Inhaltsverzeichnis

Development of
Algorithmic Constructions

20:02:33
Deutsch
28.Mar 2024

Polynom = x^2+823

0. Sequence

1. Algorithm

2. Mathematical background

3. Correctness of the algorithm

4. Infinity of the sequence

5. Sequence of the polynom with 1

6. Sequence of the polynom (only primes)

7. Distribution of the primes

8. Check for existing Integer Sequences by OEIS

0. Sequence

f(0) = 823 = 823
f(1) = 103 = 103
f(2) = 827 = 827
f(3) = 13 = 13
f(4) = 839 = 839
f(5) = 53 = 53
f(6) = 859 = 859
f(7) = 109 = 109
f(8) = 887 = 887
f(9) = 113 = 113
f(10) = 923 = 13*71
f(11) = 59 = 59
f(12) = 967 = 967
f(13) = 31 = 31
f(14) = 1019 = 1019
f(15) = 131 = 131
f(16) = 1079 = 13*83
f(17) = 139 = 139
f(18) = 1147 = 31*37
f(19) = 37 = 37
f(20) = 1223 = 1223
f(21) = 79 = 79
f(22) = 1307 = 1307
f(23) = 169 = 13*13
f(24) = 1399 = 1399
f(25) = 181 = 181
f(26) = 1499 = 1499
f(27) = 97 = 97
f(28) = 1607 = 1607
f(29) = 13 = 13
f(30) = 1723 = 1723
f(31) = 223 = 223
f(32) = 1847 = 1847
f(33) = 239 = 239
f(34) = 1979 = 1979
f(35) = 1 = 1
f(36) = 2119 = 13*163
f(37) = 137 = 137
f(38) = 2267 = 2267
f(39) = 293 = 293
f(40) = 2423 = 2423
f(41) = 313 = 313
f(42) = 2587 = 13*199
f(43) = 167 = 167
f(44) = 2759 = 31*89
f(45) = 89 = 89
f(46) = 2939 = 2939
f(47) = 379 = 379
f(48) = 3127 = 53*59
f(49) = 403 = 13*31
f(50) = 3323 = 3323
f(51) = 107 = 107
f(52) = 3527 = 3527
f(53) = 227 = 227
f(54) = 3739 = 3739
f(55) = 481 = 13*37
f(56) = 3959 = 37*107
f(57) = 509 = 509
f(58) = 4187 = 53*79
f(59) = 269 = 269
f(60) = 4423 = 4423
f(61) = 71 = 71
f(62) = 4667 = 13*359
f(63) = 599 = 599
f(64) = 4919 = 4919
f(65) = 631 = 631
f(66) = 5179 = 5179
f(67) = 83 = 83
f(68) = 5447 = 13*419
f(69) = 349 = 349
f(70) = 5723 = 59*97
f(71) = 733 = 733
f(72) = 6007 = 6007
f(73) = 769 = 769
f(74) = 6299 = 6299
f(75) = 403 = 13*31
f(76) = 6599 = 6599
f(77) = 211 = 211
f(78) = 6907 = 6907
f(79) = 883 = 883
f(80) = 7223 = 31*233
f(81) = 923 = 13*71
f(82) = 7547 = 7547
f(83) = 241 = 241
f(84) = 7879 = 7879
f(85) = 503 = 503
f(86) = 8219 = 8219
f(87) = 1049 = 1049
f(88) = 8567 = 13*659
f(89) = 1093 = 1093
f(90) = 8923 = 8923
f(91) = 569 = 569
f(92) = 9287 = 37*251
f(93) = 37 = 37
f(94) = 9659 = 13*743
f(95) = 1231 = 1231
f(96) = 10039 = 10039
f(97) = 1279 = 1279
f(98) = 10427 = 10427
f(99) = 83 = 83
f(100) = 10823 = 79*137

1. Algorithm

If you are interested in some better algorithms have a look at quadr_Sieb_x^2+1.php.

