Inhaltsverzeichnis

Development of
Algorithmic Constructions

16:59:58
Deutsch
29.Mar 2024

Polynom = x^2+136x-233

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) = 233 = 233
f(1) = 3 = 3
f(2) = 43 = 43
f(3) = 23 = 23
f(4) = 327 = 3*109
f(5) = 59 = 59
f(6) = 619 = 619
f(7) = 3 = 3
f(8) = 919 = 919
f(9) = 67 = 67
f(10) = 1227 = 3*409
f(11) = 173 = 173
f(12) = 1543 = 1543
f(13) = 213 = 3*71
f(14) = 1867 = 1867
f(15) = 127 = 127
f(16) = 2199 = 3*733
f(17) = 37 = 37
f(18) = 2539 = 2539
f(19) = 339 = 3*113
f(20) = 2887 = 2887
f(21) = 383 = 383
f(22) = 3243 = 3*23*47
f(23) = 107 = 107
f(24) = 3607 = 3607
f(25) = 237 = 3*79
f(26) = 3979 = 23*173
f(27) = 521 = 521
f(28) = 4359 = 3*1453
f(29) = 569 = 569
f(30) = 4747 = 47*101
f(31) = 309 = 3*103
f(32) = 5143 = 37*139
f(33) = 167 = 167
f(34) = 5547 = 3*43*43
f(35) = 719 = 719
f(36) = 5959 = 59*101
f(37) = 771 = 3*257
f(38) = 6379 = 6379
f(39) = 103 = 103
f(40) = 6807 = 3*2269
f(41) = 439 = 439
f(42) = 7243 = 7243
f(43) = 933 = 3*311
f(44) = 7687 = 7687
f(45) = 989 = 23*43
f(46) = 8139 = 3*2713
f(47) = 523 = 523
f(48) = 8599 = 8599
f(49) = 69 = 3*23
f(50) = 9067 = 9067
f(51) = 1163 = 1163
f(52) = 9543 = 3*3181
f(53) = 1223 = 1223
f(54) = 10027 = 37*271
f(55) = 321 = 3*107
f(56) = 10519 = 67*157
f(57) = 673 = 673
f(58) = 11019 = 3*3673
f(59) = 1409 = 1409
f(60) = 11527 = 11527
f(61) = 1473 = 3*491
f(62) = 12043 = 12043
f(63) = 769 = 769
f(64) = 12567 = 3*59*71
f(65) = 401 = 401
f(66) = 13099 = 13099
f(67) = 1671 = 3*557
f(68) = 13639 = 23*593
f(69) = 1739 = 37*47
f(70) = 14187 = 3*4729
f(71) = 113 = 113
f(72) = 14743 = 23*641
f(73) = 939 = 3*313
f(74) = 15307 = 15307
f(75) = 1949 = 1949
f(76) = 15879 = 3*67*79
f(77) = 2021 = 43*47
f(78) = 16459 = 109*151
f(79) = 1047 = 3*349
f(80) = 17047 = 17047
f(81) = 271 = 271
f(82) = 17643 = 3*5881
f(83) = 2243 = 2243
f(84) = 18247 = 71*257
f(85) = 2319 = 3*773
f(86) = 18859 = 18859
f(87) = 599 = 599
f(88) = 19479 = 3*43*151
f(89) = 1237 = 1237
f(90) = 20107 = 20107
f(91) = 2553 = 3*23*37
f(92) = 20743 = 20743
f(93) = 2633 = 2633
f(94) = 21387 = 3*7129
f(95) = 1357 = 23*59
f(96) = 22039 = 22039
f(97) = 699 = 3*233
f(98) = 22699 = 22699
f(99) = 2879 = 2879
f(100) = 23367 = 3*7789

