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[5]
Numbers with absolute value < 300 generated mainly from set of
{ 2 , 3 , 7 , 11 , 13}
Let S = { 2 , 3 , 7 , 11 , 13} and take A = { 2 ,3 , 13 } and
B = { 7 , 11 } ,
thus π A = 78 and πB = 77 , while L = 17 ^2 = 289 .
To generate n = 78 * i - 77 * j for | n | < 300 .
( cases of i = j = prime are included )
(1) 78 * 1 = 78 - 77 * - 1 = - 77 , d = 155 ( = 5*31)
(2) 78 * 1 = 78 - 77 * 1 = 77 d = 1
(3) 78 * 2 = 156 - 77 * -1 = - 77 , d = 233 (P)
(4) 78 * 2 = 156 - 77 * 1 = 77 , d = 79 (P)
(5) 78 * 2 = 156 - 77 * 5 = 385 , d = - 229 (P)
(6) 78 * 3 = 234 - 77 * 1 = 77 , d = 157 (P)
(7) 78 * 3 = 234 - 77 * 5 = 385 , d = - 151 (P)
(8) 78 * 4 = 312 - 77 * 1 = 77 , d = 235 (= 5*47)
(9) 78 * 4 = 312 - 77 * 5 = 385 , d = - 73 (P)
(10)78 * 4 = 312 - 77 * 7 = 539 , d = - 227 (P)
(11) 78 * 5 = 390 - 77 * 5 = 385 , d = 5 (P)
(12) 78 * 5 = 390 - 77 * 7 = 539 , d = - 149 (P)
(13)78 * 6 = 468 - 77 * 5 = 385 , d = 83 (P)
(14)78 * 6 = 468 - 77 * 7 = 539 , d = - 71 (P)
(15) 78 * 8 = 624 - 77 * 5 = 385 , d = 239 (P)
(16) 78 * 8 = 624 - 77 * 7 = 539 , d = 85 (= 5* 17)
(17) 78 * 8 = 624 - 77 * 11 = 847 , d = - 223
(18) 78 * 9 = 702 - 77 * 7 = 539 , d = 163 (P)
(19) 78 * 9 = 702 - 77 * 11 = 847 , d = - 145 (= 5* 29)
(20) 78 *10 = 780 - 77 * 7 = 539 , d = 241 (P)
(21) 78 * 10 = 780 - 77 * 11 = 847 , d = - 67 (P)
(22) 78 * 12 = 936 - 77 * 11 = 847 , d = 89 (P)
(23) 78 * 13 = 1014 - 77 * 11 = 847 , d = 167 (P
(24) 78 * 13 = 1014 - 77 * 17 = 1309 , d = - 295 (= 5 *59)
(25) 78 * 15 = 1170 - 77 * 17 = 1309 , d = - 139 (P)
(26) 78 * 15 = 1170 - 77 * 19 = 1463 , d = - 293 (P)
(27) 78 * 16 = 1248 - 77 * 17 = 1309 , d = - 61 (P)
(28) 78 * 16 = 1248 - 77 * 19 = 1463 , d = - 215 ( = 5 * 43)
(29)78 * 17 = 1326 - 77 * 17= 1309 , d = 17 (P)
(30)78 * 17 = 1326 - 77 * 19 = 1463 , d = - 137 (P)
(31)78 * 18 = 1404 - 77 * 17 = 1309 , d = 95 ( = 5*19)
(32)78 * 18 = 1404 - 77 * 19 = 1463 , d = - 59 (P)
(33)78 * 19 = 1482 - 77 * 17 = 1309 , d = 173 (P)
(34)78 * 19 = 1482 - 77 * 19 = 1463 , d = 19 (P)
(35) 78 * 19 = 1482 - 77 * 23 = 1771 , d = - 289 (= 17*17)
(36) 78 * 20 = 1560 - 77 * 17 = 1309 , d = 251 (P)
(37) 78 * 20 = 1560 - 77 * 19 = 1463 , d = 97 (P)
(38) 78 * 20 = 1560 - 77 * 23 = 1771 , d = - 211 (P)
(39) 78 * 23 = 1794 - 77 * 23 = 1771 , d = 23 (P)
(40) 78 * 23 = 1794 - 77 * 25 = 1925 , d = - 131 (P)
(41) 78 * 24 = 1872 - 77 * 23 = 1771 , d = 101 (P)
(42) 78 * 24 = 1872 - 77 * 25 = 1925 , d = - 53 (P)
(43) 78 * 25 = 1950 - 77 * 23 = 1771 , d = 179 (P)
(44) 78 * 25 = 1950 - 77 * 29 = 2233 , d = - 283 (P)
(45) 78 * 26 = 2028 - 77 * 23 = 1771 , d = 257 (P)
(46) 78 * 26 = 2028 - 77 * 25 = 1925 , d = 103 (P)
(47) 78 * 26 = 2028 - 77 * 29 = 2233 , d = - 205 (= 5*41)
(48)78 * 27 = 2106 - 77 * 25 = 1925 , d = 181 (P)
(49)78 * 27 = 2106 - 77 * 29 = 2233 , d = - 127 (P)
(50)78 * 27 = 2106 - 77 * 31 = 2387 , d = - 281 (P)
(51) 78 * 29 = 2262 - 77 * 29 = 2233 , d = 29 (P)
(52) 78 * 29 = 2262 - 77 * 31 = 2387 , d = - 125 (= 5 *25)
(53) 78 * 30 = 2340 - 77 * 29 = 2233 , d = 107 (P)
(54) 78 * 30 = 2340 - 77 * 31 = 2387 , d = - 47 (P)
(55) 78 * 31 = 2418 - 77 * 29 = 2233 , d = 185 (= 5*37)
(56) 78 * 31 = 2418 - 77 * 31 = 2387 , d = 31 (P)
(57) 78 * 31 = 2418 - 77 * 35 = 2695 , d = - 277 (P)
(58) 78 * 32 = 2496 - 77 * 29 = 2233 , d = 263 (P)
(59) 78 * 32 = 2496 - 77 * 31 = 2387 , d = 109 (P)
(60) 78 * 32 = 2496 - 77 * 35 = 2695 , d = - 199 (P)
(61) 78 * 34 = 2652 - 77 * 31 = 2387 , d = 265 (=5*53)
(62) 78 * 34 = 2652 - 77 * 35 = 2695 , d = - 43 (P)
(63) 78 * 34 = 2652 - 77 * 37 = 2849 , d = - 197 (P)
(64)78 * 36 = 2808 - 77 * 35 = 2695 , d = 113 (P)
(65) 78 * 36 = 2808 - 77 * 37 = 2849 , d = - 41 (P)
(66) 78 * 37 = 2886 - 77 * 35 = 2695 , d = 191(P)
(67) 78 * 37 = 2886 - 77 * 37 = 2849 , d = 37 (P)
(68) 78 * 37 = 2886 - 77 * 41 = 3157 , d = - 271 (P)
(69) 78 * 38 = 2964 - 77 * 35 = 2695 , d = 269 (P)
(70) 78 * 38 = 2964 - 77 * 37 = 2849 , d = 115 (=5*23)
(71) 78 * 38 = 2964 - 77 * 41 = 3157 , d = - 193 (P)
--------------------------------------------------------------------------------
(72)78 * 39 = 3042 - 77 * 37 = 2849 , d = 193
(73)78 * 39 = 3042 - 77 * 41 = 3157 , d = - 115 (=5*23)
(74)78 * 39 = 3042 - 77 * 43 = 3311 , d = - 269
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[ 4 A ]
To generate n = 7 - 6 * j for | n | < 500
(1) 7 - 6 * - 82 = - 492 , n = 499 (P)
(2) 7 - 6 * - 81 = - 486 , n = 493 (= 17 * 29)
(3) 7 - 6 * - 80 = - 480 , n = 487 (P)
(4) 7 - 6 * - 79 = - 474 , n = 481 ( = 13 *37)
(5) 7 - 6 * - 78 = - 468 , n = 475
(6) 7 - 6 * - 77 = - 462 , n = 469 X ( = 7 * 67 )
(7) 7 - 6 * - 76 = - 456 , n = 463 (P)
(8) 7 - 6 * - 75 = - 450 , n = 457 (P)
(9) 7 - 6 * - 74 = - 444 , n = 451 ( = 11 * 41)
(10) 7 - 6 * - 73 = - 438 , n = 445
(11) 7 - 6 * - 72 = - 432 , n = 439 (P)
(12) 7 - 6 * - 71 = - 426 , n = 433 (P)
(13) 7 - 6 * - 70 = - 420 , n = 427 X ( = 7 * 61 )
(14) 7 - 6 * - 69 = - 414 , n = 421 (P)
(15) 7 - 6 * - 68 = - 408 , n = 415
(16) 7 - 6 * - 67 = - 402 , n = 409 (P)
(17) 7 - 6 * - 66 = - 396 , n = 403 ( = 13 *31)
(18) 7 - 6 * - 65 = - 390 , n = 397 (P)
(19) 7 - 6 * - 64 = - 384 , n = 391 ( = 17 * 23)
(20) 7 - 6 * - 63 = - 378 , n = 385 X ( = 7 * 55 )
.......................
