- 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 would chime in.

>From: "julienbenney" <jpbenney@...>

Yes Many possiblt infinite. But watch out

>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??

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 - On Fri, 5 Dec 2003, julienbenney wrote:

> Using the ordinary equipment on my old home computer, I found that (13!

for a from 2 to 30 and n from 2 to 30 I get the following primes of the

> + 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??

>

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