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+136x-233

f(0)=233
f(1)=3
f(2)=43
f(3)=23
f(4)=109
f(5)=59
f(6)=619
f(7)=1
f(8)=919
f(9)=67
f(10)=409
f(11)=173
f(12)=1543
f(13)=71
f(14)=1867
f(15)=127
f(16)=733
f(17)=37
f(18)=2539
f(19)=113
f(20)=2887
f(21)=383
f(22)=47
f(23)=107
f(24)=3607
f(25)=79
f(26)=1
f(27)=521
f(28)=1453
f(29)=569
f(30)=101
f(31)=103
f(32)=139
f(33)=167
f(34)=1
f(35)=719
f(36)=1
f(37)=257
f(38)=6379
f(39)=1
f(40)=2269
f(41)=439
f(42)=7243
f(43)=311
f(44)=7687
f(45)=1
f(46)=2713
f(47)=523
f(48)=8599
f(49)=1
f(50)=9067
f(51)=1163
f(52)=3181
f(53)=1223
f(54)=271
f(55)=1
f(56)=157
f(57)=673
f(58)=3673
f(59)=1409
f(60)=11527
f(61)=491
f(62)=12043
f(63)=769
f(64)=1
f(65)=401
f(66)=13099
f(67)=557
f(68)=593
f(69)=1
f(70)=4729
f(71)=1
f(72)=641
f(73)=313
f(74)=15307
f(75)=1949
f(76)=1
f(77)=1
f(78)=151
f(79)=349
f(80)=17047
f(81)=1
f(82)=5881
f(83)=2243
f(84)=1
f(85)=773
f(86)=18859
f(87)=599
f(88)=1
f(89)=1237
f(90)=20107
f(91)=1
f(92)=20743
f(93)=2633
f(94)=7129
f(95)=1
f(96)=22039
f(97)=1
f(98)=22699
f(99)=2879

b) Substitution of the polynom
The polynom f(x)=x^2+136x-233 could be written as f(y)= y^2-4857 with x=y-68

