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-   -   Factorial and Goldbach conjecture. (https://www.mersenneforum.org/showthread.php?t=22969)

 MisterBitcoin 2018-01-26 21:29

Factorial and Goldbach conjecture.

Factorial numbers are interesting, same for the goldbach conjecture.
Let´s make something with these to themes. What turns out is a PRP search. (for the higher n!-values. ;) )

Smallest combination for 20!: 31+P19; [URL="http://factordb.com/index.php?id=1100000000036429676"]here [/URL]
Smallest combination for 21!: 31+P20; [URL="http://factordb.com/index.php?id=1100000000455761720"]here [/URL]
Smallest combination for 22!: 73+P22; [URL="http://factordb.com/index.php?id=1100000000929900911"]here [/URL]
and so on...

I´ll build an script to do this kind of "work", I won´t waste FactorDB resources.

 kar_bon 2018-01-26 23:06

Quick shot: 261 primes/PRPs for 20!-n, 1<=n<=10000

 a1call 2018-01-27 02:03

Pari-gp code:

BZC-200-A by Rashid Naimi (a1call)
Goldbach Conjecture for n!:
[URL]https://artofproblemsolving.com/wiki/index.php?title=Goldbach_Conjecture[/URL]
Every factorial greater than 2 is the sum of 2 distinct primes.
Can be proven using my Theorem 1:
[URL]http://www.mersenneforum.org/showpost.php?p=413772&postcount=39[/URL]
If it can be shown/proven that there always exists a prime P such that
n!-n^2 < P < n!
for all integers n greater than 2.
:smile:
[CODE]

print("\nBZC-200-A by Rashid Naimi (a1call)")
print("Goldbach Conjecture for n!:")
print("https://artofproblemsolving.com/wiki/index.php?title=Goldbach_Conjecture");
print("Every factorial greater than 2 is the sum of 2 distinct primes.")
print("Can be proven using my Theorem 1:")
print("http://www.mersenneforum.org/showpost.php?p=413772&postcount=39")
print("If it can be shown/proven that there always exists a prime P such that")
print("n!-n^2 < P < n!")
print("for all integers n greater than 2.")

UpperBoundLimit = 19^2 \\ Set your desired Upper Bound for the n!
for( theFactorial =3,UpperBoundLimit ,{
prime1 =nextprime(abs(theFactorial! -theFactorial ^2));
prime2 =theFactorial! - prime1;
\\print("\nn**** " ,theFactorial,"! = ",prime1 ," + ",prime2," ****" );\\\\Un-Commemt to print prime1
print("\n**** " ,theFactorial,"! = P",#digits(prime1) ," + ",prime2," ****" );
print(prime2," is guaranteed to be prime: ",isprime(prime2) );
if(!isprime(prime2),break(19););
})

[/CODE][CODE]
BZC-200-A by Rashid Naimi (a1call)
Goldbach Conjecture for n!:
https://artofproblemsolving.com/wiki/index.php?title=Goldbach_Conjecture
Every factorial greater than 2 is the sum of 2 distinct primes.
Can be proven using my Theorem 1:
http://www.mersenneforum.org/showpost.php?p=413772&postcount=39
If it can be shown/proven that there always exists a prime P such that
n!-n^2 < P < n!
for all integers n greater than 2.
361