2. Mathematical background

Lemma: If p | f(x) then also p | f(x+p) and p | f(-x-b/a) a) p | f(x) <=> ax^2 + bx + c = 0 mod p p | f(x+p) <=> a(x+p)^2 + b(x+p) + c = 0 mod p <=> ax^2 + 2axp + ap^2 + bx + bp + c = 0 mod p <=> ax^2 + bx + c = 0 mod p Thus if p | f(x) then p | f(x+p) b) if b = 0 mod a p | f(x) <=> ax^2 + bx + c = 0 mod p p | f(-x-b/a) <=> a(-x-b/a)^2 + b(-x-b/a) + c = 0 mod p <=> ax^2 + 2bx + b^2/a - bx - b^2/a + c = 0 mod p <=> ax^2 + bx + c = 0 mod p Thus if p | f(x) then p | f(-x-b/a)

3. Correctness of the algorithm

The proof for this polynom is similar to the proof for the polynom f(x)=x^2-4x+1. a) First terms for the polynom f(x) = x^2+823

f(0)=823
f(1)=103
f(2)=827
f(3)=13
f(4)=839
f(5)=53
f(6)=859
f(7)=109
f(8)=887
f(9)=113
f(10)=71
f(11)=59
f(12)=967
f(13)=31
f(14)=1019
f(15)=131
f(16)=83
f(17)=139
f(18)=37
f(19)=1
f(20)=1223
f(21)=79
f(22)=1307
f(23)=1
f(24)=1399
f(25)=181
f(26)=1499
f(27)=97
f(28)=1607
f(29)=1
f(30)=1723
f(31)=223
f(32)=1847
f(33)=239
f(34)=1979
f(35)=1
f(36)=163
f(37)=137
f(38)=2267
f(39)=293
f(40)=2423
f(41)=313
f(42)=199
f(43)=167
f(44)=89
f(45)=1
f(46)=2939
f(47)=379
f(48)=1
f(49)=1
f(50)=3323
f(51)=107
f(52)=3527
f(53)=227
f(54)=3739
f(55)=1
f(56)=1
f(57)=509
f(58)=1
f(59)=269
f(60)=4423
f(61)=1
f(62)=359
f(63)=599
f(64)=4919
f(65)=631
f(66)=5179
f(67)=1
f(68)=419
f(69)=349
f(70)=1
f(71)=733
f(72)=6007
f(73)=769
f(74)=6299
f(75)=1
f(76)=6599
f(77)=211
f(78)=6907
f(79)=883
f(80)=233
f(81)=1
f(82)=7547
f(83)=241
f(84)=7879
f(85)=503
f(86)=8219
f(87)=1049
f(88)=659
f(89)=1093
f(90)=8923
f(91)=569
f(92)=251
f(93)=1
f(94)=743
f(95)=1231
f(96)=10039
f(97)=1279
f(98)=10427
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+823 could be written as f(y)= y^2+823 with x=y+0

c) Backsubstitution Beside by backsubstitution you get an estimation for the huge of the primes with p | f(x) and p < f(x) f'(y)>(2y-1) with with y=x+0
f'(x)>2x-1 with x > 29