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+68
f'(x)>2x+135

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

233, 3, 43, 23, 109, 59, 619, 1, 919, 67, 409, 173, 1543, 71, 1867, 127, 733, 37, 2539, 113, 2887, 383, 47, 107, 3607, 79, 1, 521, 1453, 569, 101, 103, 139, 167, 1, 719, 1, 257, 6379, 1, 2269, 439, 7243, 311, 7687, 1, 2713, 523, 8599, 1, 9067, 1163, 3181, 1223, 271, 1, 157, 673, 3673, 1409, 11527, 491, 12043, 769, 1, 401, 13099, 557, 593, 1, 4729, 1, 641, 313, 15307, 1949, 1, 1, 151, 349, 17047, 1, 5881, 2243, 1, 773, 18859, 599, 1, 1237, 20107, 1, 20743, 2633, 7129, 1, 22039, 1, 22699, 2879, 7789, 2963, 24043, 1, 1, 1567, 229, 3221, 26119, 1103, 193, 1699, 9181, 1, 1229, 1193, 617, 3671, 431, 941, 709, 643, 199, 1, 227, 4049, 32779, 691, 907, 1061, 1, 1, 1, 1481, 1, 1, 12253, 1, 37579, 1583, 1, 211, 1, 1, 40087, 1, 40939, 5171, 13933, 5279, 42667, 449, 43543, 2749, 251, 1, 45319, 1907, 46219, 2917, 683, 1487, 1, 1, 2129, 1, 16633, 787, 50839, 1069, 51787, 1, 17581, 6653, 1249, 1129, 54679, 1, 18553, 7019, 1531, 2381, 977, 1, 1, 3697, 59659, 1, 60679, 7649, 307, 3889, 62743, 659, 281, 8039, 21613, 8171, 65899, 1, 1, 4219, 613, 8573, 69127, 2903, 1, 4423, 23773, 1123, 1, 3041, 293, 1, 24889, 2351, 331, 1, 1637, 9689, 26029, 9833, 727, 1663, 80407, 2531, 1, 10271, 82759, 1, 661, 1321, 1, 1, 1093, 3623, 87559, 1, 1, 1, 90007, 1, 91243, 11483, 30829, 1, 93739, 983, 94999, 1, 32089, 12113, 4241, 4091, 2671, 6217, 1451, 1, 101419, 4253, 2389, 12923, 1, 1, 105367, 1, 106699, 13421, 36013, 1, 109387, 2293, 1877, 1741, 37369, 1, 3067, 1, 114859, 1, 38749, 7309, 117643, 4931, 119047, 14969, 40153, 7573, 397, 1277, 123307, 419, 967, 15683, 1, 1, 1171, 1, 1871, 16229, 2213, 5471, 5741, 1, 947, 1049, 135019, 5657, 136519, 17159, 1, 4337, 2969, 1, 141067, 17729, 47533, 17921, 144139, 3019, 145687, 1, 49081, 18503, 2221, 1, 150379, 1181, 1, 9547, 1, 1, 929, 1, 52249, 1, 158359, 829, 159979, 1, 1, 1, 163243, 1709, 1, 10357, 1291, 20921, 1, 7043, 4591, 1, 1, 5387, 1, 7253, 1, 1, 1, 1, 178327, 3733, 180043, 22613, 60589, 1, 183499, 1, 2609, 1, 1, 1021, 1091, 7901, 190507, 5981, 1, 12073, 4513, 8123, 1297, 24593, 65881, 12409, 199447, 2087, 1, 1, 487, 1, 1, 1, 2617, 12979, 3023, 26189, 1657, 8807, 4517, 13327, 71389, 3361, 5839, 9041, 1, 1, 1559, 6899, 221719, 4639, 1979, 1, 75181, 1231, 227467, 4759, 229399, 1, 1307, 1, 2953, 1, 235243, 3691, 2137, 14887, 239179, 10007, 1, 30269, 81049, 15259, 5701, 1, 2447, 31019, 1, 31271, 251179, 1, 1, 15889, 85081, 1, 257287, 1, 259339, 16273, 87133, 1, 541, 1, 265543, 1, 89209, 2099, 269719, 5641, 4057, 1483, 91309, 1, 276043, 1, 6469, 4363, 1, 35171, 282439, 11813, 284587, 1, 95581, 17989, 288907, 1, 7867, 1, 1459, 18397, 1, 3089, 12941, 1, 1, 37619, 571, 1579, 304279, 19087, 1439, 38453, 6569, 12911, 310987, 19507, 1, 1, 8527, 1, 6761, 39863, 106681, 10037, 322327, 1, 324619, 