(21) n = 385 - 6 = 379 (P)
(22) n = 379 - 6 = 373 (P)
(23) n = 373 - 6 = 367 (P)
(24) n = 367 - 6 = 361 ( = 19 *19)
(25) n = 361 - 6 = 355
(26) n = 355 - 6 = 349 (P)
(27) n = 349 - 6 = 343 X ( = 7 * 49 )
(28) n = 343 - 6 = 337 (P)
(29) n = 337 - 6 = 331 (P)
(30) n = 331 - 6 = 325
(31) n = 325 - 6 = 319 ( = 11 * 29)
(32) n = 319 - 6 = 313 (P)
(33) n = 313 - 6 = 307 (P)
(34) n = 307 - 6 = 301 X ( = 7 * 43)
(35) n = 301 - 6 = 295
(36) n = 295 - 6 = 289 ( = 17 *17)
(37) n = 289 - 6 = 283 (P)
(38) n = 283 - 6 = 277 (P)
(39) n = 277 - 6 = 271 (P)
(40) n = 271 - 6 = 265
(41) n = 265 - 6 = 259 X ( 7 *)
(42) n = 259 - 6 = 253 ( = 11 *23)
(43) n = 253 - 6 = 247 (= 13 *19)
(44) n = 247 - 6 = 241 (P)
(45) n = 241 - 6 = 235
(46) n = 235 - 6 = 229 (P)
(47) n = 229 - 6 = 223 (P)
(48) n = 223 - 6 = 217 X (7*)
(49) n = 217 - 6 = 211 (P)
(50) n = 211 - 6 = 205
(51) n = 205 - 6 = 199 (P)
(52) n = 199 - 6 = 193 (P)
(53) n = 193 - 6 = 187 ( = 11 *17)
(54) n = 187 - 6 = 181 (P)
(55) n = 181 - 6 = 175 X
(56) n = 175 - 6 = 169 (= 13 *13)
(57) n = 169 - 6 = 163 (P)
(58) n = 163 - 6 = 157 (P)
(59) n = 157 - 6 = 151 (P)
(60) n = 151 - 6 = 145
(61) n = 145 - 6 = 139 (P)
(62) n = 139 - 6 = 133 X
(63) n = 133 - 6 = 127 (P)
(64) n = 127 - 6 = 121 ( = 11*11)
(65) n = 121 - 6 = 115
(66) n = 115 - 6 = 109 (P)
(67) n = 109 - 6 = 103 (P)
(68) n = 103 - 6 = 97 (P)
(69) n = 97 - 6 = 91 X
(70) n = 91 - 6 = 85
(71) n = 85 - 6 = 79 (P)
(72) n = 79 - 6 = 73 (P)
(73) n = 73 - 6 = 67 (P)
(74) n = 67 - 6 = 61 (P)
(75) n = 61 - 6 = 55
(76) n = 55 - 6 = 49 X
(77) n = 49 - 6 = 43 (P)
(78) n = 43 - 6 = 37 (P)
(79) n = 37 - 6 = 31 (P)
(80) n = 31 - 6 = 25
(81) n = 25 - 6 = 19 (P)
(82) n = 19 - 6 = 13 (P)
(83) n = 13 - 6 = 7 (P)
(84) n = 7 - 6 = 1
(85) n = 1 - 6 = - 5 (P)
(86) n = - 5 - 6 = - 11 (P)
(87) n = - 11 - 6 = - 17 (P)
(88) n = - 17 - 6 = - 23 (P)
(89) n = - 23 - 6 = - 29 (P)
(90) n = - 29 - 6 = - 35 X
(91) n = - 35 - 6 = - 41 (P)
(92) n = - 41 - 6 = - 47 (P)
(93) n = - 47 - 6 = - 53 (P)
(94) n = - 53 - 6 = - 59 (P)
(95) n = - 59 - 6 = - 65
(96) n = - 65 - 6 = - 71 (P)
(97) n = - 71 - 6 = - 77 X
(98) n = - 77 - 6 = - 83 (P)
(99) n = - 83 - 6 = - 89 (P)
(100) n = - 89 - 6 = - 95
(101) n = - 95 - 6 = - 101 (P)
(102) n = - 101 - 6 = - 107 (P)
(103) n = - 107 - 6 = - 113 (P)
(104) n = - 113 - 6 = - 119 X
(105) n = - 119 - 6 = - 125
(106) n = - 125 - 6 = - 131 (P)
(107) n = - 131 - 6 = - 137 (P)
(108) n = - 137 - 6 = - 143 (= 11 * 13)
(109) n = - 143 - 6 = - 149 (P)
(110) n = - 149 - 6 = - 155
(111) n = - 155 - 6 = - 161 X
(112) n = - 161 - 6 = - 167 (P)
(113) n = - 167 - 6 = - 173 (P)
(114) n = - 173 - 6 = - 179 (P)
(115) n = - 179 - 6 = - 185
(116) n = - 185 - 6 = - 191 (P)
(117) n = - 191 - 6 = - 197 (P)
(118) n = - 197 - 6 = - 203 X
(119) n = - 203 - 6 = - 209 ( = 11* 19)
(120) n = - 209 - 6 = - 215
(121) n = - 215 - 6 = - 221 ( = 13*17)
(122) n = - 221 - 6 = - 227 (P)
(123) n = - 227 - 6 = - 233 (P)
(124) n = - 233 - 6 = - 239 (P)
(125) n = - 239 - 6 = - 245
(126) n = - 245 - 6 = - 251 (P)
(127) n = - 251 - 6 = - 257 (P)
(128) n = - 257 - 6 = - 263 (P)
(129) n = - 263 - 6 = - 269 (P)
(130) n = - 269 - 6 = - 275
(131) n = - 275 - 6 = - 281 (P)
(132) n = - 281 - 6 = - 287 X
(133) n = - 287 - 6 = - 293 (P)
(134) n = - 293 - 6 = - 299 ( = 13*23)
(135) n = - 299 - 6 = - 305
(136) n = - 305 - 6 = - 311 (P)
(137) n = - 311 - 6 = - 317 (P)
(138) n = - 317 - 6 = - 323 ( = 17*19)
(139) n = - 323 - 6 = - 329 X
(140) n = - 329 - 6 = - 335
(141) n = - 335 - 6 = - 341 ( = 11* 31)
(142) n = - 341 - 6 = - 347 (P)
(143) n = - 347 - 6 = - 353 (P)
(144) n = - 353 - 6 = - 359 (P)
(145) n = - 359 - 6 = - 365
(146) n = - 365 - 6 = - 371 X
(147) n = - 371 - 6 = - 377 ( = 13*29)
(148) n = - 377 - 6 = - 383 (P)
(149) n = - 383 - 6 = - 389 (P)
(150) n = - 389 - 6 = - 395
(151) n = - 395 - 6 = - 401 (P)
(152) n = - 401 - 6 = - 407 ( = 11* 37)
(153) n = - 407 - 6 = - 413 X
(154) n = - 413 - 6 = - 419 (P)
(155) n = - 419 - 6 = - 425
(156) n = - 425 - 6 = - 431 (P)
(157) n = - 431 - 6 = - 437 ( = 19*23)
(158) n = - 437 - 6 = - 443 (P)
(159) n = - 443 - 6 = - 449 (P)
(160) n = - 449 - 6 = - 455 X
(161) n = - 455 - 6 = - 461 (P)
(162) n = - 461 - 6 = - 467 (P)
(163) n = - 467 - 6 = - 473 ( = 11*43)
(164) n = - 473 - 6 = - 479 (P)
(165) n = - 479 - 6 = - 485
(166) n = - 485 - 6 = - 491 (P)
(167) n = - 491 - 6 = - 497 X
In general , ( r ) n = 7 - 6 * ( r - 83 ) = 7 - 6 r + 498 = 505 - 6 r
The n's so generated , after excluding all multiples of 5 ,( by
inspection) and all multiples of 7 , ( with sieve , every 1 out of 7 and marked with an X ) and the composite no. not containing factors of 2 or 3 but with at least 1 factor of 11 , 13 , 17 or 19 , had covered 1 and every prime from 5 up to 499 .
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[ 6 ] To generate n = 210 - j
n = 210 - j = 2 * 3 * 5 * 7 - 1 * j ( where j is coprime with 2, 3, 5 and 7 ) for | n | < 300
(1) 210 - - 89 = 299 ( = 13*23)
(2) 210 - - 83 = 293 (P)
(3) 210 - - 79 = 289 ( = 17*17)
(4) 210 - - 73 = 283 (P)
(5) 210 - - 71 = 281 (P)
(6) 210 - - 67 = 277 (P)
(7) 210 - - 61 = 271 (P)
(8) 210 - - 59 = 269 (P)
(9) 210 - - 53 = 263 (P)
(10) 210 - - 47 = 257 (P)
(11) 210 - - 43 = 253 ( = 11*23)
(12) 210 - - 41 = 251 (P)
(13) 210 - - 37 = 247 ( = 13*19)
(14) 210 - - 31 = 241 (P)
(15) 210 - - 29 = 239 (P)
(16) 210 - - 23 = 233 (P)
(17) 210 - - 19 = 229 (P)
(18) 210 - - 17 = 227 (P)
(19) 210 - - 13 = 223 (P)
(20) 210 - - 11 = 221 ( = 13*17)
(21) 210 - - 1 = 211 (P)
(22) 210 - 1 = 209 ( = 11*19)
(23) 210 - 11 = 199 (P)
(24) 210 - 13 = 197 (P)
(25) 210 - 17 = 193 (P)
(26) 210 - 19 = 191 (P)
(27) 210 - 23 = 187 ( = 11*17)
(28) 210 - 29 = 181 (P)
(29) 210 - 31 = 179 (P)
(30) 210 - 37 = 173 (P)
(31) 210 - 41 = 169 ( = 13*13)
(32) 210 - 43 = 167 (P)
(33) 210 - 47 = 163 (P)
(34) 210 - 53 = 157 (P)
(35) 210 - 59 = 151 (P)
(36) 210 - 61 = 149 (P)
(37) 210 - 67 = 143 ( = 11* 13)
(38) 210 - 71 = 139 (P)
(39) 210 - 73 = 137 (P)
(40) 210 - 79 = 131 (P)
(41) 210 - 83 = 127 (P)
(42) 210 - 89 = 121 ( = 11*11)
(43) 210 - 97 = 113 (P)
(44) 210 - 101 = 109 (P)
(45) 210 - 103 = 107 (P)
(46) 210 - 107 = 103 (P)
(47) 210 - 109 = 101 (P)
(48) 210 - 113 = 97 (P)
(49) 210 - 121 = 89 (P)
(50) 210 - 127 = 83 (P)
(51) 210 - 131 = 79 (P)
(52) 210 - 137 = 73 (P)
(53) 210 - 139 = 71 (P)
(54) 210 - 143 = 67 (P)
(55) 210 - 149 = 61 (P)
(56) 210 - 151 = 59 (P)
(57) 210 - 157 = 53 (P)
(58) 210 - 163 = 47 (P)
(59) 210 - 167 = 43 (P)
(60) 210 - 169 = 41 (P)
(61) 210 - 173 = 37 (P)
(62) 210 - 179 = 31 (P)
(63) 210 - 181 = 29 (P)
(64) 210 - 187 = 23 (P)
(65) 210 - 191 = 19 (P)
(66) 210 - 193 = 17 (P)
(67) 210 - 197 = 13 (P)
(68) 210 - 199 = 11 (P)
(69) 210 - 209 = 1
-------------------------
(70) 210 - 211 = -1
(71) 210 - 221 = - 11
......................
