Inhaltsverzeichnis

Development of
Algorithmic Constructions

18:33:11
Deutsch
28.Mar 2024

Polynom = x^2+184x-569

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) = 569 = 569
f(1) = 3 = 3
f(2) = 197 = 197
f(3) = 1 = 1
f(4) = 183 = 3*61
f(5) = 47 = 47
f(6) = 571 = 571
f(7) = 3 = 3
f(8) = 967 = 967
f(9) = 73 = 73
f(10) = 1371 = 3*457
f(11) = 197 = 197
f(12) = 1783 = 1783
f(13) = 249 = 3*83
f(14) = 2203 = 2203
f(15) = 151 = 151
f(16) = 2631 = 3*877
f(17) = 89 = 89
f(18) = 3067 = 3067
f(19) = 411 = 3*137
f(20) = 3511 = 3511
f(21) = 467 = 467
f(22) = 3963 = 3*1321
f(23) = 131 = 131
f(24) = 4423 = 4423
f(25) = 291 = 3*97
f(26) = 4891 = 67*73
f(27) = 641 = 641
f(28) = 5367 = 3*1789
f(29) = 701 = 701
f(30) = 5851 = 5851
f(31) = 381 = 3*127
f(32) = 6343 = 6343
f(33) = 103 = 103
f(34) = 6843 = 3*2281
f(35) = 887 = 887
f(36) = 7351 = 7351
f(37) = 951 = 3*317
f(38) = 7867 = 7867
f(39) = 127 = 127
f(40) = 8391 = 3*2797
f(41) = 541 = 541
f(42) = 8923 = 8923
f(43) = 1149 = 3*383
f(44) = 9463 = 9463
f(45) = 1217 = 1217
f(46) = 10011 = 3*47*71
f(47) = 643 = 643
f(48) = 10567 = 10567
f(49) = 339 = 3*113
f(50) = 11131 = 11131
f(51) = 1427 = 1427
f(52) = 11703 = 3*47*83
f(53) = 1499 = 1499
f(54) = 12283 = 71*173
f(55) = 393 = 3*131
f(56) = 12871 = 61*211
f(57) = 823 = 823
f(58) = 13467 = 3*67*67
f(59) = 1721 = 1721
f(60) = 14071 = 14071
f(61) = 1797 = 3*599
f(62) = 14683 = 14683
f(63) = 937 = 937
f(64) = 15303 = 3*5101
f(65) = 61 = 61
f(66) = 15931 = 89*179
f(67) = 2031 = 3*677
f(68) = 16567 = 16567
f(69) = 2111 = 2111
f(70) = 17211 = 3*5737
f(71) = 137 = 137
f(72) = 17863 = 17863
f(73) = 1137 = 3*379
f(74) = 18523 = 18523
f(75) = 2357 = 2357
f(76) = 19191 = 3*6397
f(77) = 2441 = 2441
f(78) = 19867 = 19867
f(79) = 1263 = 3*421
f(80) = 20551 = 20551
f(81) = 653 = 653
f(82) = 21243 = 3*73*97
f(83) = 2699 = 2699
f(84) = 21943 = 21943
f(85) = 2787 = 3*929
f(86) = 22651 = 22651
f(87) = 719 = 719
f(88) = 23367 = 3*7789
f(89) = 1483 = 1483
f(90) = 24091 = 24091
f(91) = 3057 = 3*1019
f(92) = 24823 = 103*241
f(93) = 3149 = 47*67
f(94) = 25563 = 3*8521
f(95) = 1621 = 1621
f(96) = 26311 = 83*317
f(97) = 417 = 3*139
f(98) = 27067 = 27067
f(99) = 3431 = 47*73
f(100) = 27831 = 3*9277