1, 1847, 1783, 4637, 6883, 331543, 1, 1, 41879, 336199, 14057, 7873, 1327, 1, 21379, 343243, 1, 345607, 647, 601, 1, 15233, 1831, 2039, 1, 5147, 44543, 2141, 1, 6101, 22573, 607, 1, 1, 1, 367243, 23029, 123229, 1, 3613, 1, 1, 1, 1, 1, 1, 7933, 3709, 2083, 128173, 48221, 3617, 8089, 389527, 1, 130681, 1, 1723, 16493, 397099, 1, 133213, 25057, 1, 16811, 10939, 50753, 5903, 25537, 409879, 4283, 1, 51719, 138349, 1, 417643, 1, 3719, 26347, 2999, 1433, 3061, 17783, 4001, 26839, 2143, 1, 9221, 1, 1609, 2377, 1, 13751, 1, 1, 444043, 55673, 3463, 56009, 449419, 9391, 1, 1, 151609, 1, 6829, 19121, 460267, 7213, 154333, 29023, 465739, 19463, 20369, 58733, 157081, 1, 1, 1, 8081, 59771, 159853, 761, 482347, 5039, 6833, 1, 162649, 61169, 1, 20507, 493579, 30937, 165469, 1, 4943, 1, 502087, 62939, 4549, 1, 2237, 1, 4021, 64013, 2411, 64373, 8753, 10789, 2069, 1, 1597, 1, 525127, 1, 528043, 16547, 176989, 1, 1, 22307, 536839, 67289, 7823, 33829, 1, 5669, 2383, 68399, 182893, 68771, 551659, 1, 11801, 34759, 1, 1889, 3571, 1, 1, 35323, 4019, 1, 13249, 23801, 3793, 3121, 191929, 18041, 578839, 1, 15727, 1, 194989, 1, 588043, 1, 591127, 18521, 198073, 74471, 7561, 24953, 1, 4703, 8747, 1, 2417, 25343, 26513, 76421, 1, 1, 16651, 3217, 14401, 1, 207469, 77999, 9337, 1, 628759, 39397, 1, 79193, 5827, 1, 638347, 1, 2707, 1, 5077, 1, 648007, 81203, 217081, 1, 11093, 13669, 9817, 82421, 220333, 1, 4231, 13873, 6481, 1, 1, 84059, 2399, 1, 29453, 21221, 2203, 42649, 29741, 28571, 1, 2003, 230233, 1, 694039, 7247, 697387, 87383, 1, 87803, 1, 1, 16453, 1, 1, 89069, 714247, 1, 15269, 44959, 240349, 1, 724459, 1, 727879, 91199, 243769, 22907, 734743, 1, 1, 92489, 1, 92921, 2543, 15559, 748567, 23447, 250681, 1, 1, 1, 759019, 11887, 1, 1, 6779, 31991, 769543, 1, 257689, 1, 776599, 2027, 1, 2273, 261229, 2089, 787243, 8219, 2699, 49537, 264793, 4327, 1951, 1, 1913, 1, 1, 1, 808747, 33773, 812359, 101771, 2693, 6389, 819607, 1, 823243, 1, 2729, 1, 22447, 17341, 834199, 13063, 12143, 2441, 1, 35141, 36749, 1, 282973, 53173, 1, 35603, 12781, 107273, 1, 1, 863767, 1, 1, 1, 1, 109139, 874987, 4567, 878743, 2393, 294169, 1, 1, 1, 2689, 1, 4447, 3499, 4651, 1013, 901447, 112919, 301753, 28349, 1, 18979, 15473, 1, 305581, 114833, 1, 19219, 40193, 28949, 309433, 116279, 40529, 38921, 3049, 1, 2467, 1, 943819, 1, 947719, 118709, 317209, 1, 4951, 4987, 959467, 2557, 4523, 120671, 4861, 1, 26251, 1031, 325081, 1, 979207, 40883, 983179, 1, 329053, 30911, 991147, 41381, 4271, 124643, 333049, 15643, 1, 1, 1007179, 1, 337069, 126653, 1, 21193, 7333, 1, 14831, 1, 23893, 1, 9463, 32297, 345181, 64849, 1, 43403, 1, 3533, 3391, 65617, 1051927, 10979, 22469, 132263, 353389, 1, 10333, 2777, 1068439, 2909, 7607, 134333, 3253, 1, 1080907, 1, 1, 16987, 1089259, 1, 1, 136943, 1, 34367, 7297, 1, 10337, 1, 16091, 1, 1114507, 1, 1, 35027, 1, 140639, 1127239, 47057, 7207, 17713,

