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 2018-01-26, 21:29 #1 MisterBitcoin     "Nuri, the dragon :P" Jul 2016 Good old Germany 11001001102 Posts 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; here Smallest combination for 21!: 31+P20; here Smallest combination for 22!: 73+P22; here and so on... I´ll build an script to do this kind of "work", I won´t waste FactorDB resources. Last fiddled with by MisterBitcoin on 2018-01-27 at 11:15
 2018-01-26, 23:06 #2 kar_bon     Mar 2006 Germany 2·11·131 Posts Quick shot: 261 primes/PRPs for 20!-n, 1<=n<=10000
 2018-01-27, 02:03 #3 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 111110111002 Posts Pari-gp code: BZC-200-A by Rashid Naimi (a1call) Goldbach Conjecture for n!: https://artofproblemsolving.com/wiki...ach_Conjecture Every factorial greater than 2 is the sum of 2 distinct primes. Can be proven using my Theorem 1: http://www.mersenneforum.org/showpos...2&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. 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: 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 Last fiddled with by a1call on 2018-01-27 at 02:49
 2018-01-27, 04:41 #4 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 22×503 Posts The Primorial Version The Primorial Version Pari-gp code: BZC-230-A by Rashid Naimi (a1call) Goldbach Conjecture for n#: https://artofproblemsolving.com/wiki...ach_Conjecture Every Primorial greater than 2 is the sum of 2 distinct primes. Can be proven using my Theorem 1: http://www.mersenneforum.org/showpos...2&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 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: 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
 2018-01-27, 11:15 #5 MisterBitcoin     "Nuri, the dragon :P" Jul 2016 Good old Germany 2×13×31 Posts Thanks a1call for your informations.
2018-01-27, 22:33   #6
CRGreathouse

Aug 2006

176116 Posts

Quote:
 Originally Posted by a1call BZC-200-A by Rashid Naimi (a1call) Goldbach Conjecture for n!: https://artofproblemsolving.com/wiki...ach_Conjecture Every factorial greater than 2 is the sum of 2 distinct primes. Can be proven using my Theorem 1: http://www.mersenneforum.org/showpos...2&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.
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.

2018-01-27, 22:39   #7
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

22×503 Posts

My pleasure Mr. MisterBitcoin.
Anytime.

Couple of more runs with the default memory allocation:
Attached Files
 BZC-228-A-Factorials between 361! & 1001! expressed as sum of 2 distinct primes.txt (45.1 KB, 183 views) BZC-238-A-Primorials between 367# & 6347# expressed as sum of 2 distinct primes.txt (46.6 KB, 130 views)

 2018-01-28, 06:51 #8 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 22·503 Posts Number of primes P such that n!-n^2 < P < n! for n such that 3 < n < 352 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 I find the distribution quite interesting. Last fiddled with by a1call on 2018-01-28 at 07:16
2018-01-28, 07:57   #9
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

22×503 Posts

Quote:
 Originally Posted by CRGreathouse 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.
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.

Last fiddled with by a1call on 2018-01-28 at 07:59

 2018-01-28, 08:24 #10 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 7DC16 Posts 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. Last fiddled with by a1call on 2018-01-28 at 08:31
2018-01-28, 12:06   #11
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

838410 Posts

Quote:
 Originally Posted by a1call 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.
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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