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+184x-569

f(0)=569
f(1)=3
f(2)=197
f(3)=1
f(4)=61
f(5)=47
f(6)=571
f(7)=1
f(8)=967
f(9)=73
f(10)=457
f(11)=1
f(12)=1783
f(13)=83
f(14)=2203
f(15)=151
f(16)=877
f(17)=89
f(18)=3067
f(19)=137
f(20)=3511
f(21)=467
f(22)=1321
f(23)=131
f(24)=4423
f(25)=97
f(26)=67
f(27)=641
f(28)=1789
f(29)=701
f(30)=5851
f(31)=127
f(32)=6343
f(33)=103
f(34)=2281
f(35)=887
f(36)=7351
f(37)=317
f(38)=7867
f(39)=1
f(40)=2797
f(41)=541
f(42)=8923
f(43)=383
f(44)=9463
f(45)=1217
f(46)=71
f(47)=643
f(48)=10567
f(49)=113
f(50)=11131
f(51)=1427
f(52)=1
f(53)=1499
f(54)=173
f(55)=1
f(56)=211
f(57)=823
f(58)=1
f(59)=1721
f(60)=14071
f(61)=599
f(62)=14683
f(63)=937
f(64)=5101
f(65)=1
f(66)=179
f(67)=677
f(68)=16567
f(69)=2111
f(70)=5737
f(71)=1
f(72)=17863
f(73)=379
f(74)=18523
f(75)=2357
f(76)=6397
f(77)=2441
f(78)=19867
f(79)=421
f(80)=20551
f(81)=653
f(82)=1
f(83)=2699
f(84)=21943
f(85)=929
f(86)=22651
f(87)=719
f(88)=7789
f(89)=1483
f(90)=24091
f(91)=1019
f(92)=241
f(93)=1
f(94)=8521
f(95)=1621
f(96)=1
f(97)=139
f(98)=27067
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+184x-569 could be written as f(y)= y^2-9033 with x=y-92