4. Infinity of the sequence

The mathematical proof is analogue to the proof for the polynom f(x)=x^2+1

5. Sequence of the polynom with 1

823, 103, 827, 13, 839, 53, 859, 109, 887, 113, 71, 59, 967, 31, 1019, 131, 83, 139, 37, 1, 1223, 79, 1307, 1, 1399, 181, 1499, 97, 1607, 1, 1723, 223, 1847, 239, 1979, 1, 163, 137, 2267, 293, 2423, 313, 199, 167, 89, 1, 2939, 379, 1, 1, 3323, 107, 3527, 227, 3739, 1, 1, 509, 1, 269, 4423, 1, 359, 599, 4919, 631, 5179, 1, 419, 349, 1, 733, 6007, 769, 6299, 1, 6599, 211, 6907, 883, 233, 1, 7547, 241, 7879, 503, 8219, 1049, 659, 1093, 8923, 569, 251, 1, 743, 1231, 10039, 1279, 10427, 1, 1, 1, 1, 1429, 1, 1481, 389, 1, 12487, 397, 12923, 1, 13367, 1699, 1063, 439, 1, 907, 14747, 1873, 1171, 1933, 1, 997, 1, 257, 16699, 1, 17207, 1, 479, 281, 1, 1, 1, 2381, 19319, 1, 19867, 1259, 1571, 647, 677, 2659, 21559, 2731, 1, 701, 22727, 1439, 1, 2953, 337, 1, 463, 1553, 1, 1, 1, 1, 26423, 3343, 27067, 1, 523, 1753, 1, 1, 937, 3673, 29723, 1879, 2339, 1, 1, 3931, 31799, 4019, 32507, 1, 33223, 2099, 409, 4289, 34679, 1, 35419, 2237, 613, 571, 36923, 4663, 1, 4759, 38459, 607, 39239, 2477, 3079, 1, 40823, 5153, 41627, 1, 1, 1, 1, 1, 44087, 5563, 1, 1, 45767, 2887, 46619, 5881, 601, 1, 3719, 3049, 49223, 1, 563, 1, 3923, 1, 51899, 1, 52807, 3329, 1733, 521, 54647, 1, 55579, 1, 56519, 1, 57467, 7243, 1579, 1, 59387, 1871, 4643, 3803, 61339, 1, 62327, 7853, 4871, 3989, 64327, 1013, 1, 8231, 66359, 643, 853, 1061, 1291, 1, 69467, 673, 727, 1, 2309, 4507, 72647, 2287, 1, 9283, 1, 9419, 1069, 2389, 5923, 1, 2111, 9833, 1, 9973, 80347, 1, 1381, 641, 82619, 10399, 1, 811, 1, 1, 2777, 5417, 1229, 1, 6803, 1, 89627, 5639, 90823, 2857, 7079, 11579, 93239, 11731, 1601, 2971, 929, 1, 941, 1, 1103, 1, 2687, 1, 1213, 1583, 1051, 12823, 947, 12983, 8039, 1, 1, 6653, 107099, 13469, 1, 13633, 971, 6899, 1, 3491, 1031, 1087, 113719, 1, 115067, 3617, 116423, 1, 117787, 1, 119159, 1, 120539, 7577, 1, 1, 123323, 1, 3371, 15679, 1, 991, 1, 8017, 128987, 1, 130423, 1, 809, 8287, 133319, 1, 2543, 1303, 136247, 17123, 137723, 4327, 1301, 8747, 1, 17681, 2683, 1, 1097, 9029, 11171, 2281, 1, 18439, 148279, 1, 1, 1, 4091, 1, 4933, 19213, 154487, 1493, 156059, 9803, 157639, 4951, 159227, 1, 1, 1, 2753, 5101, 164039, 10303, 12743, 20809, 1277, 21013, 1, 1, 1, 1, 172219, 1, 1, 21839, 1, 1, 177223, 1, 178907, 22469, 3061, 1, 1, 11447, 184007, 1, 185723, 1, 14419, 23539, 1361, 5939, 2689, 11987, 192667, 1861, 977, 24413, 196187, 1, 197959, 1, 199739, 1, 1471, 25303, 203323, 3191, 1, 1, 206939, 25981, 208759, 26209, 1, 13219, 212423, 1, 5791, 1, 216119, 2087, 217979, 6841, 219847, 13799, 221723, 2141, 4219, 28069, 225499, 14153, 227399, 1, 1, 1, 231223, 29023, 1, 1, 1, 14753, 237019, 1, 238967, 1, 2707, 1163, 242887, 7621, 244859, 1, 246839, 2383, 248827, 1, 6779, 15739, 252827, 31729, 19603, 