, 6, 1037836801
13, 11, 566092801
13, 16, 389188801
14, 5, 17435658241
14, 6, 14529715201
14, 8, 10897286401
14, 16, 5448643201
14, 25, 3487131649
14, 28, 3113510401
14, 30, 2905943041
15, 7, 186810624001
15, 10, 130767436801
15, 24, 54486432001
16, 2, 10461394944001
16, 7, 2988969984001
16, 21, 996323328001
16, 22, 951035904001
17, 3, 118562476032001
17, 6, 59281238016001
17, 14, 25406244864001
17, 24, 14820309504001
17, 26, 13680285696001
17, 27, 13173608448001
18, 26, 246245142528001
19, 5, 24329020081766401
19, 15, 8109673360588801
19, 20, 6082255020441601
20, 8, 304112751022080001
20, 30, 81096733605888001
21, 4, 12772735542927360001
21, 6, 8515157028618240001
21, 11, 4644631106519040001
21, 15, 3406062811447296001
21, 19, 2688996956405760001
22, 16, 70250045486100480001
22, 25, 44960029111104307201
22, 26, 43230797222215680001
23, 3, 8617338912961658880001
23, 5, 5170403347776995328001
23, 8, 3231502092360622080001
23, 10, 2585201673888497664001
23, 15, 1723467782592331776001
23, 24, 1077167364120207360001
23, 27, 957482101440184320001
23, 30, 861733891296165888001
24, 8, 77556050216654929920001
24, 15, 41363226782215962624001
24, 22, 28202200078783610880001
24, 27, 22979570434564423680001
24, 28, 22158871490472837120001
24, 30, 20681613391107981312001
25, 4, 3877802510832746496000001
25, 7, 2215887149047283712000001
25, 26, 596585001666576384000001
26, 10, 40329146112660563558400001
28, 13, 23452949585516450807808000001
28, 28, 10888869450418352160768000001
29, 8, 1105220249217462744317952000001
29, 12, 736813499478308496211968000001
29, 15, 589450799582646796969574400001
30, 2, 132626429906095529318154240000001

trying again we have primes of the form (a!-n)/n

a, n, (a!-n)/n
3, 2, 2
4, 2, 11
4, 3, 7
4, 4, 5
4, 6, 3
4, 8, 2
5, 2, 59
5, 4, 29
5, 5, 23
5, 6, 19
5, 10, 11
5, 15, 7
5, 20, 5
5, 30, 3
6, 2, 359
6, 3, 239
6, 4, 179
6, 8, 89
6, 9, 79
6, 10, 71
6, 12, 59
6, 15, 47
6, 24, 29
6, 30, 23
7, 4, 1259
7, 6, 839
7, 7, 719
7, 10, 503
7, 12, 419
7, 14, 359
7, 20, 251
7, 21, 239
7, 28, 179
7, 30, 167
8, 4, 10079
8, 6, 6719
8, 8, 5039
8, 12, 3359
8, 14, 2879
8, 15, 2687
8, 18, 2239
8, 28, 1439
9, 2, 181439
9, 7, 51839
9, 14, 25919
9, 16, 22679
9, 20, 18143
9, 28, 12959
10, 4, 907199
10, 8, 453599
10, 12, 302399
10, 15, 241919
10, 16, 226799
10, 18, 201599
10, 20, 181439
10, 27, 134399
11, 5, 7983359
11, 6, 6652799
11, 8, 4989599
11, 10, 3991679
11, 12, 3326399
11, 15, 2661119
11, 30, 1330559
12, 3, 159667199
12, 5, 95800319
12, 14, 34214399
12, 28, 17107199
13, 10, 622702079
13, 13, 479001599
13, 15, 415134719
13, 30, 207567359
14, 6, 14529715199
14, 18, 4843238399
14, 26, 3353011199
15, 4, 326918591999
15, 9, 145297151999
15, 11, 118879487999
15, 12, 108972863999
15, 14, 93405311999
15, 15, 87178291199
15, 24, 54486431999
15, 25, 52306974719
16, 3, 6974263295999
16, 5, 4184557977599
16, 6, 3487131647999
16, 8, 2615348735999
16, 12, 1743565823999
16, 15, 1394852659199
16, 26, 804722687999
17, 6, 59281238015999
17, 9, 39520825343999
17, 10, 35568742809599
17, 25, 14227497123839
18, 4, 1600593426431999
18, 6, 1067062284287999
18, 9, 711374856191999
18, 14, 457312407551999
18, 15, 426824913715199
18, 16, 400148356607999
18, 24, 266765571071999
18, 30, 213412456857599
19, 8, 15205637551103999
19, 11, 11058645491711999
19, 18, 6758061133823999
19, 20, 6082255020441599
19, 21, 5792623828991999
19, 22, 5529322745855999
19, 28, 4344467871743999
20, 6, 405483668029439999
20, 7, 347557429739519999
20, 12, 202741834014719999
20, 13, 187146308321279999
20, 30, 81096733605887999
21, 9, 5676771352412159999
21, 14, 3649353012264959999
21, 19, 2688996956405759999
22, 25, 44960029111104307199
23, 4, 6463004184721244159999
23, 7, 3693145248412139519999
23, 18, 1436223152160276479999
23, 20, 1292600836944248831999
23, 21, 1231048416137379839999
23, 25, 1034080669555399065599
23, 28, 923286312103034879999
24, 28, 22158871490472837119999
25, 12, 1292600836944248831999999
25, 28, 553971787261820927999999
26, 11, 36662860102418694143999999
26, 22, 18331430051209347071999999
26, 24, 16803810880275234815999999
27, 9, 1209874383379816906751999999
27, 23, 473429106539928354815999999
28, 10, 30488834461171386050150399999
28, 18, 16938241367317436694527999999
28, 25, 12195533784468554420060159999
29, 3, 2947253997913233984847871999999
29, 9, 982417999304411328282623999999
29, 13, 680135537979977073426431999999
29, 18, 491208999652205664141311999999
29, 28, 315777214062132212662271999999
30, 9, 29472539979132339848478719999999
30, 27, 9824179993044113282826239999999
30, 29, 9146650338351415815045119999999

And for a = 100 and n = 2 to 100 Maple finds the following primes
of the form (a!+n)/n:

a, n, (a!+n)/n

100, 21,
4444105497330673937223773278869842880510284203065791498504426852153219047296662648044831617912215156488271329563201131345294533853183999999999999999999999999
100, 95,
982381215199412133491570935329123163060168086993490752300978567318079999928735953778331199749015981960565241271865513244749318009651199999999999999999999999
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