**** 3! = P1 + 3 ****
3 is guaranteed to be prime: 1

**** 4! = P2 + 13 ****
13 is guaranteed to be prime: 1

**** 5! = P2 + 23 ****
23 is guaranteed to be prime: 1

**** 6! = P3 + 29 ****
29 is guaranteed to be prime: 1

**** 7! = P4 + 47 ****
47 is guaranteed to be prime: 1

**** 8! = P5 + 43 ****
43 is guaranteed to be prime: 1

**** 9! = P6 + 79 ****
79 is guaranteed to be prime: 1

**** 10! = P7 + 89 ****
89 is guaranteed to be prime: 1

**** 11! = P8 + 113 ****
113 is guaranteed to be prime: 1

**** 12! = P9 + 139 ****
139 is guaranteed to be prime: 1

**** 13! = P10 + 163 ****
163 is guaranteed to be prime: 1

**** 14! = P11 + 131 ****
131 is guaranteed to be prime: 1

**** 15! = P13 + 167 ****
167 is guaranteed to be prime: 1

**** 16! = P14 + 239 ****
239 is guaranteed to be prime: 1

**** 17! = P15 + 281 ****
281 is guaranteed to be prime: 1

**** 18! = P16 + 317 ****
317 is guaranteed to be prime: 1

**** 19! = P18 + 359 ****
359 is guaranteed to be prime: 1

**** 20! = P19 + 397 ****
397 is guaranteed to be prime: 1

**** 21! = P20 + 383 ****
383 is guaranteed to be prime: 1

**** 22! = P22 + 463 ****
463 is guaranteed to be prime: 1

**** 23! = P23 + 373 ****
373 is guaranteed to be prime: 1

**** 24! = P24 + 571 ****
571 is guaranteed to be prime: 1

**** 25! = P26 + 617 ****
617 is guaranteed to be prime: 1

**** 26! = P27 + 479 ****
479 is guaranteed to be prime: 1

**** 27! = P29 + 571 ****
571 is guaranteed to be prime: 1

**** 28! = P30 + 383 ****
383 is guaranteed to be prime: 1

**** 29! = P31 + 769 ****
769 is guaranteed to be prime: 1

**** 30! = P33 + 739 ****
739 is guaranteed to be prime: 1

**** 31! = P34 + 853 ****
853 is guaranteed to be prime: 1

**** 32! = P36 + 1021 ****
1021 is guaranteed to be prime: 1

**** 33! = P37 + 523 ****
523 is guaranteed to be prime: 1

**** 34! = P39 + 1153 ****
1153 is guaranteed to be prime: 1

**** 35! = P41 + 1123 ****
1123 is guaranteed to be prime: 1

**** 36! = P42 + 1283 ****
1283 is guaranteed to be prime: 1

**** 37! = P44 + 1193 ****
1193 is guaranteed to be prime: 1

**** 38! = P45 + 1297 ****
1297 is guaranteed to be prime: 1

**** 39! = P47 + 1483 ****
1483 is guaranteed to be prime: 1

**** 40! = P48 + 1531 ****
1531 is guaranteed to be prime: 1

**** 41! = P50 + 1597 ****
1597 is guaranteed to be prime: 1

**** 42! = P52 + 1669 ****
1669 is guaranteed to be prime: 1

**** 43! = P53 + 1693 ****
1693 is guaranteed to be prime: 1

**** 44! = P55 + 1871 ****
1871 is guaranteed to be prime: 1

**** 45! = P57 + 2017 ****
2017 is guaranteed to be prime: 1

**** 46! = P58 + 1733 ****
1733 is guaranteed to be prime: 1

**** 47! = P60 + 2053 ****
2053 is guaranteed to be prime: 1

**** 48! = P62 + 2143 ****
2143 is guaranteed to be prime: 1

**** 49! = P63 + 2287 ****
2287 is guaranteed to be prime: 1

**** 50! = P65 + 2347 ****
2347 is guaranteed to be prime: 1

**** 51! = P67 + 1583 ****
1583 is guaranteed to be prime: 1

**** 52! = P68 + 2671 ****
2671 is guaranteed to be prime: 1

**** 53! = P70 + 2729 ****
2729 is guaranteed to be prime: 1

**** 54! = P72 + 2897 ****
2897 is guaranteed to be prime: 1

**** 55! = P74 + 2791 ****
2791 is guaranteed to be prime: 1

**** 56! = P75 + 3019 ****
3019 is guaranteed to be prime: 1

**** 57! = P77 + 3221 ****
3221 is guaranteed to be prime: 1

**** 58! = P79 + 2543 ****
2543 is guaranteed to be prime: 1

**** 59! = P81 + 2843 ****
2843 is guaranteed to be prime: 1

**** 60! = P82 + 3593 ****
3593 is guaranteed to be prime: 1

**** 61! = P84 + 3319 ****
3319 is guaranteed to be prime: 1

**** 62! = P86 + 3533 ****
3533 is guaranteed to be prime: 1

**** 63! = P88 + 3907 ****
3907 is guaranteed to be prime: 1

**** 64! = P90 + 4003 ****
4003 is guaranteed to be prime: 1

**** 65! = P91 + 4153 ****
4153 is guaranteed to be prime: 1

**** 66! = P93 + 4073 ****
4073 is guaranteed to be prime: 1

**** 67! = P95 + 4261 ****
4261 is guaranteed to be prime: 1

**** 68! = P97 + 4457 ****
4457 is guaranteed to be prime: 1

**** 69! = P99 + 4663 ****
4663 is guaranteed to be prime: 1

**** 70! = P101 + 4649 ****
4649 is guaranteed to be prime: 1

**** 71! = P102 + 5003 ****
5003 is guaranteed to be prime: 1

**** 72! = P104 + 5101 ****
5101 is guaranteed to be prime: 1

**** 73! = P106 + 5279 ****
5279 is guaranteed to be prime: 1

**** 74! = P108 + 5153 ****
5153 is guaranteed to be prime: 1

**** 75! = P110 + 5519 ****
5519 is guaranteed to be prime: 1

**** 76! = P112 + 5743 ****
5743 is guaranteed to be prime: 1

**** 77! = P114 + 5441 ****
5441 is guaranteed to be prime: 1

**** 78! = P116 + 5573 ****
5573 is guaranteed to be prime: 1

**** 79! = P117 + 6221 ****
6221 is guaranteed to be prime: 1

**** 80! = P119 + 6379 ****
6379 is guaranteed to be prime: 1

**** 81! = P121 + 6007 ****
6007 is guaranteed to be prime: 1

**** 82! = P123 + 6451 ****
6451 is guaranteed to be prime: 1

**** 83! = P125 + 6737 ****
6737 is guaranteed to be prime: 1

**** 84! = P127 + 7043 ****
7043 is guaranteed to be prime: 1

**** 85! = P129 + 7069 ****
7069 is guaranteed to be prime: 1

**** 86! = P131 + 6263 ****
6263 is guaranteed to be prime: 1

**** 87! = P133 + 6949 ****
6949 is guaranteed to be prime: 1

**** 88! = P135 + 7673 ****
7673 is guaranteed to be prime: 1

**** 89! = P137 + 7283 ****
7283 is guaranteed to be prime: 1

**** 90! = P139 + 7879 ****
7879 is guaranteed to be prime: 1

**** 91! = P141 + 7681 ****
7681 is guaranteed to be prime: 1

**** 92! = P143 + 8273 ****
8273 is guaranteed to be prime: 1

**** 93! = P145 + 8389 ****
8389 is guaranteed to be prime: 1

**** 94! = P147 + 8219 ****
8219 is guaranteed to be prime: 1

**** 95! = P149 + 8971 ****
8971 is guaranteed to be prime: 1

**** 96! = P150 + 9157 ****
9157 is guaranteed to be prime: 1

**** 97! = P152 + 9043 ****
9043 is guaranteed to be prime: 1

**** 98! = P154 + 8971 ****
8971 is guaranteed to be prime: 1

**** 99! = P156 + 9349 ****
9349 is guaranteed to be prime: 1

**** 100! = P158 + 9689 ****
9689 is guaranteed to be prime: 1

**** 101! = P160 + 9857 ****
9857 is guaranteed to be prime: 1

**** 102! = P162 + 10331 ****
10331 is guaranteed to be prime: 1

**** 103! = P164 + 10151 ****
10151 is guaranteed to be prime: 1

**** 104! = P167 + 10501 ****
10501 is guaranteed to be prime: 1

**** 105! = P169 + 10691 ****
10691 is guaranteed to be prime: 1

**** 106! = P171 + 11171 ****
11171 is guaranteed to be prime: 1

**** 107! = P173 + 10733 ****
10733 is guaranteed to be prime: 1

**** 108! = P175 + 11579 ****
11579 is guaranteed to be prime: 1

**** 109! = P177 + 11257 ****
11257 is guaranteed to be prime: 1

**** 110! = P179 + 11807 ****
11807 is guaranteed to be prime: 1

**** 111! = P181 + 12251 ****
12251 is guaranteed to be prime: 1

**** 112! = P183 + 12227 ****
12227 is guaranteed to be prime: 1

**** 113! = P185 + 12757 ****
12757 is guaranteed to be prime: 1

**** 114! = P187 + 12149 ****
12149 is guaranteed to be prime: 1

**** 115! = P189 + 13219 ****
13219 is guaranteed to be prime: 1

**** 116! = P191 + 12979 ****
12979 is guaranteed to be prime: 1

**** 117! = P193 + 12577 ****
12577 is guaranteed to be prime: 1

**** 118! = P195 + 13807 ****
13807 is guaranteed to be prime: 1

**** 119! = P197 + 14153 ****
14153 is guaranteed to be prime: 1

**** 120! = P199 + 14197 ****
14197 is guaranteed to be prime: 1

**** 121! = P201 + 14557 ****
14557 is guaranteed to be prime: 1

**** 122! = P203 + 13907 ****
13907 is guaranteed to be prime: 1

**** 123! = P206 + 15107 ****
15107 is guaranteed to be prime: 1

**** 124! = P208 + 13297 ****
13297 is guaranteed to be prime: 1

**** 125! = P210 + 15473 ****
15473 is guaranteed to be prime: 1

**** 126! = P212 + 15319 ****
15319 is guaranteed to be prime: 1

**** 127! = P214 + 15923 ****
15923 is guaranteed to be prime: 1

**** 128! = P216 + 15739 ****
15739 is guaranteed to be prime: 1

**** 129! = P218 + 16111 ****
16111 is guaranteed to be prime: 1

**** 130! = P220 + 16811 ****
16811 is guaranteed to be prime: 1

**** 131! = P222 + 16901 ****
16901 is guaranteed to be prime: 1

**** 132! = P225 + 17123 ****
17123 is guaranteed to be prime: 1

**** 133! = P227 + 17681 ****
17681 is guaranteed to be prime: 1

**** 134! = P229 + 17851 ****
17851 is guaranteed to be prime: 1

**** 135! = P231 + 16187 ****
16187 is guaranteed to be prime: 1

**** 136! = P233 + 18457 ****
18457 is guaranteed to be prime: 1

**** 137! = P235 + 18133 ****
18133 is guaranteed to be prime: 1

**** 138! = P237 + 18169 ****
18169 is guaranteed to be prime: 1