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+92
f'(x)>2x+183

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

569, 3, 197, 1, 61, 47, 571, 1, 967, 73, 457, 1, 1783, 83, 2203, 151, 877, 89, 3067, 137, 3511, 467, 1321, 131, 4423, 97, 67, 641, 1789, 701, 5851, 127, 6343, 103, 2281, 887, 7351, 317, 7867, 1, 2797, 541, 8923, 383, 9463, 1217, 71, 643, 10567, 113, 11131, 1427, 1, 1499, 173, 1, 211, 823, 1, 1721, 14071, 599, 14683, 937, 5101, 1, 179, 677, 16567, 2111, 5737, 1, 17863, 379, 18523, 2357, 6397, 2441, 19867, 421, 20551, 653, 1, 2699, 21943, 929, 22651, 719, 7789, 1483, 24091, 1019, 241, 1, 8521, 1621, 1, 139, 27067, 1, 9277, 3527, 28603, 1, 29383, 1861, 1, 3821, 1, 1307, 31771, 2011, 10861, 1031, 33403, 1409, 34231, 1, 11689, 1109, 35911, 757, 1, 4649, 12541, 1, 631, 811, 39367, 311, 13417, 5087, 41143, 1733, 42043, 1, 1, 2713, 43867, 1847, 953, 5657, 15241, 2887, 46663, 491, 1013, 6011, 16189, 6131, 49531, 521, 50503, 3187, 1, 1, 1, 2207, 1, 3373, 271, 859, 1, 2333, 56503, 7127, 1, 907, 58567, 1231, 59611, 7517, 277, 7649, 61723, 1297, 62791, 1979, 349, 1, 64951, 2729, 313, 2081, 22381, 4231, 1, 1, 977, 8741, 23497, 4441, 1069, 1, 72763, 1, 347, 9311, 1, 1, 76231, 4801, 25801, 9749, 78583, 3299, 331, 5023, 1, 2549, 82171, 3449, 83383, 10499, 28201, 2663, 85831, 1801, 1049, 1, 29437, 11117, 89563, 1879, 90823, 1429, 30697, 1, 1279, 3917, 94651, 1489, 31981, 6037, 2069, 4079, 709, 12401, 33289, 1, 2153, 1061, 102523, 12899, 389, 1, 105211, 1103, 1747, 6703, 35977, 13577, 109303, 4583, 733, 6961, 37357, 881, 113467, 1, 367, 14447, 1, 1, 117703, 2467, 119131, 1, 40189, 15161, 122011, 2557, 373, 3881, 41641, 1, 1303, 5297, 127867, 4019, 43117, 1, 130843, 5483, 1487, 1, 44617, 1, 135367, 1, 1, 17207, 46141, 1, 2089, 1, 141511, 8893, 463, 17981, 2371, 1, 146203, 9187, 49261, 4643, 149371, 6257, 150967, 1, 50857, 4793, 154183, 3229, 449, 19577, 1, 1, 2179, 3331, 160711, 1, 54121, 20399, 164023, 6869, 2473, 1301, 1187, 10513, 2381, 7079, 1511, 1, 1223, 10831, 499, 1823, 2477, 22091, 59197, 22307, 1741, 1877, 181063, 1, 60937, 22961, 184567, 7727, 479, 11701, 62701, 2953, 1, 7949, 191671, 24071, 64489, 3037, 195271, 1, 197083, 24749, 66301, 24977, 200731, 4201, 202567, 6359, 1117, 25667, 206263, 1, 2851, 1, 69997, 13183, 211867, 8867, 213751, 1, 71881, 13537, 2621, 1, 219451, 27551, 1, 27791, 223291, 1, 225223, 1, 75721, 28517, 1163, 9587, 231067, 14503, 1, 1, 235003, 9833, 236983, 1, 79657, 7499, 240967, 1, 242971, 30497, 1, 1, 1777, 5167, 1901, 3907, 83689, 31511, 1993, 10589, 1, 4003, 1, 16141, 1, 10847, 1, 32801, 1, 1, 265543, 2777, 267643, 33587, 89917, 33851, 271867, 2843, 1, 17191, 92041, 34649, 601, 1, 280411, 1, 1, 1, 284731, 11909, 1, 35999, 557, 2267, 291271, 6091, 293467, 36821, 1471, 37097, 3347, 6229, 4111, 9413, 563, 1, 2017, 1, 3163, 9623, 1451, 19387, 311323, 1, 1, 39341, 1, 19813, 4481, 1663, 2339, 659, 107581, 40487, 325051, 1699, 1, 20533, 109897, 41357, 5443, 13883, 334363, 1, 112237, 10559, 339067, 14177, 1, 1, 114601, 10781, 3361, 7237, 348571, 43721, 