31981, 1, 1, 258887, 1, 20071, 1, 2711, 32999, 1, 4157, 2593, 1289, 1487, 33773, 4597, 34033, 1, 1319, 275399, 1, 277499, 34819, 279607, 35083, 1667, 8837, 3593, 17807, 285979, 1, 1, 1, 2663, 1, 9433, 2293, 4993, 2843, 1777, 1201, 1283, 1, 301127, 1453, 303323, 38053, 2803, 38329, 1, 1, 1, 9721, 4397, 39163, 314423, 39443, 24359, 9931, 318919, 1, 1, 40289, 1787, 3121, 325723, 1, 328007, 1, 1, 3187, 10729, 41719, 1, 1, 337223, 21149, 26119, 42589, 341879, 1, 3217, 21587, 1, 10867, 348923, 1, 351287, 44059, 1409, 1, 356039, 1, 358427, 44953, 360823, 1, 11717, 22777, 365639, 1433, 368059, 1489, 28499, 1, 10079, 1, 375367, 23537, 29063, 47381, 2333, 1, 382747, 1, 385223, 1, 387707, 48619, 390199, 1, 392699, 1, 1741, 1, 397723, 1, 400247, 1619, 30983, 25253, 1, 6353, 1, 1, 1, 1, 412987, 6473, 415559, 26053, 1627, 1, 1, 1, 423323, 26539, 4391, 1, 428539, 53731, 1, 54059, 433787, 13597, 1, 1, 1, 55049, 14249, 55381, 34183, 1, 5023, 1, 3433, 56383, 1, 4363, 455099, 1783, 457799, 28697, 3313, 4441, 5581, 58073, 465947, 29207, 1, 1, 36263, 1, 2909, 59443, 6037, 14947, 36899, 1, 9103, 1951, 1, 1, 4561, 1, 1, 7691, 493627, 61879, 496439, 4787, 5147, 7823, 502087, 31469, 2393, 1, 1, 63649, 510619, 32003, 513479, 16091, 39719, 1, 1, 1759, 1, 16361, 524999, 2531, 17029, 66169, 530807, 66533, 533723, 1, 6793, 1, 2377, 67631, 542519, 1, 41959, 4273, 548423, 34369, 551387, 69109, 42643, 69481, 557339, 1, 2083, 1, 2357, 5431, 1, 70979, 2861, 17839, 1, 1, 15551, 1949, 1, 72493, 18757, 1, 44963, 9157, 587579, 1, 590647, 1, 1, 1, 1, 37397, 599899, 75181, 602999, 5813, 10273, 1, 1697, 19087, 5419, 5903, 615479, 1, 618619, 19381, 1, 38959, 1, 1, 3761, 2539, 631259, 1, 1, 4969, 637627, 79903, 1, 1, 1, 1, 1, 1, 2699, 81509, 653687, 6301, 656923, 1, 660167, 20681, 7993, 1, 51283, 1, 669947, 1, 673223, 42179, 4003, 1, 21929, 1, 2027, 42797, 1, 1, 689723, 86423, 18731, 2347, 696379, 1, 699719, 1, 3533, 88093, 2411, 88513, 1, 1, 713159, 1, 1, 89779, 1, 90203, 23333, 1, 3119, 1, 730139, 1, 733559, 91909, 736987, 1, 740423, 1, 10477, 93199, 747319, 1, 2791, 5879, 4463, 1, 20479, 94933, 761207, 95369, 1, 47903, 768199, 24061, 4621, 1, 4283, 1, 1, 24391, 2011, 49003, 25349, 7573, 789367, 98893, 10037, 49669, 796487, 12473, 61543, 1, 1, 1, 2579, 12641, 1, 50789, 6217, 1, 1, 102481, 1, 1, 825287, 25847, 828923, 103843, 1, 1, 836219, 26189, 839879, 1, 2879, 105673, 65171, 1, 6211, 1, 854599, 6691, 1, 1, 8369, 1, 4783, 3389, 6637, 1, 9811, 1, 1, 109849, 880667, 4243, 884423, 27697, 5449, 1, 24107, 111731, 68903, 28051, 29017, 1, 903323, 113153, 69779, 1, 910939, 1, 6581, 14323, 918587, 1, 922423, 1, 1, 14503, 930119, 4481, 933979, 116989, 937847, 1, 941723, 58979, 72739, 29611, 1, 118931, 953399, 119419, 1, 1, 1, 1, 965147, 120889, 13649, 9337, 1, 60937, 8963, 1, 7057, 1, 16693, 1, 988859, 7741, 7247, 1, 76679, 124853,