**** 139! = P239 + 18973 ****
18973 is guaranteed to be prime: 1

**** 140! = P242 + 19471 ****
19471 is guaranteed to be prime: 1

**** 141! = P244 + 19751 ****
19751 is guaranteed to be prime: 1

**** 142! = P246 + 18959 ****
18959 is guaranteed to be prime: 1

**** 143! = P248 + 19477 ****
19477 is guaranteed to be prime: 1

**** 144! = P250 + 20507 ****
20507 is guaranteed to be prime: 1

**** 145! = P252 + 20921 ****
20921 is guaranteed to be prime: 1

**** 146! = P255 + 21107 ****
21107 is guaranteed to be prime: 1

**** 147! = P257 + 20963 ****
20963 is guaranteed to be prime: 1

**** 148! = P259 + 20641 ****
20641 is guaranteed to be prime: 1

**** 149! = P261 + 22109 ****
22109 is guaranteed to be prime: 1

**** 150! = P263 + 20089 ****
20089 is guaranteed to be prime: 1

**** 151! = P265 + 22147 ****
22147 is guaranteed to be prime: 1

**** 152! = P268 + 22073 ****
22073 is guaranteed to be prime: 1

**** 153! = P270 + 23087 ****
23087 is guaranteed to be prime: 1

**** 154! = P272 + 23539 ****
23539 is guaranteed to be prime: 1

**** 155! = P274 + 23689 ****
23689 is guaranteed to be prime: 1

**** 156! = P276 + 23719 ****
23719 is guaranteed to be prime: 1

**** 157! = P279 + 23893 ****
23893 is guaranteed to be prime: 1

**** 158! = P281 + 22637 ****
22637 is guaranteed to be prime: 1

**** 159! = P283 + 25247 ****
25247 is guaranteed to be prime: 1

**** 160! = P285 + 25541 ****
25541 is guaranteed to be prime: 1

**** 161! = P287 + 25799 ****
25799 is guaranteed to be prime: 1

**** 162! = P290 + 24889 ****
24889 is guaranteed to be prime: 1

**** 163! = P292 + 26177 ****
26177 is guaranteed to be prime: 1

**** 164! = P294 + 26141 ****
26141 is guaranteed to be prime: 1

**** 165! = P296 + 25717 ****
25717 is guaranteed to be prime: 1

**** 166! = P298 + 27541 ****
27541 is guaranteed to be prime: 1

**** 167! = P301 + 27653 ****
27653 is guaranteed to be prime: 1

**** 168! = P303 + 27103 ****
27103 is guaranteed to be prime: 1

**** 169! = P305 + 28283 ****
28283 is guaranteed to be prime: 1

**** 170! = P307 + 28409 ****
28409 is guaranteed to be prime: 1

**** 171! = P310 + 28669 ****
28669 is guaranteed to be prime: 1

**** 172! = P312 + 28621 ****
28621 is guaranteed to be prime: 1

**** 173! = P314 + 29819 ****
29819 is guaranteed to be prime: 1

**** 174! = P316 + 30071 ****
30071 is guaranteed to be prime: 1

**** 175! = P319 + 30181 ****
30181 is guaranteed to be prime: 1

**** 176! = P321 + 30941 ****
30941 is guaranteed to be prime: 1

**** 177! = P323 + 31189 ****
31189 is guaranteed to be prime: 1

**** 178! = P325 + 31567 ****
31567 is guaranteed to be prime: 1

**** 179! = P328 + 31181 ****
31181 is guaranteed to be prime: 1

**** 180! = P330 + 31513 ****
31513 is guaranteed to be prime: 1

**** 181! = P332 + 32579 ****
32579 is guaranteed to be prime: 1

**** 182! = P334 + 32323 ****
32323 is guaranteed to be prime: 1

**** 183! = P337 + 33479 ****
33479 is guaranteed to be prime: 1

**** 184! = P339 + 33827 ****
33827 is guaranteed to be prime: 1

**** 185! = P341 + 33581 ****
33581 is guaranteed to be prime: 1

**** 186! = P343 + 33871 ****
33871 is guaranteed to be prime: 1

**** 187! = P346 + 33599 ****
33599 is guaranteed to be prime: 1

**** 188! = P348 + 35251 ****
35251 is guaranteed to be prime: 1

**** 189! = P350 + 35279 ****
35279 is guaranteed to be prime: 1

**** 190! = P352 + 34897 ****
34897 is guaranteed to be prime: 1

**** 191! = P355 + 35107 ****
35107 is guaranteed to be prime: 1

**** 192! = P357 + 36229 ****
36229 is guaranteed to be prime: 1

**** 193! = P359 + 37123 ****
37123 is guaranteed to be prime: 1

**** 194! = P362 + 36467 ****
36467 is guaranteed to be prime: 1

**** 195! = P364 + 37561 ****
37561 is guaranteed to be prime: 1

**** 196! = P366 + 37879 ****
37879 is guaranteed to be prime: 1

**** 197! = P369 + 38767 ****
38767 is guaranteed to be prime: 1

**** 198! = P371 + 38317 ****
38317 is guaranteed to be prime: 1

**** 199! = P373 + 38447 ****
38447 is guaranteed to be prime: 1

**** 200! = P375 + 39373 ****
39373 is guaranteed to be prime: 1

**** 201! = P378 + 39443 ****
39443 is guaranteed to be prime: 1

**** 202! = P380 + 37693 ****
37693 is guaranteed to be prime: 1

**** 203! = P382 + 40829 ****
40829 is guaranteed to be prime: 1

**** 204! = P385 + 40237 ****
40237 is guaranteed to be prime: 1

**** 205! = P387 + 41257 ****
41257 is guaranteed to be prime: 1

**** 206! = P389 + 42299 ****
42299 is guaranteed to be prime: 1

**** 207! = P392 + 41941 ****
41941 is guaranteed to be prime: 1

**** 208! = P394 + 42863 ****
42863 is guaranteed to be prime: 1

**** 209! = P396 + 43481 ****
43481 is guaranteed to be prime: 1

**** 210! = P399 + 44087 ****
44087 is guaranteed to be prime: 1

**** 211! = P401 + 44017 ****
44017 is guaranteed to be prime: 1

**** 212! = P403 + 44371 ****
44371 is guaranteed to be prime: 1

**** 213! = P406 + 44797 ****
44797 is guaranteed to be prime: 1

**** 214! = P408 + 45317 ****
45317 is guaranteed to be prime: 1

**** 215! = P410 + 43801 ****
43801 is guaranteed to be prime: 1

**** 216! = P413 + 46229 ****
46229 is guaranteed to be prime: 1

**** 217! = P415 + 45569 ****
45569 is guaranteed to be prime: 1

**** 218! = P417 + 46471 ****
46471 is guaranteed to be prime: 1

**** 219! = P420 + 47059 ****
47059 is guaranteed to be prime: 1

**** 220! = P422 + 45863 ****
45863 is guaranteed to be prime: 1

**** 221! = P424 + 47543 ****
47543 is guaranteed to be prime: 1

**** 222! = P427 + 47207 ****
47207 is guaranteed to be prime: 1

**** 223! = P429 + 49261 ****
49261 is guaranteed to be prime: 1

**** 224! = P431 + 49613 ****
49613 is guaranteed to be prime: 1

**** 225! = P434 + 50527 ****
50527 is guaranteed to be prime: 1

**** 226! = P436 + 50459 ****
50459 is guaranteed to be prime: 1

**** 227! = P438 + 51307 ****
51307 is guaranteed to be prime: 1

**** 228! = P441 + 49747 ****
49747 is guaranteed to be prime: 1

**** 229! = P443 + 49081 ****
49081 is guaranteed to be prime: 1

**** 230! = P445 + 52783 ****
52783 is guaranteed to be prime: 1

**** 231! = P448 + 51893 ****
51893 is guaranteed to be prime: 1

**** 232! = P450 + 53411 ****
53411 is guaranteed to be prime: 1

**** 233! = P452 + 53939 ****
53939 is guaranteed to be prime: 1

**** 234! = P455 + 54409 ****
54409 is guaranteed to be prime: 1

**** 235! = P457 + 54049 ****
54049 is guaranteed to be prime: 1

**** 236! = P460 + 52541 ****
52541 is guaranteed to be prime: 1

**** 237! = P462 + 55219 ****
55219 is guaranteed to be prime: 1

**** 238! = P464 + 54919 ****
54919 is guaranteed to be prime: 1

**** 239! = P467 + 51473 ****
51473 is guaranteed to be prime: 1

**** 240! = P469 + 56611 ****
56611 is guaranteed to be prime: 1

**** 241! = P471 + 57787 ****
57787 is guaranteed to be prime: 1

**** 242! = P474 + 57709 ****
57709 is guaranteed to be prime: 1

**** 243! = P476 + 58049 ****
58049 is guaranteed to be prime: 1

**** 244! = P479 + 58027 ****
58027 is guaranteed to be prime: 1

**** 245! = P481 + 58543 ****
58543 is guaranteed to be prime: 1

**** 246! = P483 + 54799 ****
54799 is guaranteed to be prime: 1

**** 247! = P486 + 59621 ****
59621 is guaranteed to be prime: 1

**** 248! = P488 + 60821 ****
60821 is guaranteed to be prime: 1

**** 249! = P491 + 57737 ****
57737 is guaranteed to be prime: 1

**** 250! = P493 + 61981 ****
61981 is guaranteed to be prime: 1

**** 251! = P495 + 62473 ****
62473 is guaranteed to be prime: 1

**** 252! = P498 + 63029 ****
63029 is guaranteed to be prime: 1

**** 253! = P500 + 62861 ****
62861 is guaranteed to be prime: 1

**** 254! = P503 + 63559 ****
63559 is guaranteed to be prime: 1

**** 255! = P505 + 64877 ****
64877 is guaranteed to be prime: 1

**** 256! = P507 + 65183 ****
65183 is guaranteed to be prime: 1

**** 257! = P510 + 60133 ****
60133 is guaranteed to be prime: 1

**** 258! = P512 + 66553 ****
66553 is guaranteed to be prime: 1

**** 259! = P515 + 66047 ****
66047 is guaranteed to be prime: 1

**** 260! = P517 + 66587 ****
66587 is guaranteed to be prime: 1

**** 261! = P519 + 66499 ****
66499 is guaranteed to be prime: 1

**** 262! = P522 + 67339 ****
67339 is guaranteed to be prime: 1

**** 263! = P524 + 63197 ****
63197 is guaranteed to be prime: 1

**** 264! = P527 + 69491 ****
69491 is guaranteed to be prime: 1

**** 265! = P529 + 67547 ****
67547 is guaranteed to be prime: 1

**** 266! = P532 + 69257 ****
69257 is guaranteed to be prime: 1