116989, 44021, 3643, 1, 355783, 2789, 1, 44927, 7673, 15077, 363067, 1423, 1669, 22921, 7829, 15383, 370423, 46457, 124297, 1, 375367, 3923, 377851, 47387, 126781, 47699, 382843, 4001, 385351, 1, 129289, 1, 390391, 16319, 392923, 1, 2161, 6199, 398011, 1, 3889, 50231, 1619, 1, 405703, 1, 408283, 51197, 1, 51521, 3659, 8641, 416071, 13043, 2083, 1, 421303, 17609, 423931, 1, 142189, 1, 429211, 17939, 431863, 1, 1, 27241, 437191, 1, 439867, 1, 147517, 55487, 1, 1, 447943, 28081, 150217, 56501, 453367, 18947, 456091, 28591, 152941, 1, 1, 19289, 464311, 58211, 155689, 14639, 5279, 9817, 2399, 1, 751, 1, 478171, 1, 480967, 7537, 1, 60647, 486583, 20333, 1, 7669, 3491, 30853, 495067, 1, 1601, 62417, 2351, 31387, 503623, 5261, 506491, 1, 169789, 1, 512251, 5351, 515143, 1, 172681, 64937, 1, 21767, 523867, 32833, 1973, 4127, 529723, 22133, 532663, 1, 178537, 1, 538567, 11251, 541531, 1, 181501, 1, 547483, 11437, 550471, 1, 184489, 69371, 1, 1, 559483, 1, 1933, 35251, 1, 23627, 1, 71261, 190537, 1, 1, 3001, 4549, 1, 193597, 1, 4457, 3049, 586951, 36781, 196681, 1, 593143, 1, 596251, 37363, 829, 1, 1, 25169, 4357, 1, 202921, 19073, 9133, 12781, 615067, 77081, 206077, 77477, 2293, 12979, 1, 1, 209257, 1, 630967, 26357, 1, 4967, 212461, 39937, 853, 26759, 7757, 80681, 215689, 40543, 1, 6791, 653563, 81899, 218941, 82307, 2383, 1, 9901, 1, 1, 83537, 669943, 27983, 1, 42181, 3697, 10597, 679867, 1, 2521, 85607, 228841, 10753, 1, 14407, 4007, 86861, 232189, 87281, 699931, 1, 2539, 22031, 2287, 88547, 1, 1, 713467, 22349, 2879, 1, 720283, 1, 10193, 90677, 242377, 45553, 730567, 1907, 4861, 91967, 245821, 92399, 740923, 1, 744391, 46633, 2801, 93701, 751351, 31379, 4217, 47287, 2237, 23753, 919, 31817, 12547, 95891, 1, 24083, 5557, 1, 775963, 1, 259837, 1, 16661, 1, 4547, 1, 263401, 98999, 16889, 33149, 1, 12487, 1, 1, 804571, 33599, 11071, 983, 270601, 1, 815431, 8513, 9203, 102611, 274237, 103067, 826363, 8627, 4637, 51991, 277897, 1, 837367, 1, 12553, 1, 281581, 3307, 848443, 1, 852151, 106751, 285289, 6701, 7607, 1, 863323, 1, 289021, 2311, 870811, 18181, 874567, 1, 292777, 2341, 882103, 36833, 885883, 27743, 296557, 1, 1, 37307, 897271, 112397, 4483, 56437, 904903, 4723, 908731, 1, 1, 114311, 916411, 4783, 920263, 57637, 308041, 115757, 2659, 38747, 1, 58363, 1, 29303, 11321, 39233, 943543, 1, 1009, 29669, 951367, 19861, 2753, 119657, 6803, 1, 963163, 20107, 967111, 1, 1, 121631, 4621, 40709, 13411, 1, 327661, 61561, 13901, 1, 990967, 124121, 5437, 62311, 998983, 10427, 1003003, 125627, 3259, 126131, 1011067, 1, 7993, 1, 1, 127649, 2239, 42719, 12377, 64333, 2707, 1, 1035451, 1, 1039543, 130199, 1051, 16339, 1047751, 21871, 1, 2803, 2687, 132257, 1060123, 22129, 1064263, 1, 356137, 133811, 1063, 44777, 1076731, 33713, 1, 1, 1, 1, 12239, 136421, 4993, 68473, 10657, 1, 1101883, 137999, 5503, 1, 1110331, 2897, 1114567, 1, 2683, 2297, 8573, 1, 1127323, 1, 377197, 1, 8291, 47417, 18691, 142787, 381481, 35831, 1093, 23977, 24533, 1, 1, 144941, 1161691, 24247, 24809, 18253, 1, 1, 1, 49037, 1, 18457,

