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

Parigp code:
BZC200A 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("\nBZC200A 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," ****" );\\\\UnCommemt 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] BZC200A 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] 
The Primorial Version
The Primorial Version
Parigp code: BZC230A 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("\nBZC230A 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," ****" );\\\\UnCommemt 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] BZC230A 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] 
Thanks a1call for your informations. :smile:

[QUOTE=a1call;478487]BZC200A 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. 
2 Attachment(s)
My pleasure Mr. MisterBitcoin.
Anytime. :smile: Couple of more runs with the default memory allocation: 
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: 
[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. Can you please provide links to the references that you mentioned. My googling did not return any relevant hits. Thank you in advance. 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. 
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=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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