**** 267! = P534 + 69847 ****
69847 is guaranteed to be prime: 1

**** 268! = P536 + 67651 ****
67651 is guaranteed to be prime: 1

**** 269! = P539 + 69929 ****
69929 is guaranteed to be prime: 1

**** 270! = P541 + 72689 ****
72689 is guaranteed to be prime: 1

**** 271! = P544 + 72031 ****
72031 is guaranteed to be prime: 1

**** 272! = P546 + 73783 ****
73783 is guaranteed to be prime: 1

**** 273! = P549 + 72931 ****
72931 is guaranteed to be prime: 1

**** 274! = P551 + 74279 ****
74279 is guaranteed to be prime: 1

**** 275! = P554 + 72043 ****
72043 is guaranteed to be prime: 1

**** 276! = P556 + 72869 ****
72869 is guaranteed to be prime: 1

**** 277! = P558 + 75323 ****
75323 is guaranteed to be prime: 1

**** 278! = P561 + 76543 ****
76543 is guaranteed to be prime: 1

**** 279! = P563 + 77773 ****
77773 is guaranteed to be prime: 1

**** 280! = P566 + 74587 ****
74587 is guaranteed to be prime: 1

**** 281! = P568 + 77969 ****
77969 is guaranteed to be prime: 1

**** 282! = P571 + 79153 ****
79153 is guaranteed to be prime: 1

**** 283! = P573 + 79201 ****
79201 is guaranteed to be prime: 1

**** 284! = P576 + 78193 ****
78193 is guaranteed to be prime: 1

**** 285! = P578 + 80191 ****
80191 is guaranteed to be prime: 1

**** 286! = P580 + 77849 ****
77849 is guaranteed to be prime: 1

**** 287! = P583 + 81547 ****
81547 is guaranteed to be prime: 1

**** 288! = P585 + 79829 ****
79829 is guaranteed to be prime: 1

**** 289! = P588 + 83071 ****
83071 is guaranteed to be prime: 1

**** 290! = P590 + 82267 ****
82267 is guaranteed to be prime: 1

**** 291! = P593 + 83689 ****
83689 is guaranteed to be prime: 1

**** 292! = P595 + 84211 ****
84211 is guaranteed to be prime: 1

**** 293! = P598 + 85661 ****
85661 is guaranteed to be prime: 1

**** 294! = P600 + 85091 ****
85091 is guaranteed to be prime: 1

**** 295! = P603 + 86843 ****
86843 is guaranteed to be prime: 1

**** 296! = P605 + 86729 ****
86729 is guaranteed to be prime: 1

**** 297! = P608 + 85667 ****
85667 is guaranteed to be prime: 1

**** 298! = P610 + 87973 ****
87973 is guaranteed to be prime: 1

**** 299! = P613 + 89273 ****
89273 is guaranteed to be prime: 1

**** 300! = P615 + 88493 ****
88493 is guaranteed to be prime: 1

**** 301! = P617 + 89839 ****
89839 is guaranteed to be prime: 1

**** 302! = P620 + 89101 ****
89101 is guaranteed to be prime: 1

**** 303! = P622 + 90971 ****
90971 is guaranteed to be prime: 1

**** 304! = P625 + 91081 ****
91081 is guaranteed to be prime: 1

**** 305! = P627 + 92831 ****
92831 is guaranteed to be prime: 1

**** 306! = P630 + 93257 ****
93257 is guaranteed to be prime: 1

**** 307! = P632 + 94117 ****
94117 is guaranteed to be prime: 1

**** 308! = P635 + 93557 ****
93557 is guaranteed to be prime: 1

**** 309! = P637 + 93979 ****
93979 is guaranteed to be prime: 1

**** 310! = P640 + 94709 ****
94709 is guaranteed to be prime: 1

**** 311! = P642 + 95651 ****
95651 is guaranteed to be prime: 1

**** 312! = P645 + 95651 ****
95651 is guaranteed to be prime: 1

**** 313! = P647 + 97607 ****
97607 is guaranteed to be prime: 1

**** 314! = P650 + 97381 ****
97381 is guaranteed to be prime: 1

**** 315! = P652 + 96527 ****
96527 is guaranteed to be prime: 1

**** 316! = P655 + 99559 ****
99559 is guaranteed to be prime: 1

**** 317! = P657 + 98809 ****
98809 is guaranteed to be prime: 1

**** 318! = P660 + 100447 ****
100447 is guaranteed to be prime: 1

**** 319! = P662 + 101503 ****
101503 is guaranteed to be prime: 1

**** 320! = P665 + 95723 ****
95723 is guaranteed to be prime: 1

**** 321! = P667 + 100669 ****
100669 is guaranteed to be prime: 1

**** 322! = P670 + 100591 ****
100591 is guaranteed to be prime: 1

**** 323! = P672 + 103123 ****
103123 is guaranteed to be prime: 1

**** 324! = P675 + 104743 ****
104743 is guaranteed to be prime: 1

**** 325! = P677 + 103319 ****
103319 is guaranteed to be prime: 1

**** 326! = P680 + 104917 ****
104917 is guaranteed to be prime: 1

**** 327! = P682 + 106213 ****
106213 is guaranteed to be prime: 1

**** 328! = P685 + 104723 ****
104723 is guaranteed to be prime: 1

**** 329! = P687 + 106781 ****
106781 is guaranteed to be prime: 1

**** 330! = P690 + 107773 ****
107773 is guaranteed to be prime: 1

**** 331! = P692 + 109433 ****
109433 is guaranteed to be prime: 1

**** 332! = P695 + 109297 ****
109297 is guaranteed to be prime: 1

**** 333! = P698 + 109663 ****
109663 is guaranteed to be prime: 1

**** 334! = P700 + 111539 ****
111539 is guaranteed to be prime: 1

**** 335! = P703 + 106681 ****
106681 is guaranteed to be prime: 1

**** 336! = P705 + 112403 ****
112403 is guaranteed to be prime: 1

**** 337! = P708 + 110557 ****
110557 is guaranteed to be prime: 1

**** 338! = P710 + 114157 ****
114157 is guaranteed to be prime: 1

**** 339! = P713 + 114679 ****
114679 is guaranteed to be prime: 1

**** 340! = P715 + 115513 ****
115513 is guaranteed to be prime: 1

**** 341! = P718 + 116089 ****
116089 is guaranteed to be prime: 1

**** 342! = P720 + 113279 ****
113279 is guaranteed to be prime: 1

**** 343! = P723 + 111533 ****
111533 is guaranteed to be prime: 1

**** 344! = P725 + 116929 ****
116929 is guaranteed to be prime: 1

**** 345! = P728 + 116047 ****
116047 is guaranteed to be prime: 1

**** 346! = P730 + 118409 ****
118409 is guaranteed to be prime: 1

**** 347! = P733 + 119689 ****
119689 is guaranteed to be prime: 1

**** 348! = P736 + 119503 ****
119503 is guaranteed to be prime: 1

**** 349! = P738 + 121379 ****
121379 is guaranteed to be prime: 1

**** 350! = P741 + 120671 ****
120671 is guaranteed to be prime: 1

**** 351! = P743 + 121727 ****
121727 is guaranteed to be prime: 1

**** 352! = P746 + 123829 ****
123829 is guaranteed to be prime: 1

**** 353! = P748 + 124067 ****
124067 is guaranteed to be prime: 1

**** 354! = P751 + 122827 ****
122827 is guaranteed to be prime: 1

**** 355! = P753 + 125219 ****
125219 is guaranteed to be prime: 1

**** 356! = P756 + 126047 ****
126047 is guaranteed to be prime: 1

**** 357! = P758 + 125789 ****
125789 is guaranteed to be prime: 1

**** 358! = P761 + 126943 ****
126943 is guaranteed to be prime: 1

**** 359! = P764 + 125929 ****
125929 is guaranteed to be prime: 1

**** 360! = P766 + 128389 ****
128389 is guaranteed to be prime: 1

**** 361! = P769 + 127807 ****
127807 is guaranteed to be prime: 1

[/CODE]

 a1call 2018-01-27 04:41

The Primorial Version

The Primorial Version

Pari-gp code:

BZC-230-A by Rashid Naimi (a1call)
Goldbach Conjecture for n#:
[URL]https://artofproblemsolving.com/wiki/index.php?title=Goldbach_Conjecture[/URL]
Every Primorial greater than 2 is the sum of 2 distinct primes.
Can be proven using my Theorem 1:
[URL]http://www.mersenneforum.org/showpost.php?p=413772&postcount=39[/URL]
If it can be shown/proven that there always exists a prime P such that
n#-n^2 < P < n#
for all integers n greater than 2.
361

[CODE]

print("\nBZC-230-A by Rashid Naimi (a1call)")
print("Goldbach Conjecture for n#:")
print("https://artofproblemsolving.com/wiki/index.php?title=Goldbach_Conjecture");
print("Every Primorial greater than 2 is the sum of 2 distinct primes.")
print("Can be proven using my Theorem 1:")
print("http://www.mersenneforum.org/showpost.php?p=413772&postcount=39")
print("If it can be shown/proven that there always exists a prime P such that")
print("n#-n^2 < P < n#")
print("for all integers n greater than 2.")

UpperBoundLimit = 19^2 \\ Set your desired Upper Bound for the n!

\\
primorial(n)=
{
my(thePrimorial=1);
forprime(p=2,n,thePrimorial=thePrimorial*p);
return (thePrimorial);
}
\\

forprime( thePrimorial =3,UpperBoundLimit ,{
evaluatedPrimorial =primorial(thePrimorial +2);
prime1 =nextprime(abs(evaluatedPrimorial -thePrimorial ^2));
prime2 =evaluatedPrimorial - prime1;
\\print("\nn**** " ,thePrimorial,"# = ",prime1 ," + ",prime2," ****" );\\\\Un-Commemt to print prime1
print("\n**** " ,thePrimorial,"# = P",#digits(prime1) ," + ",prime2," ****" );
print(prime2," is guaranteed to be prime: ",isprime(prime2) );
if(!isprime(prime2),break(19););
})

[/CODE]

[CODE]
BZC-230-A by Rashid Naimi (a1call)
Goldbach Conjecture for n#:
https://artofproblemsolving.com/wiki/index.php?title=Goldbach_Conjecture
Every Primorial greater than 2 is the sum of 2 distinct primes.
Can be proven using my Theorem 1:
http://www.mersenneforum.org/showpost.php?p=413772&postcount=39
If it can be shown/proven that there always exists a prime P such that
n#-n^2 < P < n#
for all integers n greater than 2.
361