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[ 6 A ]
To generate n = 210 - j for 300 < | n | < 500
(1) 210 - - 289 = 499 (P)
(2) 210 - - 283 = 493 ( Composite )
(3) 210 - - 281 = 491 (P)
(4) 210 - - 277 = 487 (P)
(5) 210 - - 271 = 481 (C)
(6) 210 - - 269 = 479 (P)
(7) 210 - - 263 = 473 (C)
(8) 210 - - 257 = 467 (P)
(9) 210 - - 253 = 463 (P)
(10) 210 - - 251 = 461 (P)
(11) 210 - - 247 = 457 (P)
(12) 210 - - 241 = 451 (C)
(13) 210 - - 239 = 449 (P)
(14) 210 - - 233 = 443 (P)
(15) 210 - - 229 = 439 (P)
(16) 210 - - 227 = 437 (C)
(17) 210 - - 223 = 433 (P)
(18) 210 - - 221 = 431 (P)
(19) 210 - - 211 = 421 (P)
(20) 210 - - 209 = 419 (P)
(21) 210 - - 199 = 409 (P)
(22) 210 - - 197 = 407 (C)
(23) 210 - - 193 = 403 (C)
(24) 210 - - 191 = 401 (P)
(25) 210 - - 187 = 397 (P)
(26) 210 - - 181 = 391 (C)
(27) 210 - - 179 = 389 (P)
(28) 210 - - 173 = 383 (P)
(29) 210 - - 169 = 379 (P)
(30) 210 - - 167 = 377 (C)
(31) 210 - - 163 = 373 (P)
(32) 210 - - 157 = 367 (P)
(33) 210 - - 151 = 361 (C)
(34) 210 - - 149 = 359 (P)
(35) 210 - - 143 = 353 (P)
(36) 210 - - 139 = 349 (P)
(37) 210 - - 137 = 347 (P)
(38) 210 - - 131 = 341 (C)
(39) 210 - - 127 = 337 (P)
(40) 210 - - 121 = 331 (P)
(41) 210 - - 113 = 323 (C)
(42) 210 - - 109 = 319 (C)
(43) 210 - - 107 = 317 (P)
(44) 210 - - 103 = 313 (P)
(45) 210 - - 101 = 311 (P)
(46) 210 - - 97 = 307 (P)
Thus if we are given a list of prime no. < 300 ,
together with a list of composite no. < 500 which
without factors of 2 , 3 , 5 or 7 but with at least
1 factor of 11 , 13 , 17 or 19 . ( totally 23 ones ) ,
we can generate a list of primes between 300 and 500 with the expression n = 210 - j ( where j is coprime with 210 ) after excluding the composite no. between 300 and 500 given above .
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[ 7 ] To generate primes between 1500 and 2000 .
To generate primes between 1500 and 2000 by using the expression
n = 1470 + j where 1470 = 2 * 3 * 5 * 7 ^2 and j being coprime with 1470 .
Preliminaries :
(I)
(A) Primes between 1500 - 1470 = 30 and 2000 - 1470 = 530 .
(B) Composite no . without factors of 2 , 3 , 5 or 7 but with at least
1 factor of primes between 11 and the one just < √ 530 , i.e. 23 .
from 11^2 = 121 to 530 .
The combined list of (A) and (B) will be :
31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 121* 127 131 137 139 143* 149 151 157 163 167 169* 173 179 181 187* 191 193 197 199 209* 211 221* 223 227 229 233 239 241 247* 251 253* 257 263 269 271 277 281 283 289* 293 299* 307 311 313 317 319* 323* 331 337 341* 347 349 353 359 361* 367 373 377* 379 383 389 391* 397 401 403* 407* 409 419 421 431 433 437* 439 443 449 451* 457 461 463 467 473* 479 481* 487 491 493* 499 503 509 517* 521 523 527* 529* ( * : composite )
Then j will be assigned with the above values .
(II)
(C) Composite no . without factors of 2 , 3 , 5 or 7 but with at least 1 factor of primes between 11 and the one just < √ 2000 , i.e. 43 . between 1500 and 2000 .
(1) Arranged by individual primes
11 ( from 11 * 137 = 1507 to 11 * 181 = 1991 , including 11 * 11 * 13 = 11 * 143 = 1573 .)
: 1507 , 1529 , 1573 , 1639 , 1661 , 1727 , 1793 , 1837 , 1903 , 1969 , 1991 .
13 ( from 13 * 121 = 1573 to 13 * 151 = 1963 , including 13 * 11 * 13 = 13 * 143 = 1859 .)
: 1573 , 1651 , 1703 , 1781 , 1807 , 1859 , 1937 , 1963 .
17 ( from 17 * 89 = 1513 to 17 * 113 = 1921 )
: 1513 , 1649 , 1717 , 1751 , 1819 , 1853 , 1921 .
19 ( from 19 * 79 = 1501 to 19 * 103 = 1957 )
: 1501 , 1577 , 1691 , 1843 , 1919 , 1957 .
23 ( from 23 * 67 = 1541 to 23 * 83 = 1909 )
: 1541 , 1633 , 1679 , 1817 , 1909 .
29 ( from 29 * 53 = 1537 to 29 * 67 = 1943 )
: 1537 , 1711 , 1769 , 1943 .
31 ( from 31 * 53 = 1643 to 31 * 61 = 1891 )
: 1643 , 1829 , 1891 .
37 ( from 37 * 41 = 1517 to 37 * 53 = 1961 )
: 1517 , 1591 , 1739 , 1961 .
41 ( from 41 * 37 = 1517 to 41 * 47 = 1927 )
: 1517 , 1681 , 1763 , 1927 .
43 ( from 43 * 37 = 1591 to 43 * 43 = 1849 )
: 1591 , 1763 , 1849 .
(2) Arranged by ascending order for all : (totally 51 ones , about 44 % of total n's .)
15xx : 1501 , 1507 , 1513 , 1517 , 1529 , 1537 , 1541 , 1573 , 1577 , 1591 ,
16xx : 1633 , 1639 , 1643 , 1649 , 1651 , 1661 , 1679 , 1681 , 1691 ,
17xx : 1703 , 1711 , 1717 , 1727 , 1739 , 1751 , 1763 ,1769 , 1781 , 1793 ,
18xx : 1807 , 1817 , 1819 , 1829 , 1837 , 1843 , 1849 , 1853 , 1859 , 1891 ,
19xx : 1903 , 1909 , 1919 , 1921 , 1927 , 1937 , 1943 , 1957 , 1961 , 1963 , 1969 , 1991
The above values will be excluded from the n's generated in
order to remain primes only .
n = 1470 + j between 1500 and 2000 :
(1) 1470 + 31 = 1501
(2) 1470 + 37 = 1507
(3) 1470 + 41 = 1511 (P)
(4) 1470 + 43 = 1513
(5) 1470 + 47 = 1517
(6) 1470 + 53 = 1523 (P)
(7) 1470 + 59 = 1529
(8) 1470 + 61 = 1531 (P)
(9) 1470 + 67 = 1537
(10) 1470 + 71 = 1541
(11) 1470 + 73 = 1543 (P)
(12) 1470 + 79 = 1549 (P)
(13) 1470 + 83 = 1553 (P)
(14) 1470 + 89 = 1559 (P)
(15) 1470 + 97 = 1567 (P)
(16) 1470 + 101 = 1571 (P)
(17) 1470 + 103 = 1573
(18) 1470 + 107 = 1577
(19) 1470 + 109 = 1579 (P)
(20) 1470 + 113 = 1583 (P)
(21) 1470 + 121 = 1591
(22) 1470 + 127 = 1597 (P)
(23) 1470 + 131 = 1601 (P)
(24) 1470 + 137 = 1607 (P)
(25) 1470 + 139 = 1609 (P)
(26) 1470 + 143 = 1613 (P)
(27) 1470 + 149 = 1619 (P)
(28) 1470 + 151 = 1621 (P)
(29) 1470 + 157 = 1627 (P)
(30) 1470 + 163 = 1633
(31) 1470 + 167 = 1637 (P)
(32) 1470 + 169 = 1639
(33) 1470 + 173 = 1643
(34) 1470 + 179 = 1649
(35) 1470 + 181 = 1651
(36) 1470 + 187 = 1657 (P)
(37) 1470 + 191 = 1661
(38) 1470 + 193 = 1663 (P)
(39) 1470 + 197 = 1667 (P)
(40) 1470 + 199 = 1669 (P)
(41) 1470 + 209 = 1679
(42) 1470 + 211 = 1681
(43) 1470 + 221 = 1691
(44) 1470 + 223 = 1693 (P)
(45) 1470 + 227 = 1697 (P)
(46) 1470 + 229 = 1699 (P)
(47) 1470 + 233 = 1703
(48) 1470 + 239 = 1709 (P)
(49) 1470 + 241 = 1711
(50) 1470 + 247 = 1717
(51) 1470 + 251 = 1721 (P)
(52) 1470 + 253 = 1723 (P)
(53) 1470 + 257 = 1727
(54) 1470 + 263 = 1733 (P)
(55) 1470 + 269 = 1739
(56) 1470 + 271 = 1741 (P)
(57) 1470 + 277 = 1747 (P)
(58) 1470 + 281 = 1751
(59) 1470 + 283 = 1753 (P)
(60) 1470 + 289 = 1759 (P)
(61) 1470 + 293 = 1763
(62) 1470 + 299 = 1769
(63) 1470 + 307 = 1777 (P)
(64) 1470 + 311 = 1781
(65) 1470 + 313 = 1783 (P)
(66) 1470 + 317 = 1787 (P)
(67) 1470 + 319 = 1789 (P)
(68) 1470 + 323 = 1793
(69) 1470 + 331 = 1801 (P)
(70) 1470 + 337 = 1807
(71) 1470 + 341 = 1811 (P)
(72) 1470 + 347 = 1817
(73) 1470 + 349 = 1819
(74) 1470 + 353 = 1823 (P)
(75) 1470 + 359 = 1829
(76) 1470 + 361 = 1831 (P)
(77) 1470 + 367 = 1837
(78) 1470 + 373 = 1843
(79) 1470 + 377 = 1847 (P)
(80) 1470 + 379 = 1849
(81) 1470 + 383 = 1853
(82) 1470 + 389 = 1859
(83) 1470 + 391 = 1861 (P)
(84) 1470 + 397 = 1867 (P)
(85) 1470 + 401 = 1871 (P)
(86) 1470 + 403 = 1873 (P)
(87) 1470 + 407 = 1877 (P)
(88) 1470 + 409 = 1879 (P)
(89) 1470 + 419 = 1889 (P)
(90) 1470 + 421 = 1891
(91) 1470 + 431 = 1901 (P)
(92) 1470 + 433 = 1903
(93) 1470 + 437 = 1907 (P)
(94) 1470 + 439 = 1909
(95) 1470 + 443 = 1913 (P)
(96) 1470 + 449 = 1919
(97) 1470 + 451 = 1921
(98) 1470 + 457 = 1927
(99) 1470 + 461 = 1931 (P)
(100) 1470 + 463 = 1933 (P)
(101) 1470 + 467 = 1937
(102) 1470 + 473 = 1943
(103) 1470 + 479 = 1949 (P)
(104) 1470 + 481 = 1951 (P)
(105) 1470 + 487 = 1957
(106) 1470 + 491 = 1961
(107) 1470 + 493 = 1963
(108) 1470 + 499 = 1969
(109) 1470 + 503 = 1973 (P)
(110) 1470 + 509 = 1979 (P)
(111) 1470 + 517 = 1987 (P)
(112) 1470 + 521 = 1991
(113) 1470 + 523 = 1993 (P)
(114) 1470 + 527 = 1997 (P)
(115) 1470 + 529 = 1999 (P)
Thus we have generated all the 64 primes from 1500 to 2000 .