6. Sequence of the polynom (only primes)

569, 3, 197, 61, 47, 571, 967, 73, 457, 1783, 83, 2203, 151, 877, 89, 3067, 137, 3511, 467, 1321, 131, 4423, 97, 67, 641, 1789, 701, 5851, 127, 6343, 103, 2281, 887, 7351, 317, 7867, 2797, 541, 8923, 383, 9463, 1217, 71, 643, 10567, 113, 11131, 1427, 1499, 173, 211, 823, 1721, 14071, 599, 14683, 937, 5101, 179, 677, 16567, 2111, 5737, 17863, 379, 18523, 2357, 6397, 2441, 19867, 421, 20551, 653, 2699, 21943, 929, 22651, 719, 7789, 1483, 24091, 1019, 241, 8521, 1621, 139, 27067, 9277, 3527, 28603, 29383, 1861, 3821, 1307, 31771, 2011, 10861, 1031, 33403, 1409, 34231, 11689, 1109, 35911, 757, 4649, 12541, 631, 811, 39367, 311, 13417, 5087, 41143, 1733, 42043, 2713, 43867, 1847, 953, 5657, 15241, 2887, 46663, 491, 1013, 6011, 16189, 6131, 49531, 521, 50503, 3187, 2207, 3373, 271, 859, 2333, 56503, 7127, 907, 58567, 1231, 59611, 7517, 277, 7649, 61723, 1297, 62791, 1979, 349, 64951, 2729, 313, 2081, 22381, 4231, 977, 8741, 23497, 4441, 1069, 72763, 347, 9311, 76231, 4801, 25801, 9749, 78583, 3299, 331, 5023, 2549, 82171, 3449, 83383, 10499, 28201, 2663, 85831, 1801, 1049, 29437, 11117, 89563, 1879, 90823, 1429, 30697, 1279, 3917, 94651, 1489, 31981, 6037, 2069, 4079, 709, 12401, 33289, 2153, 1061, 102523, 12899, 389, 105211, 1103, 1747, 6703, 35977, 13577, 109303, 4583, 733, 6961, 37357, 881, 113467, 367, 14447, 117703, 2467, 119131, 40189, 15161, 122011, 2557, 373, 3881, 41641, 1303, 5297, 127867, 4019, 43117, 130843, 5483, 1487, 44617, 135367, 17207, 46141, 2089, 141511, 8893, 463, 17981, 2371, 146203, 9187, 49261, 4643, 149371, 6257, 150967, 50857, 4793, 154183, 3229, 449, 19577, 2179, 3331, 160711, 54121, 20399, 164023, 6869, 2473, 1301, 1187, 10513, 2381, 7079, 1511, 1223, 10831, 499, 1823, 2477, 22091, 59197, 22307, 1741, 1877, 181063, 60937, 22961, 184567, 7727, 479, 11701, 62701, 2953, 7949, 191671, 24071, 64489, 3037, 195271, 197083, 24749, 66301, 24977, 200731, 4201, 202567, 6359, 1117, 25667, 206263, 2851, 69997, 13183, 211867, 8867, 213751, 71881, 13537, 2621, 219451, 27551, 27791, 223291, 225223, 75721, 28517, 1163, 9587, 231067, 14503, 235003, 9833, 236983, 79657, 7499, 240967, 242971, 30497, 1777, 5167, 1901, 3907, 83689, 31511, 1993, 10589, 4003, 16141, 10847, 32801, 265543, 2777, 267643, 33587, 89917, 33851, 271867, 2843, 17191, 92041, 34649, 601, 280411, 284731, 11909, 35999, 557, 2267, 291271, 6091, 293467, 36821, 1471, 37097, 3347, 6229, 4111, 9413, 563, 2017, 3163, 9623, 1451, 19387, 311323, 39341, 19813, 4481, 1663, 2339, 659, 107581, 40487, 325051, 1699, 20533, 109897, 41357, 5443, 13883, 334363, 112237, 10559, 339067, 14177, 114601, 10781, 3361, 7237, 348571, 43721, 116989, 44021, 3643, 355783, 2789, 44927, 7673, 15077, 363067, 1423, 1669, 22921, 7829, 15383, 370423, 46457, 124297, 375367, 3923, 377851, 47387, 126781, 47699, 382843, 4001, 385351, 129289, 390391, 16319, 392923, 2161, 6199, 398011, 3889, 50231, 1619, 405703, 408283, 51197, 51521, 3659, 8641, 416071, 13043, 2083, 421303, 17609, 423931, 