**** 3# = P2 + 7 ****
7 is guaranteed to be prime: 1

**** 5# = P3 + 19 ****
19 is guaranteed to be prime: 1

**** 7# = P3 + 47 ****
47 is guaranteed to be prime: 1

**** 11# = P5 + 113 ****
113 is guaranteed to be prime: 1

**** 13# = P5 + 167 ****
167 is guaranteed to be prime: 1

**** 17# = P7 + 281 ****
281 is guaranteed to be prime: 1

**** 19# = P7 + 359 ****
359 is guaranteed to be prime: 1

**** 23# = P9 + 509 ****
509 is guaranteed to be prime: 1

**** 29# = P12 + 787 ****
787 is guaranteed to be prime: 1

**** 31# = P12 + 947 ****
947 is guaranteed to be prime: 1

**** 37# = P13 + 1321 ****
1321 is guaranteed to be prime: 1

**** 41# = P17 + 1637 ****
1637 is guaranteed to be prime: 1

**** 43# = P17 + 1823 ****
1823 is guaranteed to be prime: 1

**** 47# = P18 + 2161 ****
2161 is guaranteed to be prime: 1

**** 53# = P20 + 2801 ****
2801 is guaranteed to be prime: 1

**** 59# = P24 + 3469 ****
3469 is guaranteed to be prime: 1

**** 61# = P24 + 3673 ****
3673 is guaranteed to be prime: 1

**** 67# = P25 + 4327 ****
4327 is guaranteed to be prime: 1

**** 71# = P29 + 4931 ****
4931 is guaranteed to be prime: 1

**** 73# = P29 + 5171 ****
5171 is guaranteed to be prime: 1

**** 79# = P31 + 6121 ****
6121 is guaranteed to be prime: 1

**** 83# = P33 + 6869 ****
6869 is guaranteed to be prime: 1

**** 89# = P35 + 7879 ****
7879 is guaranteed to be prime: 1

**** 97# = P37 + 9349 ****
9349 is guaranteed to be prime: 1

**** 101# = P41 + 10193 ****
10193 is guaranteed to be prime: 1

**** 103# = P41 + 10399 ****
10399 is guaranteed to be prime: 1

**** 107# = P45 + 11161 ****
11161 is guaranteed to be prime: 1

**** 109# = P45 + 11593 ****
11593 is guaranteed to be prime: 1

**** 113# = P47 + 12743 ****
12743 is guaranteed to be prime: 1

**** 127# = P49 + 16061 ****
16061 is guaranteed to be prime: 1

**** 131# = P51 + 17077 ****
17077 is guaranteed to be prime: 1

**** 137# = P56 + 18461 ****
18461 is guaranteed to be prime: 1

**** 139# = P56 + 19289 ****
19289 is guaranteed to be prime: 1

**** 149# = P60 + 21863 ****
21863 is guaranteed to be prime: 1

**** 151# = P60 + 22783 ****
22783 is guaranteed to be prime: 1

**** 157# = P62 + 24481 ****
24481 is guaranteed to be prime: 1

**** 163# = P64 + 26539 ****
26539 is guaranteed to be prime: 1

**** 167# = P66 + 27697 ****
27697 is guaranteed to be prime: 1

**** 173# = P69 + 29803 ****
29803 is guaranteed to be prime: 1

**** 179# = P73 + 31177 ****
31177 is guaranteed to be prime: 1

**** 181# = P73 + 32719 ****
32719 is guaranteed to be prime: 1

**** 191# = P78 + 36451 ****
36451 is guaranteed to be prime: 1

**** 193# = P78 + 37223 ****
37223 is guaranteed to be prime: 1

**** 197# = P82 + 38737 ****
38737 is guaranteed to be prime: 1

**** 199# = P82 + 39079 ****
39079 is guaranteed to be prime: 1

**** 211# = P85 + 44281 ****
44281 is guaranteed to be prime: 1

**** 223# = P87 + 49339 ****
49339 is guaranteed to be prime: 1

**** 227# = P92 + 51449 ****
51449 is guaranteed to be prime: 1

**** 229# = P92 + 52189 ****
52189 is guaranteed to be prime: 1

**** 233# = P94 + 54133 ****
54133 is guaranteed to be prime: 1

**** 239# = P99 + 57041 ****
57041 is guaranteed to be prime: 1

**** 241# = P99 + 57773 ****
57773 is guaranteed to be prime: 1

**** 251# = P101 + 62723 ****
62723 is guaranteed to be prime: 1

**** 257# = P104 + 65981 ****
65981 is guaranteed to be prime: 1

**** 263# = P106 + 69061 ****
69061 is guaranteed to be prime: 1

**** 269# = P111 + 72277 ****
72277 is guaranteed to be prime: 1

**** 271# = P111 + 73417 ****
73417 is guaranteed to be prime: 1

**** 277# = P113 + 76597 ****
76597 is guaranteed to be prime: 1

**** 281# = P118 + 78797 ****
78797 is guaranteed to be prime: 1

**** 283# = P118 + 79757 ****
79757 is guaranteed to be prime: 1

**** 293# = P121 + 85639 ****
85639 is guaranteed to be prime: 1

**** 307# = P123 + 93871 ****
93871 is guaranteed to be prime: 1

**** 311# = P128 + 96443 ****
96443 is guaranteed to be prime: 1

**** 313# = P128 + 97879 ****
97879 is guaranteed to be prime: 1

**** 317# = P131 + 100447 ****
100447 is guaranteed to be prime: 1

**** 331# = P133 + 109519 ****
109519 is guaranteed to be prime: 1

**** 337# = P136 + 113489 ****
113489 is guaranteed to be prime: 1

**** 347# = P141 + 120167 ****
120167 is guaranteed to be prime: 1

**** 349# = P141 + 121727 ****
121727 is guaranteed to be prime: 1

**** 353# = P143 + 123817 ****
123817 is guaranteed to be prime: 1

**** 359# = P146 + 128153 ****
128153 is guaranteed to be prime: 1

[/CODE]

 MisterBitcoin 2018-01-27 11:15

Thanks a1call for your informations. :smile:

 CRGreathouse 2018-01-27 22:33

[QUOTE=a1call;478487]BZC-200-A by Rashid Naimi (a1call)
Goldbach Conjecture for n!:
[URL]https://artofproblemsolving.com/wiki/index.php?title=Goldbach_Conjecture[/URL]
Every factorial greater than 2 is the sum of 2 distinct primes.
Can be proven using my Theorem 1:
[URL]http://www.mersenneforum.org/showpost.php?p=413772&postcount=39[/URL]
If it can be shown/proven that there always exists a prime P such that
n!-n^2 < P < n!
for all integers n greater than 2.[/QUOTE]

What you call Theorem 1 is an old observation of Guy, Lacampagne, & Selfridge. (See also Agoh, Erdős, & Granville who have a more advanced version.) But it doesn't help you here at all. Sure, the result does obtain on the hypothesis that there is a prime p with n! - n^2 < p < n!-1 (not n! as you had), but then you don't need the theorem at all.

 a1call 2018-01-27 22:39

2 Attachment(s)
My pleasure Mr. MisterBitcoin.
Anytime. :smile:

Couple of more runs with the default memory allocation:

 a1call 2018-01-28 06:51

Number of primes [B]P[/B] such that

[B]n!-n^2 < P < n![/B]

for n such that

[B]3 < n < 352[/B]

Note that any of these primes will yield a 2 primes sum equal to the factorial as per my Theorem 1 referenced earlier.

[CODE]