6. Sequence of the polynom (only primes)

823, 103, 827, 13, 839, 53, 859, 109, 887, 113, 71, 59, 967, 31, 1019, 131, 83, 139, 37, 1223, 79, 1307, 1399, 181, 1499, 97, 1607, 1723, 223, 1847, 239, 1979, 163, 137, 2267, 293, 2423, 313, 199, 167, 89, 2939, 379, 3323, 107, 3527, 227, 3739, 509, 269, 4423, 359, 599, 4919, 631, 5179, 419, 349, 733, 6007, 769, 6299, 6599, 211, 6907, 883, 233, 7547, 241, 7879, 503, 8219, 1049, 659, 1093, 8923, 569, 251, 743, 1231, 10039, 1279, 10427, 1429, 1481, 389, 12487, 397, 12923, 13367, 1699, 1063, 439, 907, 14747, 1873, 1171, 1933, 997, 257, 16699, 17207, 479, 281, 2381, 19319, 19867, 1259, 1571, 647, 677, 2659, 21559, 2731, 701, 22727, 1439, 2953, 337, 463, 1553, 26423, 3343, 27067, 523, 1753, 937, 3673, 29723, 1879, 2339, 3931, 31799, 4019, 32507, 33223, 2099, 409, 4289, 34679, 35419, 2237, 613, 571, 36923, 4663, 4759, 38459, 607, 39239, 2477, 3079, 40823, 5153, 41627, 44087, 5563, 45767, 2887, 46619, 5881, 601, 3719, 3049, 49223, 563, 3923, 51899, 52807, 3329, 1733, 521, 54647, 55579, 56519, 57467, 7243, 1579, 59387, 1871, 4643, 3803, 61339, 62327, 7853, 4871, 3989, 64327, 1013, 8231, 66359, 643, 853, 1061, 1291, 69467, 673, 727, 2309, 4507, 72647, 2287, 9283, 9419, 1069, 2389, 5923, 2111, 9833, 9973, 80347, 1381, 641, 82619, 10399, 811, 2777, 5417, 1229, 6803, 89627, 5639, 90823, 2857, 7079, 11579, 93239, 11731, 1601, 2971, 929, 941, 1103, 2687, 1213, 1583, 1051, 12823, 947, 12983, 8039, 6653, 107099, 13469, 13633, 971, 6899, 3491, 1031, 1087, 113719, 115067, 3617, 116423, 117787, 119159, 120539, 7577, 123323, 3371, 15679, 991, 8017, 128987, 130423, 809, 8287, 133319, 2543, 1303, 136247, 17123, 137723, 4327, 1301, 8747, 17681, 2683, 1097, 9029, 11171, 2281, 18439, 148279, 4091, 4933, 19213, 154487, 1493, 156059, 9803, 157639, 4951, 159227, 2753, 5101, 164039, 10303, 12743, 20809, 1277, 21013, 172219, 21839, 177223, 178907, 22469, 3061, 11447, 184007, 185723, 14419, 23539, 1361, 5939, 2689, 11987, 192667, 1861, 977, 24413, 196187, 197959, 199739, 1471, 25303, 203323, 3191, 206939, 25981, 208759, 26209, 13219, 212423, 5791, 216119, 2087, 217979, 6841, 219847, 13799, 221723, 2141, 4219, 28069, 225499, 14153, 227399, 231223, 29023, 14753, 237019, 238967, 2707, 1163, 242887, 7621, 244859, 246839, 2383, 248827, 6779, 15739, 252827, 31729, 19603, 31981, 258887, 20071, 2711, 32999, 4157, 2593, 1289, 1487, 33773, 4597, 34033, 1319, 275399, 277499, 34819, 279607, 35083, 1667, 8837, 3593, 17807, 285979, 2663, 9433, 2293, 4993, 2843, 1777, 1201, 1283, 301127, 1453, 303323, 38053, 2803, 38329, 9721, 4397, 39163, 314423, 39443, 24359, 9931, 318919, 40289, 1787, 3121, 325723, 328007, 3187, 10729, 41719, 337223, 21149, 26119, 42589, 341879, 3217, 21587, 10867, 348923, 351287, 44059, 1409, 356039, 358427, 44953, 360823, 11717, 22777, 365639, 1433, 368059, 1489, 28499, 10079, 375367, 23537, 29063, 47381, 2333, 382747, 385223, 387707, 48619, 390199, 392699, 1741, 397723, 400247, 1619, 30983, 25253, 6353, 412987, 6473, 415559, 26053, 1627, 423323, 26539, 4391, 428539, 53731, 54059, 433787, 13597, 55049, 14249, 55381, 34183, 5023, 3433, 56383, 4363, 455099, 1783, 457799, 28697, 3313, 4441, 5581, 58073, 465947, 29207, 36263, 2909, 59443, 6037, 14947, 36899, 9103, 1951, 4561, 7691, 493627, 61879, 496439, 4787, 5147, 7823, 502087, 31469, 2393, 63649, 510619, 32003, 513479, 16091, 39719, 1759, 16361, 524999, 2531, 17029, 66169, 530807, 66533, 533723, 6793, 2377, 67631, 542519, 41959, 4273, 548423, 34369, 551387, 69109, 42643, 69481, 557339, 2083, 2357, 5431, 70979, 2861, 17839, 15551, 1949, 72493, 18757, 44963, 9157, 587579, 590647, 37397, 599899, 75181, 602999, 5813, 10273, 1697, 19087, 5419, 5903, 615479, 618619, 19381, 38959, 3761, 2539, 631259, 4969, 637627, 79903, 2699, 81509, 653687, 6301, 656923, 660167, 20681, 7993, 51283, 669947, 673223, 42179, 4003, 21929, 2027, 42797, 689723, 86423, 18731, 2347, 696379, 699719, 3533, 88093, 2411, 88513, 713159, 89779, 90203, 23333, 3119, 730139, 733559, 91909, 736987, 740423, 10477, 93199, 747319, 2791, 5879, 4463, 20479, 94933, 761207, 95369, 47903, 768199, 24061, 4621, 4283, 24391, 2011, 49003, 25349, 7573, 789367, 98893, 10037, 49669, 796487, 12473, 61543, 2579, 12641, 50789, 6217, 102481, 825287, 25847, 828923, 103843, 836219, 26189, 839879, 2879, 105673, 65171, 6211, 854599, 6691, 8369, 4783, 3389, 6637, 9811, 109849, 880667, 4243, 884423, 27697, 5449, 24107, 111731, 68903, 28051, 29017, 903323, 113153, 69779, 910939, 6581, 14323, 918587, 922423, 14503, 930119, 4481, 933979, 116989, 937847, 941723, 58979, 72739, 29611, 118931, 953399, 119419, 965147, 120889, 13649, 9337, 60937, 8963, 7057, 16693, 988859, 7741, 7247, 76679, 124853,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2x+823 and
the reducible primes which appear as divisor for the first time
p | x^2x+823 and p < x^2x+823

