Any integer greater than 1 is called a prime number if and only if its positive factors are 1 and the number p itself.

The basic ideology involved in this post is flawed and the post has now been moved to Archives.

– The Editor

mistaken
via xkcd

Prime Generating Formulas

We all know how hard it is to predict a formula for prime numbers! They have extremely uncertain pattern at various number ranges. Some prime numbers may have difference of hundreds, while few others are as close as possible. Mathematicians and computer scientists have worked very well for a long time to discover more prime numbers and tried hard to compute a unique formula. But as there are no certain patterns, it remains difficult to discover exactly the next prime number after a certain prime. There are still some ideas available which help mathematics majors predict patterns in prime numbers.

 

I have already discussed two such (flawed) ideas about prime numbers generation. You can revisit them:

The third idea, originally commented by Huen Yeong Kong,$   forms an accurate sequence of prime numbers. It is very extensive and appears to be true at all my computational limits. The prime factorization of product of all 2i ‘s ($i=1, 2, 3, \ldots $ ), i.e., factor ( $\displaystyle{\prod_{i=1}^n} 2i , n \in \mathbb{Z}^+$ ), gives all consecutive prime numbers in a sequence.

As an example, take $n=1$, we have $\displaystyle{\prod_{i=1}^1} 2i=2$. Therefore, $\text{factor}\prod_{i=1}^1 2i$ =2.

Similarly, for n=2, as $\displaystyle{\prod_{i=1}^2} 2i=8$, we have $\text{factor}\prod_{i=1}^2 2i$= 2.

Again, for n=3, as $\displaystyle{\prod_{i=1}^3} 2i=48$, we have $\text{factor}\prod_{i=1}^3 2i$= 2,3.

On continuing the same for different increasing values of n,  factors containing sequence of consecutive primes: 2, 3, 5, 7, 11, 13, … were obtained. The greatest (final) terms of these sequences were always less than or equal to value of chosen n.

With Maxima for Windows, a calculation beyond n=~14000 isn’t possible for this expression. For n=13000, the prime generating product $\displaystyle{\prod_{i=1}^{13000}} 2i$ was factored by writing

factor(product(2*i,i,1,100000));

in Maxima software, which yielded the following  prime factors of the number $\displaystyle{\prod_{i=1}^{13000}} 2i$:

2^25994, 3^6494, 5^3248, 7^2164, 11^1297, 13^1081, 17^810, 19^721, 23^590, 29^463, 31^432, 37^360, 41^324, 43^309, 47^281, 53^249, 59^223, 61^216, 67^196, 71^185, 73^180, 79^166, 83^157, 89^147, 97^135, 101^129, 103^127, 107^122, 109^120, 113^116, 127^102, 131^99, 137^94, 139^93, 149^87, 151^86, 157^82, 163^79, 167^77, 173^75, 179^72, 181^71, 191^68, 193^67, 197^65, 199^65, 211^61, 223^58, 227^57, 229^56, 233^55, 239^54, 241^53, 251^51, 257^50, 263^49, 269^48, 271^47, 277^46, 281^46, 283^45, 293^44, 307^42, 311^41, 313^41, 317^41, 331^39, 337^38, 347^37, 349^37, 353^36, 359^36, 367^35, 373^34, 379^34, 383^33, 389^33, 397^32, 401^32, 409^31, 419^31, 421^30, 431^30, 433^30, 439^29, 443^29, 449^28, 457^28, 461^28, 463^28, 467^27, 479^27, 487^26, 491^26, 499^26, 503^25, 509^25, 521^24, 523^24, 541^24, 547^23, 557^23, 563^23, 569^22, 571^22, 577^22, 587^22, 593^21, 599^21, 601^21, 607^21, 613^21, 617^21, 619^21, 631^20, 641^20, 643^20, 647^20, 653^19, 659^19, 661^19, 673^19, 677^19, 683^19, 691^18, 701^18, 709^18, 719^18, 727^17, 733^17, 739^17, 743^17, 751^17, 757^17, 761^17, 769^16, 773^16, 787^16, 797^16, 809^16, 811^16, 821^15, 823^15, 827^15, 829^15, 839^15, 853^15, 857^15, 859^15, 863^15, 877^14, 881^14, 883^14, 887^14, 907^14, 911^14, 919^14, 929^13, 937^13, 941^13, 947^13, 953^13, 967^13, 971^13, 977^13, 983^13, 991^13, 997^13, 1009^12, 1013^12, 1019^12, 1021^12, 1031^12, 1033^12, 1039^12, 1049^12, 1051^12, 1061^12, 1063^12, 1069^12, 1087^11, 1091^11, 1093^11, 1097^11, 1103^11, 1109^11, 1117^11, 1123^11, 1129^11, 1151^11, 1153^11, 1163^11, 1171^11, 1181^11, 1187^10, 1193^10, 1201^10, 1213^10, 1217^10, 1223^10, 1229^10, 1231^10, 1237^10, 1249^10, 1259^10, 1277^10, 1279^10, 1283^10, 1289^10, 1291^10, 1297^10, 1301^9, 1303^9, 1307^9, 1319^9, 1321^9, 1327^9, 1361^9, 1367^9, 1373^9, 1381^9, 1399^9, 1409^9, 1423^9, 1427^9, 1429^9, 1433^9, 1439^9, 1447^8, 1451^8, 1453^8, 1459^8, 1471^8, 1481^8, 1483^8, 1487^8, 1489^8, 1493^8, 1499^8, 1511^8, 1523^8, 1531^8, 1543^8, 1549^8, 1553^8, 1559^8, 1567^8, 1571^8, 1579^8, 1583^8, 1597^8, 1601^8, 1607^8, 1609^8, 1613^8, 1619^8, 1621^8, 1627^7, 1637^7, 1657^7, 1663^7, 1667^7, 1669^7, 1693^7, 1697^7, 1699^7, 1709^7, 1721^7, 1723^7, 1733^7, 1741^7, 1747^7, 1753^7, 1759^7, 1777^7, 1783^7, 1787^7, 1789^7, 1801^7, 1811^7, 1823^7, 1831^7, 1847^7, 1861^6, 1867^6, 1871^6, 1873^6, 1877^6, 1879^6, 1889^6, 1901^6, 1907^6, 1913^6, 1931^6, 1933^6, 1949^6, 1951^6, 1973^6, 1979^6, 1987^6, 1993^6, 1997^6, 1999^6, 2003^6, 2011^6, 2017^6, 2027^6, 2029^6, 2039^6, 2053^6, 2063^6, 2069^6, 2081^6, 2083^6, 2087^6, 2089^6, 2099^6, 2111^6, 2113^6, 2129^6, 2131^6, 2137^6, 2141^6, 2143^6, 2153^6, 2161^6, 2179^5, 2203^5, 2207^5, 2213^5, 2221^5, 2237^5, 2239^5, 2243^5, 2251^5, 2267^5, 2269^5, 2273^5, 2281^5, 2287^5, 2293^5, 2297^5, 2309^5, 2311^5, 2333^5, 2339^5, 2341^5, 2347^5, 2351^5, 2357^5, 2371^5, 2377^5, 2381^5, 2383^5, 2389^5, 2393^5, 2399^5, 2411^5, 2417^5, 2423^5, 2437^5, 2441^5, 2447^5, 2459^5, 2467^5, 2473^5, 2477^5, 2503^5, 2521^5, 2531^5, 2539^5, 2543^5, 2549^5, 2551^5, 2557^5, 2579^5, 2591^5, 2593^5, 2609^4, 2617^4, 2621^4, 2633^4, 2647^4, 2657^4, 2659^4, 2663^4, 2671^4, 2677^4, 2683^4, 2687^4, 2689^4, 2693^4, 2699^4, 