**** Number of primes P such that:
4!-4^2 < P < 4!
** 5 primes

**** Number of primes P such that:
5!-5^2 < P < 5!
** 6 primes

**** Number of primes P such that:
6!-6^2 < P < 6!
** 4 primes

**** Number of primes P such that:
7!-7^2 < P < 7!
** 8 primes

**** Number of primes P such that:
8!-8^2 < P < 8!
** 3 primes

**** Number of primes P such that:
9!-9^2 < P < 9!
** 4 primes

**** Number of primes P such that:
10!-10^2 < P < 10!
** 7 primes

**** Number of primes P such that:
11!-11^2 < P < 11!
** 8 primes

**** Number of primes P such that:
12!-12^2 < P < 12!
** 9 primes

**** Number of primes P such that:
13!-13^2 < P < 13!
** 8 primes

**** Number of primes P such that:
14!-14^2 < P < 14!
** 8 primes

**** Number of primes P such that:
15!-15^2 < P < 15!
** 10 primes

**** Number of primes P such that:
16!-16^2 < P < 16!
** 9 primes

**** Number of primes P such that:
17!-17^2 < P < 17!
** 4 primes

**** Number of primes P such that:
18!-18^2 < P < 18!
** 14 primes

**** Number of primes P such that:
19!-19^2 < P < 19!
** 10 primes

**** Number of primes P such that:
20!-20^2 < P < 20!
** 12 primes

**** Number of primes P such that:
21!-21^2 < P < 21!
** 11 primes

**** Number of primes P such that:
22!-22^2 < P < 22!
** 13 primes

**** Number of primes P such that:
23!-23^2 < P < 23!
** 10 primes

**** Number of primes P such that:
24!-24^2 < P < 24!
** 15 primes

**** Number of primes P such that:
25!-25^2 < P < 25!
** 7 primes

**** Number of primes P such that:
26!-26^2 < P < 26!
** 12 primes

**** Number of primes P such that:
27!-27^2 < P < 27!
** 10 primes

**** Number of primes P such that:
28!-28^2 < P < 28!
** 7 primes

**** Number of primes P such that:
29!-29^2 < P < 29!
** 20 primes

**** Number of primes P such that:
30!-30^2 < P < 30!
** 12 primes

**** Number of primes P such that:
31!-31^2 < P < 31!
** 14 primes

**** Number of primes P such that:
32!-32^2 < P < 32!
** 12 primes

**** Number of primes P such that:
33!-33^2 < P < 33!
** 11 primes

**** Number of primes P such that:
34!-34^2 < P < 34!
** 10 primes

**** Number of primes P such that:
35!-35^2 < P < 35!
** 15 primes

**** Number of primes P such that:
36!-36^2 < P < 36!
** 15 primes

**** Number of primes P such that:
37!-37^2 < P < 37!
** 15 primes

**** Number of primes P such that:
38!-38^2 < P < 38!
** 14 primes

**** Number of primes P such that:
39!-39^2 < P < 39!
** 18 primes

**** Number of primes P such that:
40!-40^2 < P < 40!
** 16 primes

**** Number of primes P such that:
41!-41^2 < P < 41!
** 21 primes

**** Number of primes P such that:
42!-42^2 < P < 42!
** 20 primes

**** Number of primes P such that:
43!-43^2 < P < 43!
** 17 primes

**** Number of primes P such that:
44!-44^2 < P < 44!
** 18 primes

**** Number of primes P such that:
45!-45^2 < P < 45!
** 13 primes

**** Number of primes P such that:
46!-46^2 < P < 46!
** 13 primes

**** Number of primes P such that:
47!-47^2 < P < 47!
** 13 primes

**** Number of primes P such that:
48!-48^2 < P < 48!
** 19 primes

**** Number of primes P such that:
49!-49^2 < P < 49!
** 19 primes

**** Number of primes P such that:
50!-50^2 < P < 50!
** 19 primes

**** Number of primes P such that:
51!-51^2 < P < 51!
** 11 primes

**** Number of primes P such that:
52!-52^2 < P < 52!
** 18 primes

**** Number of primes P such that:
53!-53^2 < P < 53!
** 21 primes

**** Number of primes P such that:
54!-54^2 < P < 54!
** 21 primes

**** Number of primes P such that:
55!-55^2 < P < 55!
** 25 primes

**** Number of primes P such that:
56!-56^2 < P < 56!
** 20 primes

**** Number of primes P such that:
57!-57^2 < P < 57!
** 19 primes

**** Number of primes P such that:
58!-58^2 < P < 58!
** 16 primes

**** Number of primes P such that:
59!-59^2 < P < 59!
** 16 primes

**** Number of primes P such that:
60!-60^2 < P < 60!
** 18 primes

**** Number of primes P such that:
61!-61^2 < P < 61!
** 14 primes

**** Number of primes P such that:
62!-62^2 < P < 62!
** 20 primes

**** Number of primes P such that:
63!-63^2 < P < 63!
** 21 primes

**** Number of primes P such that:
64!-64^2 < P < 64!
** 18 primes

**** Number of primes P such that:
65!-65^2 < P < 65!
** 19 primes

**** Number of primes P such that:
66!-66^2 < P < 66!
** 21 primes

**** Number of primes P such that:
67!-67^2 < P < 67!
** 17 primes

**** Number of primes P such that:
68!-68^2 < P < 68!
** 19 primes

**** Number of primes P such that:
69!-69^2 < P < 69!
** 18 primes

**** Number of primes P such that:
70!-70^2 < P < 70!
** 24 primes

**** Number of primes P such that:
71!-71^2 < P < 71!
** 21 primes

**** Number of primes P such that:
72!-72^2 < P < 72!
** 22 primes

**** Number of primes P such that:
73!-73^2 < P < 73!
** 18 primes

**** Number of primes P such that:
74!-74^2 < P < 74!
** 29 primes

**** Number of primes P such that:
75!-75^2 < P < 75!
** 31 primes

**** Number of primes P such that:
76!-76^2 < P < 76!
** 25 primes

**** Number of primes P such that:
77!-77^2 < P < 77!
** 23 primes

**** Number of primes P such that:
78!-78^2 < P < 78!
** 18 primes

**** Number of primes P such that:
79!-79^2 < P < 79!
** 27 primes

**** Number of primes P such that:
80!-80^2 < P < 80!
** 30 primes

**** Number of primes P such that:
81!-81^2 < P < 81!
** 21 primes

**** Number of primes P such that:
82!-82^2 < P < 82!
** 18 primes

**** Number of primes P such that:
83!-83^2 < P < 83!
** 19 primes

**** Number of primes P such that:
84!-84^2 < P < 84!
** 26 primes

**** Number of primes P such that:
85!-85^2 < P < 85!
** 20 primes

**** Number of primes P such that:
86!-86^2 < P < 86!
** 24 primes

**** Number of primes P such that:
87!-87^2 < P < 87!
** 20 primes

**** Number of primes P such that:
88!-88^2 < P < 88!
** 34 primes

**** Number of primes P such that:
89!-89^2 < P < 89!
** 22 primes

**** Number of primes P such that:
90!-90^2 < P < 90!
** 32 primes

**** Number of primes P such that:
91!-91^2 < P < 91!
** 26 primes

**** Number of primes P such that:
92!-92^2 < P < 92!
** 25 primes

**** Number of primes P such that:
93!-93^2 < P < 93!
** 33 primes

**** Number of primes P such that:
94!-94^2 < P < 94!
** 23 primes

**** Number of primes P such that:
95!-95^2 < P < 95!
** 25 primes

**** Number of primes P such that:
96!-96^2 < P < 96!
** 29 primes

**** Number of primes P such that:
97!-97^2 < P < 97!
** 27 primes

**** Number of primes P such that:
98!-98^2 < P < 98!
** 19 primes

**** Number of primes P such that:
99!-99^2 < P < 99!
** 20 primes

**** Number of primes P such that:
100!-100^2 < P < 100!
** 21 primes

**** Number of primes P such that:
101!-101^2 < P < 101!
** 32 primes

**** Number of primes P such that:
102!-102^2 < P < 102!
** 37 primes

**** Number of primes P such that:
103!-103^2 < P < 103!
** 29 primes

**** Number of primes P such that:
104!-104^2 < P < 104!
** 21 primes

**** Number of primes P such that:
105!-105^2 < P < 105!
** 32 primes

**** Number of primes P such that:
106!-106^2 < P < 106!
** 40 primes

**** Number of primes P such that:
107!-107^2 < P < 107!
** 26 primes

**** Number of primes P such that:
108!-108^2 < P < 108!
** 22 primes

**** Number of primes P such that:
109!-109^2 < P < 109!
** 23 primes

**** Number of primes P such that:
110!-110^2 < P < 110!
** 39 primes

**** Number of primes P such that:
111!-111^2 < P < 111!
** 25 primes

**** Number of primes P such that:
112!-112^2 < P < 112!
** 34 primes

**** Number of primes P such that:
113!-113^2 < P < 113!
** 27 primes

**** Number of primes P such that:
114!-114^2 < P < 114!
** 22 primes

**** Number of primes P such that:
115!-115^2 < P < 115!
** 32 primes

**** Number of primes P such that:
116!-116^2 < P < 116!
** 29 primes

**** Number of primes P such that:
117!-117^2 < P < 117!
** 30 primes

**** Number of primes P such that:
118!-118^2 < P < 118!
** 34 primes

**** Number of primes P such that:
119!-119^2 < P < 119!
** 25 primes

**** Number of primes P such that:
120!-120^2 < P < 120!
** 33 primes

**** Number of primes P such that:
121!-121^2 < P < 121!
** 27 primes

**** Number of primes P such that:
122!-122^2 < P < 122!
** 32 primes

**** Number of primes P such that:
123!-123^2 < P < 123!
** 32 primes

**** Number of primes P such that:
124!-124^2 < P < 124!
** 26 primes

**** Number of primes P such that:
125!-125^2 < P < 125!
** 34 primes

**** Number of primes P such that:
126!-126^2 < P < 126!
** 27 primes

**** Number of primes P such that:
127!-127^2 < P < 127!
** 26 primes

**** Number of primes P such that:
128!-128^2 < P < 128!
** 39 primes

**** Number of primes P such that:
129!-129^2 < P < 129!