Offline
[ 8 ] To generate primes < 500
To generate primes from 1 to 500 using the expression
n = 252 + j where 252 = 2 ^2 * 3 ^2 * 7 and j being coprime with 252 .
Preliminaries :
(I) For j :
(A) +ve (1 and primes ) < 500 - 252 = 248 excluding 2 , 3 and 7
and -ve (1 and primes ) with absolute value < 252 excluding - 7 , - 3 and - 2 .
The list will be :
- 251 - 241 -239 - 233 - 229 - 227 -223 - 211 - 199 - 197 - 193
- 191 - 181 - 179 - 173 - 167 - 163 - 157 - 151 - 149 - 139
- 137 - 131 - 127 - 113 - 109 - 107 - 103 - 101 - 97 - 89
- 83 - 79 - 73 - 71 - 67 - 61 - 59 - 53 - 47 - 43 - 41 - 37
- 31 - 29 - 23 - 19 - 17 - 13 - 11 - 5 -1 1 5 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 .
(B) Composite no . < 248 and without factors of 2 , 3 or 7 but
(1) with only 1 factor of 5 and the other factor being ± ( 1 or primes ) < 49.6 , i.e. ± 47
The list will be :
± 5* 47 = 235 , 5* 43 = 215 , 5 * 41 = 205 , 5 * 37 = 185 ,
5 * 31 = 155 , 5 * 29 = 145 , 5 * 23 = 115 , 5 * 19 = 95 ,
5 * 17 = 85 , 5 * 13 = 65 , 5 * 11 = 55
or (2) with exactly 2 factors of 5 , ( i.e. with 1 factor of 25 ) and the other factor being ± 1 .
The list will be :
- 25 , 25
or (3) with exactly 3 factors of 5 , ( i.e. with 1 factor of 125 ) and the other factor being ±1 .
The list will be :
- 125 , 125
or (4) with no factor of 5 , and with at least 1 factor of primes between 11 and the one just < √ 248 , i.e. 13 . from 11^2 = 121 to 248 .
The list will be :
±
* | 11 | 13 | 17 | 19 ||
11 | 121 | 143 | 187 | 209 ||
13 | / | 169 | 221 | 247 ||
Arranged in ascending order
±
121 , 143 , 169 , 187 , 209 , 221 , 247 ,
The combined list of (A) and (B) will be :
- 251 - 247 * - 241 -239 - 235 * - 233 - 229 - 227 -223 - 221 * - 215 * - 211 - 209 * - 205 * - 199 - 197 - 193 - 191 - 187 * - 185 * - 181 - 179 - 173 - 169 * - 167 - 163 - 157 - 155 * - 151 - 149 - 145 * - 143 * - 139 - 137 - 131 - 127 - 125 * - 121 * - 115 * - 113 - 109 - 107 - 103 - 101 - 97 - 95 * - 89 - 85 *
- 83 - 79 - 73 - 71 - 67 - 65 * - 61 - 59 - 55 * - 53 - 47 - 43 - 41 - 37 - 31 - 29 - 25 * - 23 - 19 - 17 - 13 - 11 - 5 -1 1 5 11 13 17 19 23 25 * 29 31 37 41 43 47 53 55 * 59 61 65 * 67 71 73 79 83 85 * 89 95 * 97 101 103 107 109 113 115 * 121 * 125 * 127 131 137 139 143 * 145 * 149 151 155 * 157 163 167 169 * 173 179 181 185 * 187 * 191 193 197 199 205 * 209 * 211 215 * 221 * 223 227 229 233 235 * 239 241 247 * .
( * : composite )
Then j will be assigned with the above values .
(II) For n :
(C)
(1) Composite no . < 500 and without factors of 2 , 3 , 7 or 5 but with at least 1 factor of primes between 11 and the one just < √ 500 , i.e. 19 .
the list arranged by individual primes will be :
11 : ( Since 500 / 11 > 45 , thus n will be from 11 * 11 = 121 to
11 * 43 = 473 . )
13 : ( Since 500 / 13 > 38 , thus n will be from 13 * 13 = 169 to 13 * 37 = 481 )
17 : ( Since 500 / 17 > 29 , thus n will be from 17 * 17 = 289 to 17 * 29 = 493 . )
19 : ( Since 500 / 19 > 26 , thus n will be from 19 * 19 = 361 to 19 * 23 = 437 . )
( i ) Arranged in products of prime factors
* | 11 | 13 | 17 | 19 || 23 | 29 | 31 | 37 | 41 | 43 |
11 | 121 | 143 | 187 | 209 || 253 | 319 | 341 | 407 | 451 | 473 |
13 | / | 169 | 221 | 247 || 299 | 377 | 403 | 481 | / | / |
17 | / | / | 289 | 323 || 391 | 493 | / | / | / | / |
19 | / | / | / | 361 || 437 | / | / | / | / | / |
(ii) Arranged by ascending order for all :
1xx : 121 , 143 , 169 , 187
2xx : 209 , 221 , 247 , 253 , 289 , 299
3xx : 319 , 323 , 341 , 361 , 377 , 391
4xx : 403 , 407 , 437 ,451 , 473 , 481 , 493
The above values will be excluded from the n's generated in
order to remain primes only .
(2) Composite no. < 500 and without factors of 2 , 3 or 7 but with
at least 1 factor of 5 . These n. will all ends in 5 , there is no need to list all these n's , those n's end with 5 will all be excluded . (except 5 )
( in fact , for j with negative values ending in 7 , or positive
values ending in 3 , values of n ending in 5 will be generated .
These values of j may be deleted previously except for j = - 247 which yields n = 5 . )
n = 252 + j from 1 to 500 :
(1) 252 + - 251 = 1
(2) 252 + - 247 = 5 (P)
(3) 252 + - 241 = 11 (P)
(4) 252 + - 239 = 13 (P)
(5) 252 + - 235 = 17 (P)
(6) 252 + - 233 = 19 (P)
(7) 252 + - 229 = 23 (P)
(8) 252 + - 227 = 25
(9) 252 + - 223 = 29 (P)
(10) 252 + - 221 = 31 (P)
(11) 252 + - 215 = 37 (P)
(12) 252 + - 211 = 41 (P)
(13) 252 + - 209 = 43 (P)
(14) 252 + - 205 = 47 (P)
(15) 252 + - 199 = 53 (P)
(16) 252 + - 197 = 55
(17) 252 + - 193 = 59 (P)
(18) 252 + - 191 = 61 (P)
(19) 252 + - 187 = 65
(20) 252 + - 185 = 67 (P)
(21) 252 + - 181 = 71 (P)
(22) 252 + - 179 = 73 (P)
(23) 252 + - 173 = 79 (P)
(24) 252 + - 169 = 83 (P)
(25) 252 + - 167 = 85
(26) 252 + - 163 = 89 (P)
(27) 252 + - 157 = 95
(28) 252 + - 155 = 97 (P)
(29) 252 + - 151 = 101 (P)
(30) 252 + - 149 = 103 (P)
(31) 252 + - 145 = 107 (P)
(32) 252 + - 143 = 109 (P)
(33) 252 + - 139 = 113 (P)
(34) 252 + - 137 = 115
(35) 252 + - 131 = 121
(36) 252 + - 127 = 125
(37) 252 + - 125 = 127 (P)
(38) 252 + - 121 = 131 (P)
(39) 252 + - 115 = 137 (P)
(40) 252 + - 113 = 139 (P)
(41) 252 + - 109 = 143
(42) 252 + - 107 = 145
(43) 252 + - 103 = 149 (P)
(44) 252 + - 101 = 151 (P)
(45) 252 + - 97 = 155
(46) 252 + - 95 = 157 (P)
(47) 252 + - 89 = 163 (P)
(48) 252 + - 85 = 167 (P)
(49) 252 + - 83 = 169
(50) 252 + - 79 = 173 (P)
(51) 252 + - 73 = 179 (P)
(52) 252 + - 71 = 181 (P)
(53) 252 + - 67 = 185
(54) 252 + - 65 = 187
(55) 252 + - 61 = 191 (P)
(56) 252 + - 59 = 193 (P)
(57) 252 + - 55 = 197 (P)
(58) 252 + - 53 = 199 (P)
(59) 252 + - 47 = 205
(60) 252 + - 43 = 209
(61) 252 + - 41 = 211 (P)
(62) 252 + - 37 = 215
(63) 252 + - 31 = 221
(64) 252 + - 29 = 223 (P)
(65) 252 + - 25 = 227 (P)
(66) 252 + - 23 = 229 (P)
(67) 252 + - 19 = 233 (P)
(68) 252 + - 17 = 235
(69) 252 + - 13 = 239 (P)
(70) 252 + - 11 = 241 (P)
(71) 252 + - 5 = 247
(72) 252 + - 1 = 251 (P)
(73) 252 + 1 = 253
(74) 252 + 5 = 257 (P)
(75) 252 + 11 = 263 (P)
(76) 252 + 13 = 265
(77) 252 + 17 = 269 (P)
(78) 252 + 19 = 271 (P)
(79) 252 + 23 = 275
(80) 252 + 25 = 277 (P)
(81) 252 + 29 = 281 (P)
(82) 252 + 31 = 283 (P)
(83) 252 + 37 = 289
(84) 252 + 41 = 293 (P)
(85) 252 + 43 = 295
(86) 252 + 47 = 299
(87) 252 + 53 = 305
(88) 252 + 55 = 307 (P)
(89) 252 + 59 = 311 (P)
(90) 252 + 61 = 313 (P)
(91) 252 + 65 = 317 (P)
(92) 252 + 67 = 319
(93) 252 + 71 = 323
(94) 252 + 73 = 325
(95) 252 + 79 = 331 (P)
(96) 252 + 83 = 335
(97) 252 + 85 = 337 (P)
(98) 252 + 89 = 341
(99) 252 + 95 = 347 (P)
(100) 252 + 97 = 349 (P)
(101) 252 + 101 = 353 (P)
(102) 252 + 103 = 355
(103) 252 + 107 = 359 (P)
(104) 252 + 109 = 361
(105) 252 + 113 = 365
(106) 252 + 115 = 367 (P)
(107) 252 + 121 = 373 (P)
(108) 252 + 125 = 377
(109) 252 + 127 = 379 (P)
(110) 252 + 131 = 383 (P)
(111) 252 + 137 = 389 (P)
(112) 252 + 139 = 391
(113) 252 + 143 = 395
(114) 252 + 145 = 397 (P)
(115) 252 + 149 = 401 (P)
(116) 252 + 151 = 403
(117) 252 + 155 = 407
(118) 252 + 157 = 409 (P)
(119) 252 + 163 = 415
(120) 252 + 167 = 419 (P)
(121) 252 + 169 = 421 (P)
(122) 252 + 173 = 425
(123) 252 + 179 = 431 (P)
(124) 252 + 181 = 433 (P)
(125) 252 + 185 = 437
(126) 252 + 187 = 439 (P)
(127) 252 + 191 = 443 (P)
(128) 252 + 193 = 445
(129) 252 + 197 = 449 (P)
(131) 252 + 199 = 451
(132) 252 + 205 = 457 (P)
(133) 252 + 209 = 461 (P)
(134) 252 + 211 = 463 (P)
(135) 252 + 215 = 467 (P)
(136) 252 + 221 = 473
(137) 252 + 223 = 475
(138) 252 + 227 = 479 (P)
(139) 252 + 229 = 481
(140) 252 + 233 = 485
(141) 252 + 235 = 487 (P)
(142) 252 + 239 = 491 (P)
(143) 252 + 241 = 493
(144) 252 + 247 = 499 (P)
Thus after we are given the list of primes between 1 and 252 ,
( about 1/2 of 500 .) we can generate further the other 1/2
by using the expression n = 252 + j .