2707^4, 2711^4, 2713^4, 2719^4, 2729^4, 2731^4, 2741^4, 2749^4, 2753^4, 2767^4, 2777^4, 2789^4, 2791^4, 2797^4, 2801^4, 2803^4, 2819^4, 2833^4, 2837^4, 2843^4, 2851^4, 2857^4, 2861^4, 2879^4, 2887^4, 2897^4, 2903^4, 2909^4, 2917^4, 2927^4, 2939^4, 2953^4, 2957^4, 2963^4, 2969^4, 2971^4, 2999^4, 3001^4, 3011^4, 3019^4, 3023^4, 3037^4, 3041^4, 3049^4, 3061^4, 3067^4, 3079^4, 3083^4, 3089^4, 3109^4, 3119^4, 3121^4, 3137^4, 3163^4, 3167^4, 3169^4, 3181^4, 3187^4, 3191^4, 3203^4, 3209^4, 3217^4, 3221^4, 3229^4, 3251^3, 3253^3, 3257^3, 3259^3, 3271^3, 3299^3, 3301^3, 3307^3, 3313^3, 3319^3, 3323^3, 3329^3, 3331^3, 3343^3, 3347^3, 3359^3, 3361^3, 3371^3, 3373^3, 3389^3, 3391^3, 3407^3, 3413^3, 3433^3, 3449^3, 3457^3, 3461^3, 3463^3, 3467^3, 3469^3, 3491^3, 3499^3, 3511^3, 3517^3, 3527^3, 3529^3, 3533^3, 3539^3, 3541^3, 3547^3, 3557^3, 3559^3, 3571^3, 3581^3, 3583^3, 3593^3, 3607^3, 3613^3, 3617^3, 3623^3, 3631^3, 3637^3, 3643^3, 3659^3, 3671^3, 3673^3, 3677^3, 3691^3, 3697^3, 3701^3, 3709^3, 3719^3, 3727^3, 3733^3, 3739^3, 3761^3, 3767^3, 3769^3, 3779^3, 3793^3, 3797^3, 3803^3, 3821^3, 3823^3, 3833^3, 3847^3, 3851^3, 3853^3, 3863^3, 3877^3, 3881^3, 3889^3, 3907^3, 3911^3, 3917^3, 3919^3, 3923^3, 3929^3, 3931^3, 3943^3, 3947^3, 3967^3, 3989^3, 4001^3, 4003^3, 4007^3, 4013^3, 4019^3, 4021^3, 4027^3, 4049^3, 4051^3, 4057^3, 4073^3, 4079^3, 4091^3, 4093^3, 4099^3, 4111^3, 4127^3, 4129^3, 4133^3, 4139^3, 4153^3, 4157^3, 4159^3, 4177^3, 4201^3, 4211^3, 4217^3, 4219^3, 4229^3, 4231^3, 4241^3, 4243^3, 4253^3, 4259^3, 4261^3, 4271^3, 4273^3, 4283^3, 4289^3, 4297^3, 4327^3, 4337^2, 4339^2, 4349^2, 4357^2, 4363^2, 4373^2, 4391^2, 4397^2, 4409^2, 4421^2, 4423^2, 4441^2, 4447^2, 4451^2, 4457^2, 4463^2, 4481^2, 4483^2, 4493^2, 4507^2, 4513^2, 4517^2, 4519^2, 4523^2, 4547^2, 4549^2, 4561^2, 4567^2, 4583^2, 4591^2, 4597^2, 4603^2, 4621^2, 4637^2, 4639^2, 4643^2, 4649^2, 4651^2, 4657^2, 4663^2, 4673^2, 4679^2, 4691^2, 4703^2, 4721^2, 4723^2, 4729^2, 4733^2, 4751^2, 4759^2, 4783^2, 4787^2, 4789^2, 4793^2, 4799^2, 4801^2, 4813^2, 4817^2, 4831^2, 4861^2, 4871^2, 4877^2, 4889^2, 4903^2, 4909^2, 4919^2, 4931^2, 4933^2, 4937^2, 4943^2, 4951^2, 4957^2, 4967^2, 4969^2, 4973^2, 4987^2, 4993^2, 4999^2, 5003^2, 5009^2, 5011^2, 5021^2, 5023^2, 5039^2, 5051^2, 5059^2, 5077^2, 5081^2, 5087^2, 5099^2, 5101^2, 5107^2, 5113^2, 5119^2, 5147^2, 5153^2, 5167^2, 5171^2, 5179^2, 5189^2, 5197^2, 5209^2, 5227^2, 5231^2, 5233^2, 5237^2, 5261^2, 5273^2, 5279^2, 5281^2, 5297^2, 5303^2, 5309^2, 5323^2, 5333^2, 5347^2, 5351^2, 5381^2, 5387^2, 5393^2, 5399^2, 5407^2, 5413^2, 5417^2, 5419^2, 5431^2, 5437^2, 5441^2, 5443^2, 5449^2, 5471^2, 5477^2, 5479^2, 5483^2, 5501^2, 5503^2, 5507^2, 5519^2, 5521^2, 5527^2, 5531^2, 5557^2, 5563^2, 5569^2, 5573^2, 5581^2, 5591^2, 5623^2, 5639^2, 5641^2, 5647^2, 5651^2, 5653^2, 5657^2, 5659^2, 5669^2, 5683^2, 5689^2, 5693^2, 5701^2, 5711^2, 5717^2, 5737^2, 5741^2, 5743^2, 5749^2, 5779^2, 5783^2, 5791^2, 5801^2, 5807^2, 5813^2, 5821^2, 5827^2, 5839^2, 5843^2, 5849^2, 5851^2, 5857^2, 5861^2, 5867^2, 5869^2, 5879^2, 5881^2, 5897^2, 5903^2, 5923^2, 5927^2, 5939^2, 5953^2, 5981^2, 5987^2, 6007^2, 6011^2, 6029^2, 6037^2, 6043^2, 6047^2, 6053^2, 6067^2, 6073^2, 6079^2, 