** 42 primes

**** Number of primes P such that:
130!-130^2 < P < 130!
** 29 primes

**** Number of primes P such that:
131!-131^2 < P < 131!
** 35 primes

**** Number of primes P such that:
132!-132^2 < P < 132!
** 43 primes

**** Number of primes P such that:
133!-133^2 < P < 133!
** 39 primes

**** Number of primes P such that:
134!-134^2 < P < 134!
** 25 primes

**** Number of primes P such that:
135!-135^2 < P < 135!
** 25 primes

**** Number of primes P such that:
136!-136^2 < P < 136!
** 30 primes

**** Number of primes P such that:
137!-137^2 < P < 137!
** 35 primes

**** Number of primes P such that:
138!-138^2 < P < 138!
** 35 primes

**** Number of primes P such that:
139!-139^2 < P < 139!
** 37 primes

**** Number of primes P such that:
140!-140^2 < P < 140!
** 42 primes

**** Number of primes P such that:
141!-141^2 < P < 141!
** 47 primes

**** Number of primes P such that:
142!-142^2 < P < 142!
** 34 primes

**** Number of primes P such that:
143!-143^2 < P < 143!
** 42 primes

**** Number of primes P such that:
144!-144^2 < P < 144!
** 39 primes

**** Number of primes P such that:
145!-145^2 < P < 145!
** 40 primes

**** Number of primes P such that:
146!-146^2 < P < 146!
** 44 primes

**** Number of primes P such that:
147!-147^2 < P < 147!
** 31 primes

**** Number of primes P such that:
148!-148^2 < P < 148!
** 30 primes

**** Number of primes P such that:
149!-149^2 < P < 149!
** 41 primes

**** Number of primes P such that:
150!-150^2 < P < 150!
** 50 primes

**** Number of primes P such that:
151!-151^2 < P < 151!
** 44 primes

**** Number of primes P such that:
152!-152^2 < P < 152!
** 38 primes

**** Number of primes P such that:
153!-153^2 < P < 153!
** 42 primes

**** Number of primes P such that:
154!-154^2 < P < 154!
** 37 primes

**** Number of primes P such that:
155!-155^2 < P < 155!
** 43 primes

**** Number of primes P such that:
156!-156^2 < P < 156!
** 42 primes

**** Number of primes P such that:
157!-157^2 < P < 157!
** 32 primes

**** Number of primes P such that:
158!-158^2 < P < 158!
** 36 primes

**** Number of primes P such that:
159!-159^2 < P < 159!
** 31 primes

**** Number of primes P such that:
160!-160^2 < P < 160!
** 39 primes

**** Number of primes P such that:
161!-161^2 < P < 161!
** 42 primes

**** Number of primes P such that:
162!-162^2 < P < 162!
** 38 primes

**** Number of primes P such that:
163!-163^2 < P < 163!
** 45 primes

**** Number of primes P such that:
164!-164^2 < P < 164!
** 38 primes

**** Number of primes P such that:
165!-165^2 < P < 165!
** 39 primes

**** Number of primes P such that:
166!-166^2 < P < 166!
** 47 primes

**** Number of primes P such that:
167!-167^2 < P < 167!
** 43 primes

**** Number of primes P such that:
168!-168^2 < P < 168!
** 59 primes

**** Number of primes P such that:
169!-169^2 < P < 169!
** 40 primes

**** Number of primes P such that:
170!-170^2 < P < 170!
** 40 primes

**** Number of primes P such that:
171!-171^2 < P < 171!
** 50 primes

**** Number of primes P such that:
172!-172^2 < P < 172!
** 54 primes

**** Number of primes P such that:
173!-173^2 < P < 173!
** 49 primes

**** Number of primes P such that:
174!-174^2 < P < 174!
** 38 primes

**** Number of primes P such that:
175!-175^2 < P < 175!
** 49 primes

**** Number of primes P such that:
176!-176^2 < P < 176!
** 56 primes

**** Number of primes P such that:
177!-177^2 < P < 177!
** 49 primes

**** Number of primes P such that:
178!-178^2 < P < 178!
** 44 primes

**** Number of primes P such that:
179!-179^2 < P < 179!
** 46 primes

**** Number of primes P such that:
180!-180^2 < P < 180!
** 48 primes

**** Number of primes P such that:
181!-181^2 < P < 181!
** 33 primes

**** Number of primes P such that:
182!-182^2 < P < 182!
** 39 primes

**** Number of primes P such that:
183!-183^2 < P < 183!
** 36 primes

**** Number of primes P such that:
184!-184^2 < P < 184!
** 40 primes

**** Number of primes P such that:
185!-185^2 < P < 185!
** 49 primes

**** Number of primes P such that:
186!-186^2 < P < 186!
** 41 primes

**** Number of primes P such that:
187!-187^2 < P < 187!
** 43 primes

**** Number of primes P such that:
188!-188^2 < P < 188!
** 42 primes

**** Number of primes P such that:
189!-189^2 < P < 189!
** 58 primes

**** Number of primes P such that:
190!-190^2 < P < 190!
** 37 primes

**** Number of primes P such that:
191!-191^2 < P < 191!
** 45 primes

**** Number of primes P such that:
192!-192^2 < P < 192!
** 41 primes

**** Number of primes P such that:
193!-193^2 < P < 193!
** 36 primes

**** Number of primes P such that:
194!-194^2 < P < 194!
** 36 primes

**** Number of primes P such that:
195!-195^2 < P < 195!
** 42 primes

**** Number of primes P such that:
196!-196^2 < P < 196!
** 38 primes

**** Number of primes P such that:
197!-197^2 < P < 197!
** 42 primes

**** Number of primes P such that:
198!-198^2 < P < 198!
** 44 primes

**** Number of primes P such that:
199!-199^2 < P < 199!
** 34 primes

**** Number of primes P such that:
200!-200^2 < P < 200!
** 52 primes

**** Number of primes P such that:
201!-201^2 < P < 201!
** 52 primes

**** Number of primes P such that:
202!-202^2 < P < 202!
** 36 primes

**** Number of primes P such that:
203!-203^2 < P < 203!
** 49 primes

**** Number of primes P such that:
204!-204^2 < P < 204!
** 53 primes

**** Number of primes P such that:
205!-205^2 < P < 205!
** 53 primes

**** Number of primes P such that:
206!-206^2 < P < 206!
** 59 primes

**** Number of primes P such that:
207!-207^2 < P < 207!
** 42 primes

**** Number of primes P such that:
208!-208^2 < P < 208!
** 39 primes

**** Number of primes P such that:
209!-209^2 < P < 209!
** 46 primes

**** Number of primes P such that:
210!-210^2 < P < 210!
** 45 primes

**** Number of primes P such that:
211!-211^2 < P < 211!
** 37 primes

**** Number of primes P such that:
212!-212^2 < P < 212!
** 48 primes

**** Number of primes P such that:
213!-213^2 < P < 213!
** 52 primes

**** Number of primes P such that:
214!-214^2 < P < 214!
** 48 primes

**** Number of primes P such that:
215!-215^2 < P < 215!
** 56 primes

**** Number of primes P such that:
216!-216^2 < P < 216!
** 43 primes

**** Number of primes P such that:
217!-217^2 < P < 217!
** 63 primes

**** Number of primes P such that:
218!-218^2 < P < 218!
** 51 primes

**** Number of primes P such that:
219!-219^2 < P < 219!
** 53 primes

**** Number of primes P such that:
220!-220^2 < P < 220!
** 42 primes

**** Number of primes P such that:
221!-221^2 < P < 221!
** 45 primes

**** Number of primes P such that:
222!-222^2 < P < 222!
** 43 primes

**** Number of primes P such that:
223!-223^2 < P < 223!
** 47 primes

**** Number of primes P such that:
224!-224^2 < P < 224!
** 49 primes

**** Number of primes P such that:
225!-225^2 < P < 225!
** 39 primes

**** Number of primes P such that:
226!-226^2 < P < 226!
** 53 primes

**** Number of primes P such that:
227!-227^2 < P < 227!
** 54 primes

**** Number of primes P such that:
228!-228^2 < P < 228!
** 47 primes

**** Number of primes P such that:
229!-229^2 < P < 229!
** 54 primes

**** Number of primes P such that:
230!-230^2 < P < 230!
** 56 primes

**** Number of primes P such that:
231!-231^2 < P < 231!
** 47 primes

**** Number of primes P such that:
232!-232^2 < P < 232!
** 49 primes

**** Number of primes P such that:
233!-233^2 < P < 233!
** 47 primes

**** Number of primes P such that:
234!-234^2 < P < 234!
** 60 primes

**** Number of primes P such that:
235!-235^2 < P < 235!
** 46 primes

**** Number of primes P such that:
236!-236^2 < P < 236!
** 56 primes

**** Number of primes P such that:
237!-237^2 < P < 237!
** 54 primes

**** Number of primes P such that:
238!-238^2 < P < 238!
** 51 primes

**** Number of primes P such that:
239!-239^2 < P < 239!
** 36 primes

**** Number of primes P such that:
240!-240^2 < P < 240!
** 45 primes

**** Number of primes P such that:
241!-241^2 < P < 241!
** 44 primes

**** Number of primes P such that:
242!-242^2 < P < 242!
** 54 primes

**** Number of primes P such that:
243!-243^2 < P < 243!
** 60 primes

**** Number of primes P such that:
244!-244^2 < P < 244!
** 41 primes

**** Number of primes P such that:
245!-245^2 < P < 245!
** 40 primes

**** Number of primes P such that:
246!-246^2 < P < 246!
** 48 primes

**** Number of primes P such that:
247!-247^2 < P < 247!
** 49 primes

**** Number of primes P such that:
248!-248^2 < P < 248!
** 60 primes

**** Number of primes P such that:
249!-249^2 < P < 249!
** 57 primes

**** Number of primes P such that:
250!-250^2 < P < 250!
** 66 primes

**** Number of primes P such that:
251!-251^2 < P < 251!