Last edited by mr.wong (2017-01-25 15:35:09)
Offline
Product table of nos. coprime with 2 , 3 , 5 and 7 .
Let P denotes primes relatively prime with 2 , 3 , 5 and 7 ,
i.e. P being primes ≧ 11 and N denotes nos. relatively prime
with 2 , 3 , 5 and 7 , where N may be prime or composite .
The following table shows the products of P and N ( both ≧ 11 ) .
P from 11 to 163 and N from 11 to 223 including composite nos.
121 , 143 , 169 , 187 , 209 and 221 .
This table with products of P * N arranged in ascending order will be useful in generating primes .
121 1003 2021 3007 4009 5017 6031 7003
143 1007 2033 3013 4031 5029 6059 7009
169 1027 2041 3043 4033 5041 6061 7031
187 1037 2047 3053 4061 5053 6077 7067
209 1067 2057 3071 4087 5063 6107 7081
221 1073 2059 3077 4117 5069 6109 7093
247 1079 2071 3097 4141 5083 6119 7097
253 1081 2077 3103 4147 5123 6149 7139
289 1111 2101 3127 4163 5129 6157 7139
299 1121 2117 3131 4171 5141 6161 7141
319 1133 2123 3139 4181 5143 6169 7169
323 1139 2147 3149 4183 5177 6179 7171
341 1147 2159 3151 4187 5183 6191 7181
361 1157 2167 3161 4189 5191 6241 7261
377 1159 2171 3173 4199 5203 6253 7267
391 1177 2173 3179 4223 5207 6283 7289
403 1189 2183 3193 4237 5239 6313 7303
407 1199 2189 3197 4247 5249 6319 7313
437 1207 2197 3211 4301 5251 6401 7339
451 1219 2197 3233 4307 5291 6407 7363
473 1241 2201 3239 4309 5293 6409 7367
481 1243 2209 3247 4321 5311 6413 7373
493 1247 2227 3277 4331 5329 6413 7379
517 1261 2231 3281 4343 5353 6431 7381
527 1271 2249 3287 4379 5363 6437 7381
529 1273 2257 3289 4387 5371 6439 7387
533 1313 2263 3293 4393 5423 6467 7421
551 1331 2279 3317 4399 5429 6479 7439
559 1333 2291 3337 4429 5459 6493 7519
583 1339 2299 3349 4433 5461 6497 7571
589 1343 2321 3379 4439 5513 6499 7579
611 1349 2323 3383 4453 5539 6527 7597
629 1357 2327 3397 4469 5549 6533 7661
649 1363 2329 3401 4477 5561 6541 7663
667 1369 2353 3403 4489 5587 6557 7667
671 1387 2363 3427 4531 5597 6623 7697
689 1391 2369 3431 4553 5609 6649 7729
697 1397 2407 3439 4559 5611 6667 7733
703 1403 2413 3473 4577 5617 6683 7739
713 1411 2419 3481 4601 5633 6697 7783
731 1417 2431 3503 4619 5671 6721 7807
737 1441 2449 3509 4633 5687 6751 7811
767 1457 2453 3551 4661 5699 6767 7831
779 1469 2461 3553 4681 5717 6847 7849
781 1501 2479 3569 4687 5723 6851 7897
793 1507 2483 3587 4699 5756 6887 7913
799 1513 2489 3589 4717 5771 6889 7921
803 1517 2491 3599 4727 5777 6893 7943
817 1529 2501 3611 4747 5797 6901 7957
841 1537 2507 3629 4757 5809 6913 7979
851 1541 2509 3649 4807 5863 6919 7991
869 1573 2533 3667 4819 5891 6929
871 1577 2537 3683 4841 5893 6943
893 1591 2561 3713 4843 5917
899 1633 2567 3721 4847 5921
901 1639 2573 3737 4853 5959
913 1643 2581 3743 4859 5963
923 1649 2587 3749 4867 5977
943 1651 2599 3751 4891 5983
949 1661 2603 3757 4897 5989
961 1679 2623 3763 4901
979 1681 2627 3781 4961
989 1691 2641 3791
1703 2669 3799
1711 2701 3811
1717 2717 3827
1727 2743 3841
1739 2747 3869
1751 2759 3887
1763 2763 3901
1769 2771 3937
1781 2773 3953
1793 2783 3959
1807 2809 3971
1817 2813 3973
1819 2831 3977
1829 2839 3979
1837 2867
1843 2869
1849 2873
1853 2881
1859 2899
1891 2911
1903 2921
1909 2923
1919 2929
1921 2941
1927 2983
1937 2987
1943 2993
1957
1961
1963
1969
1991
8003 9017 10001 11009 12017 13031 14003
8023 9047 10033 11021 12091 13039 14017
8041 9061 10043 11023 12127 13067 14027
8051 9071 10057 11033 12139 13081 14039
8077 9073 10117 11041 12191 13157 14089
8083 9089 10123 11077 12193 13189 14111
8107 9143 10147 11147 12221 13193 14129
8107 9167 10153 11183 12283 13199 14137
8131 9169 10187 11189 12317 13213 14141
8137 9179 10201 11227 12319 13231 14299
8159 9211 10207 11269 12331 13261 14317
8177 9259 10229 11297 12337 13277 14351
8201 9263 10309 11303 12367 13289 14359
8213 9271 10349 11323 12371 13333 14381
8249 9301 10379 11371 12403 13351 14443
8251 9313 10387 11387 12449 13439 14453
8299 9353 10403 11407 12463 13481 14507
8321 9379 10439 11413 12529 13483 14527
8357 9409 10441 11449 12533 13493 14659
8383 9487 10481 11461 12629 13529 14729
8413 9503 10519 11537 12707 13561 14773
8437 9523 10541 11573 12709 13589 14803
8453 9559 10547 11591 12727 13603 14807
8471 9563 10553 11623 12749 13651 14839
8479 9577 10561 11639 12769 13667 14857
8507 9581 10573 11651 12797 13673 14863
8509 9589 10579 11659 12827 13677 14873
8549 9593 10609 11663 12851 13703 14933
8557 9617 10679 11713 12871 13837 14941
8569 9701 10721 11737 12877 13843 14981
8591 9727 10769 11741 12931 13861
8611 9797 10807 11771 12947 13871
8633 9823 10823 11773 13943
8639 9853 10873 11819 13973
8651 9869 10877 11857 13987
8723 9911 10919 11869
8777 9917 10921 11881
8789 9943 10961 11881
8791 9971 10981 11899
8833 9983 11929
8851 9991 11993
8881 11999
8909
8927
8957
8977
8987
8989
15023 16019 17063 18029 19043 20009 21037
15041 16109 17069 18079 19097 20099 21079
15049 16129 17113 18083 19109 20227 21109
15089 16133 17161 18139 19153 20273 21131
15151 16157 17177 18161 19177 20291 21209
15229 16159 17201 18203 19261 20383 21293
15247 16171 17347 18209 19291 20413 21307
15251 16199 17363 18271 19303 20437 21311
15257 16241 17399 18281 19321 20453 21353
15301 16279 17407 18343 19367 20467 21437
15347 16351 17441 18419 19493 20497 21463
15367 16393 17459 18421 19511 20567 21473
15397 16459 17473 18437 19519 20651 21509
15403 16463 17513 18509 19549 20687 21527
15481 16511 17533 18511 19591 20701 21583
15521 16517 17557 18527 19669 20711 21593
15553 16577 17617 18601 19673 20819 21631
15563 16637 17653 18643 19723 20989 21691
15587 16643 17711 18721 19729 21733
15691 16669 17741 18733 19781 21809
15707 16781 17767 18769 19847 21823
15721 16789 17819 18779 19877 21877
15811 16799 17869 18857 19879 21971
15833 16819 17947 18871 19897
15851 16837 18887 19939
15853 16867 18923
15857 16999 18997
15931
15943
22139 23153 24047 25019 26069 27029 28067
22201 23213 24089 25021 26123 27161 28103
22261 23309 24257 25159 26167 27221 28199
22321 23393 24287 25181 26219 27263 28237
22331 23449 24307 25199 26441 27331 28321
22363 23491 24497 25217 26533 27379 28417
22451 23617 24511 25273 26543 27383 28459
22487 23647 24523 25283 26549 27547 28633
22499 23701 24613 25519 26569 27641 28757
22523 23707 24649 25591 26671 27661 28841
22577 23711 24797 25619 26797 27863 28907
22657 23749 24881 25777 26827 28951
22663 23843 24883 25807 26969
22733 23861 24973 25993 26989
22781
22801
22879
22969
22987
22999
29051 30049
29143 30277
29177 30301
29213 30481
29329 30551
29353 30719
29359 30929
29503 30997
29651 31133
29747 31141
29987 31243
31439
31459
31559
31861
32111
32437
32813
32929
33127
33227
33371
33673
34067
34393
34697
35011
36023
36349
Offline
[ 9 ] To generate primes between 20000 and 21000
To generate primes between 20000 and 21000 using the expression
n = 21000 - j where 21000 = 3 * 7 * ( 2 * 5 ) ^ 3 and j being coprime with 21000 .