142189, 429211, 17939, 431863, 27241, 437191, 439867, 147517, 55487, 447943, 28081, 150217, 56501, 453367, 18947, 456091, 28591, 152941, 19289, 464311, 58211, 155689, 14639, 5279, 9817, 2399, 751, 478171, 480967, 7537, 60647, 486583, 20333, 7669, 3491, 30853, 495067, 1601, 62417, 2351, 31387, 503623, 5261, 506491, 169789, 512251, 5351, 515143, 172681, 64937, 21767, 523867, 32833, 1973, 4127, 529723, 22133, 532663, 178537, 538567, 11251, 541531, 181501, 547483, 11437, 550471, 184489, 69371, 559483, 1933, 35251, 23627, 71261, 190537, 3001, 4549, 193597, 4457, 3049, 586951, 36781, 196681, 593143, 596251, 37363, 829, 25169, 4357, 202921, 19073, 9133, 12781, 615067, 77081, 206077, 77477, 2293, 12979, 209257, 630967, 26357, 4967, 212461, 39937, 853, 26759, 7757, 80681, 215689, 40543, 6791, 653563, 81899, 218941, 82307, 2383, 9901, 83537, 669943, 27983, 42181, 3697, 10597, 679867, 2521, 85607, 228841, 10753, 14407, 4007, 86861, 232189, 87281, 699931, 2539, 22031, 2287, 88547, 713467, 22349, 2879, 720283, 10193, 90677, 242377, 45553, 730567, 1907, 4861, 91967, 245821, 92399, 740923, 744391, 46633, 2801, 93701, 751351, 31379, 4217, 47287, 2237, 23753, 919, 31817, 12547, 95891, 24083, 5557, 775963, 259837, 16661, 4547, 263401, 98999, 16889, 33149, 12487, 804571, 33599, 11071, 983, 270601, 815431, 8513, 9203, 102611, 274237, 103067, 826363, 8627, 4637, 51991, 277897, 837367, 12553, 281581, 3307, 848443, 852151, 106751, 285289, 6701, 7607, 863323, 289021, 2311, 870811, 18181, 874567, 292777, 2341, 882103, 36833, 885883, 27743, 296557, 37307, 897271, 112397, 4483, 56437, 904903, 4723, 908731, 114311, 916411, 4783, 920263, 57637, 308041, 115757, 2659, 38747, 58363, 29303, 11321, 39233, 943543, 1009, 29669, 951367, 19861, 2753, 119657, 6803, 963163, 20107, 967111, 121631, 4621, 40709, 13411, 327661, 61561, 13901, 990967, 124121, 5437, 62311, 998983, 10427, 1003003, 125627, 3259, 126131, 1011067, 7993, 127649, 2239, 42719, 12377, 64333, 2707, 1035451, 1039543, 130199, 1051, 16339, 1047751, 21871, 2803, 2687, 132257, 1060123, 22129, 1064263, 356137, 133811, 1063, 44777, 1076731, 33713, 12239, 136421, 4993, 68473, 10657, 1101883, 137999, 5503, 1110331, 2897, 1114567, 2683, 2297, 8573, 1127323, 377197, 8291, 47417, 18691, 142787, 381481, 35831, 1093, 23977, 24533, 144941, 1161691, 24247, 24809, 18253, 49037, 18457,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+184x-569 and
the reducible primes which appear as divisor for the first time
p | x^2+184x-569 and p < x^2+184x-569

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

8. Check for existing Integer Sequences by OEIS

Found in Database : 569, 3, 197, 1, 61, 47, 571, 1, 967, 73, 457, 1, 1783, 83, 2203, 151, 877, 89, 3067, 137,
Found in Database : 569, 3, 197, 61, 47, 571, 967, 73, 457, 1783, 83, 2203, 151, 877, 89, 3067, 137, 3511, 467, 1321, 131, 4423, 97, 67, 641, 1789, 701, 5851, 127, 6343, 103, 2281, 887, 7351, 317, 7867,
Found in Database : 3, 47, 61, 67, 71, 73, 83, 89, 97, 103, 113, 127, 131, 137, 139,