6. Sequence of the polynom (only primes)

233, 3, 43, 23, 109, 59, 619, 919, 67, 409, 173, 1543, 71, 1867, 127, 733, 37, 2539, 113, 2887, 383, 47, 107, 3607, 79, 521, 1453, 569, 101, 103, 139, 167, 719, 257, 6379, 2269, 439, 7243, 311, 7687, 2713, 523, 8599, 9067, 1163, 3181, 1223, 271, 157, 673, 3673, 1409, 11527, 491, 12043, 769, 401, 13099, 557, 593, 4729, 641, 313, 15307, 1949, 151, 349, 17047, 5881, 2243, 773, 18859, 599, 1237, 20107, 20743, 2633, 7129, 22039, 22699, 2879, 7789, 2963, 24043, 1567, 229, 3221, 26119, 1103, 193, 1699, 9181, 1229, 1193, 617, 3671, 431, 941, 709, 643, 199, 227, 4049, 32779, 691, 907, 1061, 1481, 12253, 37579, 1583, 211, 40087, 40939, 5171, 13933, 5279, 42667, 449, 43543, 2749, 251, 45319, 1907, 46219, 2917, 683, 1487, 2129, 16633, 787, 50839, 1069, 51787, 17581, 6653, 1249, 1129, 54679, 18553, 7019, 1531, 2381, 977, 3697, 59659, 60679, 7649, 307, 3889, 62743, 659, 281, 8039, 21613, 8171, 65899, 4219, 613, 8573, 69127, 2903, 4423, 23773, 1123, 3041, 293, 24889, 2351, 331, 1637, 9689, 26029, 9833, 727, 1663, 80407, 2531, 10271, 82759, 661, 1321, 1093, 3623, 87559, 90007, 91243, 11483, 30829, 93739, 983, 94999, 32089, 12113, 4241, 4091, 2671, 6217, 1451, 101419, 4253, 2389, 12923, 105367, 106699, 13421, 36013, 109387, 2293, 1877, 1741, 37369, 3067, 114859, 38749, 7309, 117643, 4931, 119047, 14969, 40153, 7573, 397, 1277, 123307, 419, 967, 15683, 1171, 1871, 16229, 2213, 5471, 5741, 947, 1049, 135019, 5657, 136519, 17159, 4337, 2969, 141067, 17729, 47533, 17921, 144139, 3019, 145687, 49081, 18503, 2221, 150379, 1181, 9547, 929, 52249, 158359, 829, 159979, 163243, 1709, 10357, 1291, 20921, 7043, 4591, 5387, 7253, 178327, 3733, 180043, 22613, 60589, 183499, 2609, 1021, 1091, 7901, 190507, 5981, 12073, 4513, 8123, 1297, 24593, 65881, 12409, 199447, 2087, 487, 2617, 12979, 3023, 26189, 1657, 8807, 4517, 13327, 71389, 3361, 5839, 9041, 1559, 6899, 221719, 4639, 1979, 75181, 1231, 227467, 4759, 229399, 1307, 2953, 235243, 3691, 2137, 14887, 239179, 10007, 30269, 81049, 15259, 5701, 2447, 31019, 31271, 251179, 15889, 85081, 257287, 259339, 16273, 87133, 541, 265543, 89209, 2099, 269719, 5641, 4057, 1483, 91309, 276043, 6469, 4363, 35171, 282439, 11813, 284587, 95581, 17989, 288907, 7867, 1459, 18397, 3089, 12941, 37619, 571, 1579, 304279, 19087, 1439, 38453, 6569, 12911, 310987, 19507, 8527, 6761, 39863, 106681, 10037, 322327, 324619, 1847, 1783, 4637, 6883, 331543, 41879, 336199, 14057, 7873, 1327, 21379, 343243, 345607, 647, 601, 15233, 1831, 2039, 5147, 44543, 2141, 6101, 22573, 607, 367243, 23029, 123229, 3613, 7933, 3709, 2083, 128173, 48221, 3617, 8089, 389527, 130681, 1723, 16493, 397099, 133213, 25057, 16811, 10939, 50753, 5903, 25537, 409879, 4283, 51719, 138349, 417643, 3719, 26347, 2999, 1433, 3061, 17783, 4001, 26839, 2143, 9221, 1609, 2377, 13751, 444043, 55673, 3463, 56009, 449419, 9391, 151609, 6829, 19121, 460267, 7213, 154333, 29023, 465739, 19463, 20369, 58733, 157081, 8081, 59771, 159853, 761, 482347, 5039, 6833, 162649, 61169, 20507, 493579, 30937, 165469, 4943, 502087, 62939, 4549, 2237, 4021, 64013, 2411, 64373, 8753, 10789, 2069, 1597, 525127, 528043, 16547, 176989, 22307, 536839, 67289, 7823, 33829, 5669, 2383, 68399, 182893, 68771, 551659, 11801, 34759, 1889, 3571, 35323, 4019, 13249, 23801, 3793, 3121, 191929, 18041, 578839, 15727, 194989, 588043, 591127, 18521, 198073, 74471, 7561, 24953, 4703, 8747, 2417, 25343, 26513, 76421, 16651, 3217, 14401, 207469, 77999, 9337, 628759, 39397, 79193, 5827, 638347, 2707, 5077, 648007, 81203, 217081, 11093, 13669, 9817, 82421, 220333, 4231, 13873, 6481, 84059, 2399, 29453, 21221, 2203, 42649, 29741, 28571, 2003, 230233, 694039, 7247, 697387, 87383, 87803, 16453, 89069, 714247, 15269, 44959, 240349, 724459, 727879, 91199, 243769, 22907, 734743, 92489, 92921, 2543, 15559, 748567, 23447, 250681, 759019, 11887, 6779, 31991, 769543, 257689, 776599, 2027, 2273, 261229, 2089, 787243, 8219, 2699, 49537, 264793, 4327, 1951, 1913, 808747, 33773, 812359, 101771, 2693, 6389, 819607, 823243, 2729, 22447, 17341, 834199, 13063, 12143, 2441, 35141, 36749, 282973, 53173, 35603, 12781, 107273, 863767, 109139, 874987, 4567, 878743, 2393, 294169, 2689, 4447, 3499, 4651, 1013, 901447, 112919, 301753, 28349, 18979, 15473, 305581, 114833, 19219, 40193, 28949, 309433, 116279, 40529, 38921, 3049, 2467, 943819, 947719, 118709, 317209, 4951, 4987, 959467, 2557, 4523, 120671, 4861, 26251, 1031, 325081, 979207, 40883, 983179, 329053, 30911, 991147, 41381, 4271, 124643, 333049, 15643, 1007179, 337069, 126653, 21193, 7333, 14831, 23893, 9463, 32297, 345181, 64849, 43403, 3533, 3391, 65617, 1051927, 10979, 22469, 132263, 353389, 10333, 2777, 1068439, 2909, 7607, 134333, 3253, 1080907, 16987, 1089259, 136943, 34367, 7297, 10337, 16091, 1114507, 35027, 140639, 1127239, 47057, 7207, 17713,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+136x-233 and
the reducible primes which appear as divisor for the first time
p | x^2+136x-233 and p < x^2+136x-233

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 : 233, 3, 43, 23, 109, 59, 619, 1, 919, 67, 409, 173, 1543, 71, 1867, 127, 733, 37, 2539, 113,
Found in Database : 233, 3, 43, 23, 109, 59, 619, 919, 67, 409, 173, 1543, 71, 1867, 127, 733, 37, 2539, 113, 2887, 383, 47, 107, 3607, 79, 521, 1453, 569, 101, 103, 139, 167, 719, 257, 6379,
Found in Database : 3, 23, 37, 43, 47, 59, 67, 71, 79, 101, 103, 107, 109, 113, 127, 139,