Preliminaries :
(I) For j
(A) 1 and primes < 21000 - 20000 = 1000 excluding 2 , 3 , 5 and 7 :
1 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997
(B) Composite no. with no factor of 2 , 3 , 5 or 7 but with at least one prime factor between 11 and the one just < √ 1000 , i.e. 31 from 11^2 = 121 to 1000 arranged in ascending order : ( Taken from product table . )
1xx : 121 , 143 , 169 , 187 ,
2xx : 209 , 221 , 247 , 253 , 289 , 299 ,
3xx : 319 , 323 , 341 , 361 , 377 , 391 ,
4xx : 403 , 407 , 437 , 451 , 473 , 481 , 493 ,
5xx : 517 , 527 , 529 , 533 , 551 , 559 , 583 , 589,
6xx : 611 , 629 , 649 , 667 , 671 , 689 , 697 ,
7xx : 703 , 713 , 731 , 737 , 767 , 779 , 781 , 793 , 799 ,
8xx : 803 , 817 , 841 , 851 , 869 , 871 , 893 , 899 ,
9xx : 901 , 913 , 923 , 943 , 949 , 961 , 979 , 989 .
The combined list of (A) and (B) will be :
1 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 121 * 127 131 137 139 143 * 149 151 157 163 167 169 * 173 179 181 187 * 191 193 197 199 209 * 211 221 * 223 227 229 233 239 241 247 * 251 253 * 257 263 269 271 277 281 283 289 * 293 299 * 307 311 313 317 319 * 323 * 331 337 341 * 347 349 353 359 361 * 367 373 377 * 379 383 389 391 * 397 401 403 * 407 * 409 419 421 431 433 437 * 439 443 449 451 * 457 461 463 467 473 * 479 481 * 487 491 493 * 499 503 509 517 * 521 523 527 * 529 * 533 * 541 547 551 * 557 559 * 563 569 571 577 583 * 587 589 * 593 599 601 607 611 * 613 617 619 629 * 631 641 643 647 649 * 653 659 661 667 * 671 * 673 677 683 689 * 691 697 * 701 703 * 709 713 * 719 727 731 * 733 737 * 739 743 751 757 761 767 * 769 773 779 * 781 * 787 793 * 797 799 * 803 * 809 811 817 * 821 823 827 829 839 841 * 851 * 853 857 859 863 869 * 871 * 877 881 883 887 893 * 899 * 901 * 907 911 913 * 919 923 * 929 937 941 943 * 947 949 * 953 961 * 967 971 977 979 * 983 989 * 991 997
( * : composite )
Then j will be assigned with the above values .
(II) For n :
(C) Composite no . without factors of 2 , 3 , 5 or 7 but with at least 1 factor of primes between 11 and the one just < √ 21000 , i.e. 139 between 20000 and 21000 .
(i) Arranged by individual primes :
[11] Since 20000 / 11 = 1818.1 and 21000 / 11 = 1909 .09 , thus the quotients ( Q ) will be between
1819 and 1909 .
(1) For n = 11 * P ( P : prime ) , then Q will be from 1823 to 1907 . ( Taken from prime table )
The list of n will be :
11 * 1823 = 20053
11 * 1831 = 20141
11 * 1847 = 20317
11 * 1861 = 20471
11* 1867 = 20537
11 * 1871 = 20581
11 * 1873 = 20603
11 * 1877 = 20647
11 * 1879 = 20669
11 * 1889 = 20779
11 * 1901 = 20911
11 * 1907 = 20977
(2) For n = 11 * C ( C : composite ) , then Q will be from 1819 to 1909 . ( Taken from product table )
The list of n will be :
11 * 1819 = 20009
11 * 1829 = 20119
11 * 1837 = 20207
11 * 1843 = 20273
11 * 1849 = 20339
11 * 1853 = 20383
11 * 1859 = 20449
11 * 1891 = 20801
11 * 1903 = 20933
11 * 1909 = 20999
Thus the combined list of n will be :
20009 , 20053 , 20119 , 20141 , 20207 , 20273 , 20317 , 20339 , 20383 , 20449 ,
20471 , 20537 , 20581 , 20603 , 20647 , 20669 , 20779 , 20801 , 20911 , 20933 ,
20977 , 20999 .
[13] Since 20000 / 13 = 1538 .4 and 21000 / 13 = 1615 .3 , thus Q will be
between 1539 and 1615 .
(1) For n = 13 * P , then Q will be from 1543 to 1613 .
The list of n will be :
13 * 1543 = 20059
13 * 1549 = 20137
13 * 1553 = 20189
13 * 1559 = 20267
13 * 1567 = 20371
13 * 1571 = 20423
13 * 1579 = 20527
13 * 1583 = 20579
13 * 1597 = 20761
13 * 1601 = 20813
13 * 1607 = 20891
13 * 1609 = 20917
13 * 1613 = 20969
(2) For n = 13 * C , then Q will be from 1541 to 1591 .
The list of n will be :
13 * 1541 = 20033
13 * 1573 = 20449
13 * 1577 = 20501
13 * 1591 = 20683
Thus the combined list of n will be :
20033 , 20059 , 20137 , 20189 , 20267 , 20371 , 20423 , 20449 , 20501 , 20527 , 20579 , 20683 , 20761 , 20813 , 20891 , 20917 , 20969 .
[17] Since 20000 / 17 = 1176 .4 and 21000 / 17 = 1235.2 ,
thus Q will be between 1177 and 1235 .
(1) For n = 17 * P , then Q will be from 1181 to 1231 .
The list of n will be :
17 * 1181 = 20077
17 * 1187 = 20179
17 * 1193 = 20281
17 * 1201 = 20417
17 * 1213 = 20621
17 * 1217 = 20689
17 * 1223 = 20791
17 * 1229 = 20893
17 * 1231 = 20927
(2) For n = 17 * C , then Q will be from 1177 to 1219 .
The list of n will be :
17 * 1177 = 20009
17 * 1189 = 20213
17 * 1199 = 20383
17 * 1207 = 20519
17 * 1219 = 20723
Thus the combined list of n will be :
20009 , 20077 , 20179 , 20213 , 20281 , 20383 , 20417 , 20519 , 20621 , 20689 , 20723 , 20791 , 20893 , 20927 .
[19] Since 20000 / 19 =1052 .6 and 21000 / 19 = 1105.2 ,
thus Q will be between 1053 and 1105 .
(1) For n = 19 * P , then Q will be from 1061 to 1103 .
The list of n will be :
19 * 1061 = 20159
19 * 1063 = 20197
19 * 1069 = 20311
19 * 1087 = 20653
19 * 1091 = 20729
19 * 1093 = 20767
19 * 1097 = 20843
19 * 1103 = 20957
(2) For n = 19 * C , then Q will from 1067 to 1081 :
The list of n will be :
19 * 1067 = 20273
19 * 1073 = 20387
19 * 1079 = 20501
19 * 1081 = 20539
Thus the combined list of n will be :
20159 , 20197 , 20273 , 20311 , 20387 , 20501 , 20539 , 20653 ,
20729 , 20767 , 20843 , 20957 .
[23] Since 20000 / 23 = 869.5 and 21000 / 23 = 913.04 ,
thus Q will be between 870 and 913 .
(1) For n = 23 * P , then Q will be from 877 to 911 .
The list of n will be :
23 * 877 = 20171
23 * 881 = 20263
23 * 883 = 20309
23 * 887 = 20401
23 * 907 = 20861
23 * 911 = 20953
(2) For n = 23 * C , then Q will be from 871 to 913 .
The list of n will be :
23 * 871 = 20033
23 * 893 = 20539
23 * 899 = 20677
23 * 901 = 20723
23 * 913 = 20999
Thus the combined list of n will be :
20033 , 20171 , 20263 , 20309 , 20401 , 20539 , 20677 , 20723 ,
20861 , 20953 , 20999 .
[29] Since 20000 / 29 = 689.6 and 21000 / 29 = 724.1 , thus Q will be between 690 and 724 .
(1) For n = 29 * P , then Q will be from 691 to 719 .
The list of n will be :
29 * 691 = 20039
29 * 701 = 20329
29 * 709 = 20561
29 * 719 = 20851
(2) For n = 29 * C , then Q will be from 697 to 713 .
The list of n will be :
29 * 697 = 20213
29 * 703 = 20387
29 * 713 = 20677
Thus the combined list of n will be :
20039 , 20213 , 20329 , 20387 , 20561 , 20677 , 20851 .
[31] Since 20000 / 31 = 645.1 and 21000 / 31 = 677.4 , thus Q will be between 646 and 677 .
(1) For n = 31 * P , then Q will be from 647 to 677 .
The list of n will be :
31 * 647 = 20057
31 * 653 = 20243
31 * 659 = 20429
31 * 661 = 20491
31 * 673 = 20863
31 * 677 = 20987
(2) For n = 31 * C , then Q will be from 649 to 671 .
The list of n will be :
31 * 649 = 20119
31 * 667 = 20677
31 * 671 = 20801
Thus the combined list of n will be :
20057 , 20119 , 20243 , 20429 , 20491 , 20677 , 20801 , 20863 ,
20987 .
[37] Since 20000 / 37 = 540.5 and 21000 / 37 = 567.5 , thus Q will be between 541 and 567 .
(1) For n = 37 * P , then Q will be from 541 to 563 .
The list of n will be :
37 * 541 = 20017
37 * 547 = 20239
37 * 557 = 20609
37 * 563 = 20831
(2) For n = 37 * C , then Q will be from 551 to 559 .
The list of n will be :
37 * 551 = 20387
37 * 559 = 20683
.
Thus the combined list of n will be :
20017 , 20239 , 20387 , 20609 , 20683 , 20831
[41] Since 20000 / 41 = 487.8 and 21000 / 41 = 512.1 , thus Q will be between 488 and 512 .