6089^2, 6091^2, 6101^2, 6113^2, 6121^2, 6131^2, 6133^2, 6143^2, 6151^2, 6163^2, 6173^2, 6197^2, 6199^2, 6203^2, 6211^2, 6217^2, 6221^2, 6229^2, 6247^2, 6257^2, 6263^2, 6269^2, 6271^2, 6277^2, 6287^2, 6299^2, 6301^2, 6311^2, 6317^2, 6323^2, 6329^2, 6337^2, 6343^2, 6353^2, 6359^2, 6361^2, 6367^2, 6373^2, 6379^2, 6389^2, 6397^2, 6421^2, 6427^2, 6449^2, 6451^2, 6469^2, 6473^2, 6481^2, 6491^2, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973, 10007, 10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099, 10103, 10111, 10133, 10139, 10141, 10151, 10159, 10163, 10169, 10177, 10181, 10193, 10211, 10223, 10243, 10247, 10253, 10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321, 10331, 10333, 10337, 10343, 10357, 10369, 10391, 10399, 10427, 10429, 10433, 10453, 10457, 10459, 10463, 10477, 10487, 10499, 10501, 10513, 10529, 10531, 10559, 10567, 10589, 10597, 10601, 10607, 10613, 10627, 10631, 10639, 10651, 10657, 10663, 10667, 10687, 10691, 10709, 10711, 10723, 10729, 10733, 10739, 10753, 10771, 10781, 10789, 10799, 10831, 10837, 10847, 10853, 10859, 10861, 10867, 10883, 10889, 10891, 10903, 10909, 10937, 10939, 10949, 10957, 10973, 10979, 10987, 10993, 11003, 11027, 11047, 11057, 11059, 11069, 11071, 11083, 11087, 11093, 11113, 11117, 11119, 11131, 11149, 11159, 11161, 11171, 11173, 11177, 11197, 11213, 11239, 11243, 11251, 11257, 11261, 11273, 11279, 11287, 11299, 11311, 11317, 11321, 11329, 11351, 11353, 11369, 11383, 11393, 11399, 11411, 11423, 11437, 11443, 11447, 11467, 11471, 11483, 11489, 11491, 11497, 11503, 11519, 11527, 11549, 11551, 11579, 11587, 11593, 11597, 11617, 11621, 11633, 11657, 11677, 11681, 11689, 11699, 11701, 11717, 11719, 11731, 11743, 11777, 11779, 11783, 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833, 11839, 11863, 11867, 11887, 11897, 11903, 11909, 11923, 11927, 11933, 11939, 11941, 11953, 11959, 11969, 11971, 11981, 11987, 12007, 12011, 12037, 12041, 12043, 12049, 12071, 12073, 12097, 12101, 12107, 12109, 12113, 12119, 12143, 12149, 12157, 12161, 12163, 12197, 12203, 12211, 12227, 12239, 12241, 12251, 12253, 12263, 12269, 12277, 12281, 12289, 12301, 12323, 12329, 12343, 12347, 12373, 12377, 12379, 12391, 12401, 12409, 12413, 12421, 12433, 12437, 12451, 12457, 12473, 12479, 12487, 12491, 12497, 12503, 12511, 12517, 12527, 12539, 12541, 12547, 12553, 12569, 12577, 12583, 12589, 12601, 12611, 12613, 12619, 12637, 12641, 12647, 12653, 12659, 12671, 12689, 12697, 12703, 12713, 12721, 12739, 12743, 12757, 12763, 12781, 12791, 12799, 12809, 12821, 12823, 12829, 12841, 12853, 12889, 12893, 12899, 12907, 12911, 12917, 12919, 12923, 12941, 12953, 12959, 12967, 12973, 12979, 12983.

 

Above sequence contains each and every prime from 1 to 12985. This prime generating product may fail to produce such sequences at higher limits but still it is way better than Euler’s prime generating polynomial.

Any comments?

 

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2 comments
  1. Factorization of 1*2*3*4*5*6*7*8*9*10*…*n also yields every prime number in that interval. You put prime numbers in a product and they’re really in there, how fascinating!

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