** 67 primes

**** Number of primes P such that:
252!-252^2 < P < 252!
** 65 primes

**** Number of primes P such that:
253!-253^2 < P < 253!
** 56 primes

**** Number of primes P such that:
254!-254^2 < P < 254!
** 57 primes

**** Number of primes P such that:
255!-255^2 < P < 255!
** 65 primes

**** Number of primes P such that:
256!-256^2 < P < 256!
** 58 primes

**** Number of primes P such that:
257!-257^2 < P < 257!
** 49 primes

**** Number of primes P such that:
258!-258^2 < P < 258!
** 57 primes

**** Number of primes P such that:
259!-259^2 < P < 259!
** 57 primes

**** Number of primes P such that:
260!-260^2 < P < 260!
** 47 primes

**** Number of primes P such that:
261!-261^2 < P < 261!
** 60 primes

**** Number of primes P such that:
262!-262^2 < P < 262!
** 56 primes

**** Number of primes P such that:
263!-263^2 < P < 263!
** 41 primes

**** Number of primes P such that:
264!-264^2 < P < 264!
** 52 primes

**** Number of primes P such that:
265!-265^2 < P < 265!
** 62 primes

**** Number of primes P such that:
266!-266^2 < P < 266!
** 55 primes

**** Number of primes P such that:
267!-267^2 < P < 267!
** 54 primes

**** Number of primes P such that:
268!-268^2 < P < 268!
** 54 primes

**** Number of primes P such that:
269!-269^2 < P < 269!
** 61 primes

**** Number of primes P such that:
270!-270^2 < P < 270!
** 46 primes

**** Number of primes P such that:
271!-271^2 < P < 271!
** 52 primes

**** Number of primes P such that:
272!-272^2 < P < 272!
** 60 primes

**** Number of primes P such that:
273!-273^2 < P < 273!
** 48 primes

**** Number of primes P such that:
274!-274^2 < P < 274!
** 49 primes

**** Number of primes P such that:
275!-275^2 < P < 275!
** 64 primes

**** Number of primes P such that:
276!-276^2 < P < 276!
** 68 primes

**** Number of primes P such that:
277!-277^2 < P < 277!
** 65 primes

**** Number of primes P such that:
278!-278^2 < P < 278!
** 58 primes

**** Number of primes P such that:
279!-279^2 < P < 279!
** 65 primes

**** Number of primes P such that:
280!-280^2 < P < 280!
** 53 primes

**** Number of primes P such that:
281!-281^2 < P < 281!
** 53 primes

**** Number of primes P such that:
282!-282^2 < P < 282!
** 61 primes

**** Number of primes P such that:
283!-283^2 < P < 283!
** 57 primes

**** Number of primes P such that:
284!-284^2 < P < 284!
** 47 primes

**** Number of primes P such that:
285!-285^2 < P < 285!
** 71 primes

**** Number of primes P such that:
286!-286^2 < P < 286!
** 66 primes

**** Number of primes P such that:
287!-287^2 < P < 287!
** 57 primes

**** Number of primes P such that:
288!-288^2 < P < 288!
** 52 primes

**** Number of primes P such that:
289!-289^2 < P < 289!
** 68 primes

**** Number of primes P such that:
290!-290^2 < P < 290!
** 56 primes

**** Number of primes P such that:
291!-291^2 < P < 291!
** 68 primes

**** Number of primes P such that:
292!-292^2 < P < 292!
** 50 primes

**** Number of primes P such that:
293!-293^2 < P < 293!
** 60 primes

**** Number of primes P such that:
294!-294^2 < P < 294!
** 60 primes

**** Number of primes P such that:
295!-295^2 < P < 295!
** 58 primes

**** Number of primes P such that:
296!-296^2 < P < 296!
** 68 primes

**** Number of primes P such that:
297!-297^2 < P < 297!
** 70 primes

**** Number of primes P such that:
298!-298^2 < P < 298!
** 68 primes

**** Number of primes P such that:
299!-299^2 < P < 299!
** 64 primes

**** Number of primes P such that:
300!-300^2 < P < 300!
** 50 primes

**** Number of primes P such that:
301!-301^2 < P < 301!
** 61 primes

**** Number of primes P such that:
302!-302^2 < P < 302!
** 52 primes

**** Number of primes P such that:
303!-303^2 < P < 303!
** 62 primes

**** Number of primes P such that:
304!-304^2 < P < 304!
** 62 primes

**** Number of primes P such that:
305!-305^2 < P < 305!
** 63 primes

**** Number of primes P such that:
306!-306^2 < P < 306!
** 56 primes

**** Number of primes P such that:
307!-307^2 < P < 307!
** 60 primes

**** Number of primes P such that:
308!-308^2 < P < 308!
** 69 primes

**** Number of primes P such that:
309!-309^2 < P < 309!
** 68 primes

**** Number of primes P such that:
310!-310^2 < P < 310!
** 61 primes

**** Number of primes P such that:
311!-311^2 < P < 311!
** 68 primes

**** Number of primes P such that:
312!-312^2 < P < 312!
** 62 primes

**** Number of primes P such that:
313!-313^2 < P < 313!
** 61 primes

**** Number of primes P such that:
314!-314^2 < P < 314!
** 73 primes

**** Number of primes P such that:
315!-315^2 < P < 315!
** 65 primes

**** Number of primes P such that:
316!-316^2 < P < 316!
** 50 primes

**** Number of primes P such that:
317!-317^2 < P < 317!
** 56 primes

**** Number of primes P such that:
318!-318^2 < P < 318!
** 69 primes

**** Number of primes P such that:
319!-319^2 < P < 319!
** 73 primes

**** Number of primes P such that:
320!-320^2 < P < 320!
** 74 primes

**** Number of primes P such that:
321!-321^2 < P < 321!
** 70 primes

**** Number of primes P such that:
322!-322^2 < P < 322!
** 62 primes

**** Number of primes P such that:
323!-323^2 < P < 323!
** 75 primes

**** Number of primes P such that:
324!-324^2 < P < 324!
** 71 primes

**** Number of primes P such that:
325!-325^2 < P < 325!
** 76 primes

**** Number of primes P such that:
326!-326^2 < P < 326!
** 78 primes

**** Number of primes P such that:
327!-327^2 < P < 327!
** 70 primes

**** Number of primes P such that:
328!-328^2 < P < 328!
** 72 primes

**** Number of primes P such that:
329!-329^2 < P < 329!
** 74 primes

**** Number of primes P such that:
330!-330^2 < P < 330!
** 60 primes

**** Number of primes P such that:
331!-331^2 < P < 331!
** 84 primes

**** Number of primes P such that:
332!-332^2 < P < 332!
** 68 primes

**** Number of primes P such that:
333!-333^2 < P < 333!
** 57 primes

**** Number of primes P such that:
334!-334^2 < P < 334!
** 65 primes

**** Number of primes P such that:
335!-335^2 < P < 335!
** 78 primes

**** Number of primes P such that:
336!-336^2 < P < 336!
** 75 primes

**** Number of primes P such that:
337!-337^2 < P < 337!
** 62 primes

**** Number of primes P such that:
338!-338^2 < P < 338!
** 85 primes

**** Number of primes P such that:
339!-339^2 < P < 339!
** 71 primes

**** Number of primes P such that:
340!-340^2 < P < 340!
** 60 primes

**** Number of primes P such that:
341!-341^2 < P < 341!
** 75 primes

**** Number of primes P such that:
342!-342^2 < P < 342!
** 66 primes

**** Number of primes P such that:
343!-343^2 < P < 343!
** 76 primes

**** Number of primes P such that:
344!-344^2 < P < 344!
** 62 primes

**** Number of primes P such that:
345!-345^2 < P < 345!
** 57 primes

**** Number of primes P such that:
346!-346^2 < P < 346!
** 69 primes

**** Number of primes P such that:
347!-347^2 < P < 347!
** 61 primes

**** Number of primes P such that:
348!-348^2 < P < 348!
** 77 primes

**** Number of primes P such that:
349!-349^2 < P < 349!
** 86 primes

**** Number of primes P such that:
350!-350^2 < P < 350!
** 72 primes

**** Number of primes P such that:
351!-351^2 < P < 351!
** 58 primes

[/CODE]

I find the distribution quite interesting.
:smile:

 a1call 2018-01-28 07:57

[QUOTE=CRGreathouse;478572]What you call Theorem 1 is an old observation of Guy, Lacampagne, & Selfridge. (See also Agoh, Erdős, & Granville who have a more advanced version.) But it doesn't help you here at all. Sure, the result does obtain on the hypothesis that there is a prime p with n! - n^2 < p < n!-1 (not n! as you had), but then you don't need the theorem at all.[/QUOTE]

Ok I can see the reason why it should be -1, because P=n!-1 will yield 1 and 1 is not s prime.
My googling did not return any relevant hits.
PS the number of primes above might be off by one for appropriate primes if n!-1 is a prime such as for n=4.

 a1call 2018-01-28 08:24

Actually it seems to me that the correct optimised range is neither

n!-n^2 < P < n!

Nor

n!-n^2 < P < n!-1

But rather

n!-n^2 < P < n!-n
Since none of the integers m such that
n!-n <= m < n! -1
Can be prime.
:smile:

 science_man_88 2018-01-28 12:06

[QUOTE=a1call;478597]Actually it seems to me that the correct optimised range is neither

n!-n^2 < P < n!

Nor

n!-n^2 < P < n!-1

But rather

n!-n^2 < P < n!-n
Since none of the integers m such that
n!-n <= m < n! -1
Can be prime.
:smile:[/QUOTE]
Depends on n if odd neither will n!-(n+1) as it will be even. If n is 2 mod 3 then n+1 is 0 mod 3 and the result of subtracting it if n is more than 3 will be 0 mod 3.

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