(1) For n = 41 * P , then Q will be from 491 to 509 .
The list of n will be :
41 * 491 = 20131
41 * 499 = 20459
41 * 503 = 20623
41 * 509 = 20869
(2) For n = 41 * C , then Q will be 493 only .
The list of n will be :
41 * 493 = 20213
Thus the combined list of n will be :
20131 , 20213 , 20459 , 20623 , 20869 .
[ 43 ] Since 20000 / 43 = 465.1 and 21000 / 43 = 488.3 , thus Q will be between 466 and 488 .
(1) For n = 43 * P , then Q will be from 467 to 487 .
The list of n will be :
43 * 467 = 20081
43 * 479 = 20597
43 * 487 = 20941
(2) For n = 43 * C , then Q will be from 473 to 481 .
The list of n will be :
43 * 473 = 20339
43 * 481 = 20683
Thus the combined list of n will be :
20081 , 20339 , 20597 , 20683 , 20941 .
[ 47 ] Since 20000 / 47 = 425.5 and 21000 / 47 = 446.8 , thus Q will be between 426 and 446 .
(1) For n = 47 * P , then Q will be from 431 to 443 .
The list of n will be :
47 * 431 = 20257
47 * 433 = 20351
47 * 439 = 20633
47 * 443 = 20821
(2) For n = 47 * C , then Q will be 437 only .
The list of n will be :
47 * 437 = 20539
Thus the combined list of n will be :
20257 , 20351 , 20539 , 20633, 20821 .
[ 53 ] Since 20000 / 53 = 377.3 and 21000 / 53 = 396.2 , thus Q will be between 378 and 396.
(1) For n = 53 * P , then Q will be from 379 to 389 .
The list of n will be :
53 * 379 = 20087
53 * 383 = 20299
53 * 389 = 20617
(2) For n = 53 * C , then Q will be 391 only .
The list of n will be :
53 * 391 = 20723
Thus the combined list of n will be :
20087 , 20299 , 20617 , 20723 .
[ 59 ] Since 20000 / 59 = 338.9 and 21000 / 59 = 355.9 , thus Q will be between 339 and 355 .
(1) For n = 59 * P , then Q will be from 347 to 353 .
The list of n will be :
59 * 347 = 20473
59 * 349 = 20591
59 * 353 = 20827
(2) For n = 59 * C , then Q will be 341 only .
The list of n will be :
59 * 341 = 20119
Thus the combined list of n will be :
20119 , 20473 , 20591 , 20827 .
[ 61 ] Since 20000 / 61 = 327.8 and 21000 / 61 = 344.2, thus Q will be between 328 and 344 .
(1) For n = 61 * P , then Q will be from 331 to 337 .
The list of n will be :
61 * 331 = 20191
61 * 337 = 20557
(2) For n = 61 * C , then Q will be 341 only .
The list of n will be :
61 * 341 = 20801
Thus the combined list of n will be :
20191 , 20557 , 20801 .
[ 67 ] Since 20000 / 67 = 298.5 and 21000 / 67 = 313.4 , thus Q will be between 299 and 313 .
(1) For n = 67 * P , then Q will be from 307 to 313 .
The list of n will be :
67 * 307 = 20569
67 * 311 = 20837
67 * 313 = 20971
(2) For n = 67 * C , then Q will be 299 only .
The list of n will be :
67 * 299 = 20033
Thus the combined list of n will be :
20033 , 20569 , 20837 , 20971 .
[ 71 ] Since 20000 / 71 = 281.6 and 21000 / 71 = 295.7 , thus Q will be between 282 and 295 .
(1) For n = 71 * P , then Q will be from 283 to 293.
The list of n will be :
71 * 283 = 20093
71 * 293 = 20803
(2) For n = 71 * C , then Q will be 289 only .
The list of n will be :
71 * 289 = 20519
Thus the combined list of n will be :
20093 , 20519 , 20803 .
[ 73 ] Since 20000 / 73 = 273.9 and 21000 / 73 = 287.6 , thus Q will be between 274 and 287 .
(1) For n = 73 * P , then Q will be from 277 to 283.
The list of n will be :
73 * 277 = 20221
73 * 281 = 20513
73 * 283 = 20659
(2) For n = 73 * C , then no Q will be qualified .
Thus the combined list of n will be :
20221 , 20513 , 20659 .
[ 79 ] Since 20000 / 79 = 253.1 and 21000 / 79 = 265.8 , thus Q will be between 254 and 265 .
(1) For n = 79 * P , then Q will be from 257 to 263.
The list of n will be :
79 * 257 = 20303
79 * 263 = 20777
(2) For n = 79 * C , then no Q will be qualified .
Thus the combined list of n will be :
20303 , 20777 .
[ 83 ] Since 20000 / 83 = 240.9 and 21000 / 83 = 253.01 , thus Q will be between 241 and 253 .
(1) For n = 83 * P , then Q will be from 241 to 251.
The list of n will be :
83 * 241 = 20003
83 * 251 = 20833
(2) For n = 83 * C , then Q will be from 247 to 253.
The list of n will be :
83 * 247 = 20501
83 * 253 = 20999
Thus the combined list of n will be :
20003 , 20501 , 20833 , 20999 .
[ 89 ] Since 20000 / 89 = 224.7 and 21000 / 89 = 235.9 , thus Q will be between 225 and 235 .
(1) For n = 89 * P , then Q will be from 227 to 233 .
The list of n will be :
89 * 227 = 20203
89 * 229 = 20381
89 * 233 = 20737
(2) For n = 89 * C , then no Q will be qualified .
Thus the combined list of n will be :
20203 , 20381 , 20737 .
[ 97 ] Since 20000 / 97 = 206.1 and 21000 / 97 = 216.4 , thus Q will be between 207 and 216 .
(1) For n = 97 * P , then Q will be 211 only .
The list of n will be :
97 * 211 = 20467
(2) For n = 97 * C , then Q will be 209 only
The list of n will be :
97 * 209 = 20273
Thus the combined list of n will be :
20273 , 20467 .
[ 101 ] Since 20000 / 101 = 198.01 and 21000 / 101 = 207.9 , thus Q will be between 199 and 207 .
(1 ) For n = 101 * P , then Q will be 199 only .
The list of n will be :
101 * 199 = 20099 .
(2) For n = 101 * C , then no Q will be qualified .
Thus the combined list of n will be : 20099
[103 ] Since 20000 / 103 = 194.1 and 21000 / 103 = 203.8 , thus Q will be between 195 and 203 .
(1 ) For n = 103 * P , then Q will be from 197 to 199 .
The list of n will be :
103 * 197 = 20291
103 * 199 = 20497
(2) For n = 103 * C , then no Q will be qualified .
Thus the combined list of n will be :
20291 , 20497
[107 ] Since 20000 / 107 = 186.9 and 21000 / 107 = 196.2 , thus Q will be between 187 and 196 .
(1 ) For n = 107 * P , then Q will be from 191 to 193 .
The list of n will be :
107 * 191 = 20437
107 * 193 = 20651
(2) For n = 107 * C , then Q will be 187 only .
The list of n will be :
107 * 187 = 20009
Thus the combined list of n will be :
20009 , 20437 , 20651 .
[ 109 ] Since 20000 / 109 = 183.4 and 21000 / 109 = 192.6 , thus Q will be between 184 and 192 .
(1 ) For n = 109 * P , then Q will be 191 only .
The list of n will be :
109 * 191 =20819
(2) For n = 109 * C , then Q will be 187 only .
The list of n will be :
109 * 187 = 20383
Thus the combined list of n will be :
20383 , 20819 .
[ 113 ] Since 20000 / 113 = 176.9 and 21000 / 113 = 185.8 , thus Q will be between 177 and 185.
(1 ) For n = 113 * P , then Q will be from 179 to 181 .
The list of n will be :
113 * 179 = 20227
113 * 181 = 20453
(2) For n = 113 * C , then no Q will be qualified .
Thus the combined list of n will be :
20227 , 20453
[ 127 ] Since 20000 / 127 = 157.4 and 21000 / 127 = 165.3 , thus Q will be between 158 and 165.
(1 ) For n = 127 * P , then Q will be 163 only.
The list of n will be :
127 * 163 = 20701
(2) For n = 127 * C , then no Q will be qualified .
Thus the combined list of n will be :
20701
[ 131 ] Since 20000 / 131 = 152.6 and 21000 / 131 = 160.3, thus Q will be between 153 and 160.
(1 ) For n = 131 * P , then Q will be 157 only.
The list of n will be :
131 * 157 = 20567
(2) For n = 131 * C , then no Q will be qualified .
Thus the combined list of n will be :
20567
[ 137 ] Since 20000 / 137 = 145.9 and 21000 / 137 = 153.2 , thus Q will be between 146 and 153.
(1 ) For n = 137 * P , then Q will be from 149 to 151 .
The list of n will be :
137 * 149 = 20413
137 * 151 = 20687
(2) For n = 137 * C , then no Q will be qualified
Thus the combined list of n will be :
20413 , 20687
[ 139 ] Since 20000 / 139 = 143.8 and 21000 / 139 = 151.07 , thus Q will be between 144 and 151.
(1 ) For n = 139 * P , then Q will be from 149 to 151 .