To avoid confusion with the number of primes:
I did not count the primes <= A
but I counted the primes appending the x and therefore the x <= A

A B C D E F G H
exponent
=log2 (x)
<=x number
of all primes
number of primes
p = f(x)
number of primes
p | f(x)
C / x D / x E / x
1 2 3 2 1 1.5 1 0.5
2 4 5 3 2 1.25 0.75 0.5
3 8 9 5 4 1.125 0.625 0.5
4 16 17 7 10 1.0625 0.4375 0.625
5 32 30 14 16 0.9375 0.4375 0.5
6 64 54 23 31 0.84375 0.359375 0.484375
7 128 102 40 62 0.796875 0.3125 0.484375
8 256 191 72 119 0.74609375 0.28125 0.46484375
9 512 370 126 244 0.72265625 0.24609375 0.4765625
10 1024 720 220 500 0.703125 0.21484375 0.48828125
11 2048 1427 396 1031 0.69677734 0.19335938 0.50341797
12 4096 2851 701 2150 0.69604492 0.17114258 0.52490234
13 8192 5704 1287 4417 0.69628906 0.15710449 0.53918457
14 16384 11359 2386 8973 0.69329834 0.14562988 0.54766846
15 32768 22710 4397 18313 0.6930542 0.13418579 0.55886841
16 65536 45441 8161 37280 0.69337463 0.12452698 0.56884766
17 131072 90967 15171 75796 0.69402313 0.11574554 0.57827759
18 262144 181997 28311 153686 0.69426346 0.10799789 0.58626556
19 524288 364045 53145 310900 0.69436073 0.10136604 0.59299469
20 1048576 727940 100650 627290 0.69421768 0.09598732 0.59823036
21 2097152 1455989 190911 1265078 0.69426966 0.09103346 0.6032362
22 4194304 2911925 363219 2548706 0.69425702 0.08659816 0.60765886
23 8388608 5823506 692360 5131146 0.69421601 0.08253574 0.61168027
24 16777216 11646793 1321565 10325228 0.69420296 0.07877141 0.61543155


8. Check for existing Integer Sequences by OEIS

Found in Database : 823, 103, 827, 13, 839, 53, 859, 109, 887, 113, 71, 59, 967, 31, 1019, 131, 83, 139, 37, 1,
Found in Database : 823, 103, 827, 13, 839, 53, 859, 109, 887, 113, 71, 59, 967, 31, 1019, 131, 83, 139, 37, 1223, 79, 1307, 1399, 181, 1499, 97, 1607, 1723, 223, 1847, 239, 1979, 163, 137, 2267, 293,
Found in Database : 13, 31, 37, 53, 59, 71, 79, 83, 89, 97, 103, 107, 109, 113, 131, 137, 139,