The list of n will be :
139 * 149 = 20711
139 * 151 = 20989
(2) For n = 139 * C , then no Q will be qualified
Thus the combined list of n will be :
20711 , 20989
(ii)
Thus the total list of n from primes 11 to 139 to be excluded will be : ( in descending order )
20999 20893 20791 20689
20989 20891 20779 20687
20987 20869 20777 20683
20977 20863 20767 20677
20971 20861 20761 20669
20969 20851 20737 20659
20957 20843 20729 20653
20953 20837 20723 20651
20941 20833 20711 20647
20933 20831 20701 20633
20927 20827 20623
20917 20821 20621
20911 20819 20617
20813 20609
20803 20603
20801
20597 20497 20387
20591 20491 20383
20581 20473 20381
20579 20471 20371
20569 20467 20351
20567 20459 20339
20561 20453 20329
20557 20449 20317
20539 20437 20311
20537 20429 20309
20527 20423 20303
20519 20417
20513 20413
20501 20401
20299 20197 20099
20291 20191 20093
20281 20189 20087
20273 20179 20081
20267 20171 20077
20263 20159 20059
20257 20141 20057
20243 20137 20053
20239 20131 20039
20227 20119 20033
20221 20017
20213 20009
20207 20003
20203
( to be continued )
Last edited by mr.wong (2017-02-15 17:06:01)
Offline
( continued from # 33 )
To generate primes between 20000 and 21000 ( in descending order ) using the expression n = 21000 - j :
(1) 21000 - 1 = 20999 (C)
(2) 21000 - 11 = 20989 (C)
(3) 21000 - 13 = 20987 (C)
(4) 21000 - 17 = 20983
(5) 21000 - 19 = 20981
(6) 21000 - 23 = 20977 (C)
(7) 21000 - 29 = 20971 (C)
(8) 21000 - 31 = 20969 (C)
(9) 21000 - 37 = 20963
(10) 21000 - 41 = 20959
(11) 21000 - 43 = 20957 (C)
(12) 21000 - 47 = 20953 (C)
(13) 21000 - 53 = 20947
(14) 21000 - 59 = 20941 (C)
(15) 21000 - 61 = 20939
(16) 21000 - 67 = 20933 (C)
(17) 21000 - 71 = 20929
(18) 21000 - 73 = 20927 (C)
(19) 21000 - 79 = 20921
(20) 21000 - 83 = 20917 (C)
(21) 21000 - 89 = 20911 (C)
(22) 21000 - 97 = 20903
(23) 21000 - 101 = 20899
(24) 21000 - 103 = 20897
(25) 21000 - 107 = 20893 (C)
(26) 21000 - 109 = 20891 (C)
(27) 21000 - 113 = 20887
(28) 21000 - 121 = 20879
(29) 21000 - 127 = 20873
(30) 21000 - 131 = 20869 (C)
(31) 21000 - 137 = 20863 (C)
(32) 21000 - 139 = 20861 (C)
(33) 21000 - 143 = 21857
(34) 21000 - 149 = 20851 (C)
(35) 21000 - 151 = 20849
(36) 21000 - 157 = 20843 (C)
(37) 21000 - 163 = 20837 (C)
(38) 21000 - 167 = 20833 (C)
(39) 21000 - 169 = 20831 (C)
(40) 21000 - 173 = 20827 (C)
(41) 21000 - 179 = 20821 (C)
(42) 21000 - 181 = 20819 (C)
(43) 21000 - 187 = 20813 (C)
(44) 21000 - 191 = 20809
(45) 21000 - 193 = 20807
(46) 21000 - 197 = 20803 (C)
(47) 21000 - 199 = 20801 (C)
(48) 21000 - 209 = 20791 (C)
(49) 21000 - 211 = 20789
(50) 21000 - 221 = 20779 (C)
(51) 21000 - 223 = 20777 (C)
(52) 21000 - 227 = 20773
(53) 21000 - 229 = 20771
(54) 21000 - 233 = 20767 (C)
(55) 21000 - 239 = 20761 (C)
(56) 21000 - 241 = 20759
(57) 21000 - 247 = 20753
(58) 21000 - 251 = 20749
(59) 21000 - 253 = 20747
(60) 21000 - 257 = 20743
(61) 21000 - 263 = 20737 (C)
(62) 21000 - 269 = 20731
(63) 21000 - 271 = 20729 (C)
(64) 21000 - 277 = 20723 (C)
(65) 21000 - 281 = 20719
(66) 21000 - 283 = 20717
(67) 21000 - 289 = 20711 (C)
(68) 21000 - 293 = 20707
(69) 21000 - 299 = 20701 (C)
(70) 21000 - 307 = 20693
(71) 21000 - 311 = 20689 (C)
(72) 21000 - 313 = 20687 (C)
(73) 21000 - 317 = 20683 (C)
(74) 21000 - 319 = 20681
(75) 21000 - 323 = 20677 (C)
(76) 21000 - 331 = 20669 (C)
(77) 21000 - 337 = 20663
(78) 21000 - 341 = 20659 (C)
(79) 21000 - 347 = 20653 (C)
(80) 21000 - 349 = 20651 (C)
(81) 21000 - 353 = 20647 (C)
(82) 21000 - 359 = 20641
(83) 21000 - 361 = 20639
(84) 21000 - 367 = 20633 (C)
(85) 21000 - 373 = 20627
(86) 21000 - 377 = 20623 (C)
(87) 21000 - 379 = 20621 (C)
(88) 21000 - 383 = 20617 (C)
(89) 21000 - 389 = 20611
(90) 21000 - 391 = 20609 (C)
(91) 21000 - 397 = 20603 (C)
(92) 21000 - 401 = 20599
(93) 21000 - 403 = 20597 (C)
(94) 21000 - 407 = 20593
(95) 21000 - 409 = 20591 (C)
(96) 21000 - 419 = 20581 (C)
(97) 21000 - 421 = 20579 (C)
(98) 21000 - 431 = 20569 (C)
(99) 21000 - 433 = 20567 (C)
(100) 21000 - 437 = 20563
(101) 21000 - 439 = 20561 (C)
(102) 21000 - 443 = 20557 (C)
(103) 21000 - 449 = 20551
(104) 21000 - 451 = 20549
(105) 21000 - 457 = 20543
(106) 21000 - 461 = 20539 (C)
(107) 21000 - 463 = 20537 (C)
(108) 21000 - 467 = 20533
(109) 21000 - 473 = 20527 (C)
(110) 21000 - 479 = 20521
(111) 21000 - 481 = 20519 (C)
(112) 21000 - 487 = 20513 (C)
(113) 21000 - 491 = 20509
(114) 21000 - 493 = 20507
(115) 21000 - 499 = 20501 (C)
(116) 21000 - 503 = 20497 (C)
(117) 21000 - 509 = 20491 (C)
(118) 21000 - 517 = 20483
(119) 21000 - 521 = 20479
(120) 21000 - 523 = 20477
(121) 21000 - 527 = 20473 (C)
(122) 21000 - 529 = 20471 (C)
(123) 21000 - 533 = 20467 (C)
(124) 21000 - 541 = 20459 (C)
(125) 21000 - 547 = 20453 (C)
(126) 21000 - 551 = 20449 (C)
(127) 21000 - 557 = 20443
(128) 21000 - 559 = 20441
(129) 21000 - 563 = 20437 (C)
(130) 21000 - 569 = 20431
(131) 21000 - 571 = 20429 (C)
(132) 21000 - 577 = 20423 (C)
(133) 21000 - 583 = 20417 (C)
(134) 21000 - 587 = 20413 (C)
(135) 21000 - 589 = 20411
(136) 21000 - 593 = 20407
(137) 21000 - 599 = 20401 (C)
(138) 21000 - 601 = 20399
(139) 21000 - 607 = 20393
(140) 21000 - 611 = 20389
(141) 21000 - 613 = 20387 (C)
(142) 21000 - 617 = 20383 (C)
(143) 21000 - 619 = 20381 (C)
(144) 21000 - 629 = 20371 (C)
(145) 21000 - 631 = 20369
(146) 21000 - 641 = 20359
(147) 21000 - 643 = 20357
(148) 21000 - 647 = 20353
(149) 21000 - 649 = 20351 (C)
(150) 21000 - 653 = 20347
(151) 21000 - 659 = 20341
(152) 21000 - 661 = 20339 (C)
(153) 21000 - 667 = 20333
(154) 21000 - 671 = 20329 (C)
(155) 21000 - 673 = 20327
(156) 21000 - 677 = 20323
(157) 21000 - 683 = 20317 (C)
(158) 21000 - 689 = 20311 (C)
(159) 21000 - 691 = 20309 (C)
(160) 21000 - 697 = 20303 (C)
(161) 21000 - 701 = 20299 (C)
(162) 21000 - 703 = 20297
(163) 21000 - 709 = 20291 (C)
(164) 21000 - 713 = 20287
(165) 21000 - 719 = 20281 (C)
(166) 21000 - 727 = 20273 (C)
(167) 21000 - 731 = 20269
(168) 21000 - 733 = 20267 (C)
(169) 21000 - 737 = 20263 (C)
(170) 21000 - 739 = 20261
(171) 21000 - 743 = 20257 (C)
(172) 21000 - 751 = 20249
(173) 21000 - 757 = 20243 (C)
(174) 21000 - 761 = 20239 (C)
(175) 21000 - 767 = 20233
(176) 21000 - 769 = 20231
(177) 21000 - 773 = 20227 (C)
(178) 21000 - 779 = 20221 (C)
(179) 21000 - 781 = 20219
(180) 21000 - 787 = 20213 (C)
(181) 21000 - 793 = 20207 (C)
(182) 21000 - 797 = 20203 (C)
(183) 21000 - 799 = 20201
(184) 21000 - 803 = 20197 (C)
(185) 21000 - 809 = 20191 (C)
(186) 21000 - 811 = 20189 (C)
(187) 21000 - 817 = 20183
(188) 21000 - 821 = 20179 (C)
(189) 21000 - 823 = 20177
(190) 21000 - 827 = 20173
(191) 21000 - 829 = 20171 (C)
(192) 21000 - 839 = 20161
(193) 21000 - 841 = 20159 (C)
(194) 21000 - 851 = 20149
(195) 21000 - 853 = 20147
(196) 21000 - 857 = 20143
(197) 21000 - 859 = 20141 (C)
(198) 21000 - 863 = 20137 (C)
(199) 21000 - 869 = 20131 (C)
(200) 21000 - 871 = 20129
(201) 21000 - 877 = 20123
(202) 21000 - 881 = 20119 (C)
(203) 21000 - 883 = 20117
(204) 21000 - 887 = 20113
(205) 21000 - 893 = 20107
(206) 21000 - 899 = 20101
(207) 21000 - 901 = 20099 (C)
(208) 21000 - 907 = 20093 (C)
(209) 21000 - 911 = 20089
(210) 21000 - 913 = 20087 (C)
(211) 21000 - 919 = 20081 (C)
(212) 21000 - 923 = 20077 (C)
(213) 21000 - 929 = 20071
(214) 21000 - 937 = 20063
(215) 21000 - 941 = 20059 (C)
(216) 21000 - 943 = 20057 (C)
(217) 21000 - 947 = 20053 (C)
(218) 21000 - 949 = 20051
(219) 21000 - 953 = 20047
(220) 21000 - 961 = 20039 (C)
(221) 21000 - 967 = 20033 (C)
(222) 21000 - 971 = 20029
(223) 21000 - 977 = 20023
(224) 21000 - 979 = 20021
(225) 21000 - 983 = 20017 (C)
(226) 21000 - 989 = 20011
(227) 21000 - 991 = 20009 (C)
(228) 21000 - 997 = 20003 (C)
Thus after excluding the composite no. from
the list of n's obtained , all the primes between
20000 and 21000 had been generated .
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