Inhaltsverzeichnis

Development of
Algorithmic Constructions

06:32:45
Deutsch
29.Mar 2024

Polynom = x^2+60x-1277

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) = 1277 = 1277
f(1) = 19 = 19
f(2) = 1153 = 1153
f(3) = 17 = 17
f(4) = 1021 = 1021
f(5) = 119 = 7*17
f(6) = 881 = 881
f(7) = 101 = 101
f(8) = 733 = 733
f(9) = 41 = 41
f(10) = 577 = 577
f(11) = 31 = 31
f(12) = 413 = 7*59
f(13) = 41 = 41
f(14) = 241 = 241
f(15) = 19 = 19
f(16) = 61 = 61
f(17) = 1 = 1
f(18) = 127 = 127
f(19) = 7 = 7
f(20) = 323 = 17*19
f(21) = 53 = 53
f(22) = 527 = 17*31
f(23) = 79 = 79
f(24) = 739 = 739
f(25) = 53 = 53
f(26) = 959 = 7*137
f(27) = 67 = 67
f(28) = 1187 = 1187
f(29) = 163 = 163
f(30) = 1423 = 1423
f(31) = 193 = 193
f(32) = 1667 = 1667
f(33) = 7 = 7
f(34) = 1919 = 19*101
f(35) = 1 = 1
f(36) = 2179 = 2179
f(37) = 289 = 17*17
f(38) = 2447 = 2447
f(39) = 323 = 17*19
f(40) = 2723 = 7*389
f(41) = 179 = 179
f(42) = 3007 = 31*97
f(43) = 197 = 197
f(44) = 3299 = 3299
f(45) = 431 = 431
f(46) = 3599 = 59*61
f(47) = 469 = 7*67
f(48) = 3907 = 3907
f(49) = 127 = 127
f(50) = 4223 = 41*103
f(51) = 137 = 137
f(52) = 4547 = 4547
f(53) = 589 = 19*31
f(54) = 4879 = 7*17*41
f(55) = 631 = 631
f(56) = 5219 = 17*307
f(57) = 337 = 337
f(58) = 5567 = 19*293
f(59) = 359 = 359
f(60) = 5923 = 5923
f(61) = 763 = 7*109
f(62) = 6287 = 6287
f(63) = 809 = 809
f(64) = 6659 = 6659
f(65) = 107 = 107
f(66) = 7039 = 7039
f(67) = 113 = 113
f(68) = 7427 = 7*1061
f(69) = 953 = 953
f(70) = 7823 = 7823
f(71) = 1003 = 17*59
f(72) = 8227 = 19*433
f(73) = 527 = 17*31
f(74) = 8639 = 53*163
f(75) = 553 = 7*79
f(76) = 9059 = 9059
f(77) = 1159 = 19*61
f(78) = 9487 = 53*179
f(79) = 1213 = 1213
f(80) = 9923 = 9923
f(81) = 317 = 317
f(82) = 10367 = 7*1481
f(83) = 331 = 331
f(84) = 10819 = 31*349
f(85) = 1381 = 1381
f(86) = 11279 = 11279
f(87) = 1439 = 1439
f(88) = 11747 = 17*691
f(89) = 749 = 7*107
f(90) = 12223 = 17*719
f(91) = 779 = 19*41
f(92) = 12707 = 97*131
f(93) = 1619 = 1619
f(94) = 13199 = 67*197
f(95) = 1681 = 41*41
f(96) = 13699 = 7*19*103
f(97) = 109 = 109
f(98) = 14207 = 14207
f(99) = 113 = 113
f(100) = 14723 = 14723

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+60x-1277

f(0)=1277
f(1)=19
f(2)=1153
f(3)=17
f(4)=1021
f(5)=7
f(6)=881
f(7)=101
f(8)=733
f(9)=41
f(10)=577
f(11)=31
f(12)=59
f(13)=1
f(14)=241
f(15)=1
f(16)=61
f(17)=1
f(18)=127
f(19)=1
f(20)=1
f(21)=53
f(22)=1
f(23)=79
f(24)=739
f(25)=1
f(26)=137
f(27)=67
f(28)=1187
f(29)=163
f(30)=1423
f(31)=193
f(32)=1667
f(33)=1
f(34)=1
f(35)=1
f(36)=2179
f(37)=1
f(38)=2447
f(39)=1
f(40)=389
f(41)=179
f(42)=97
f(43)=197
f(44)=3299
f(45)=431
f(46)=1
f(47)=1
f(48)=3907
f(49)=1
f(50)=103
f(51)=1
f(52)=4547
f(53)=1
f(54)=1
f(55)=631
f(56)=307
f(57)=337
f(58)=293
f(59)=359
f(60)=5923
f(61)=109
f(62)=6287
f(63)=809
f(64)=6659
f(65)=107
f(66)=7039
f(67)=113
f(68)=1061
f(69)=953
f(70)=7823
f(71)=1
f(72)=433
f(73)=1
f(74)=1
f(75)=1
f(76)=9059
f(77)=1
f(78)=1
f(79)=1213
f(80)=9923
f(81)=317
f(82)=1481
f(83)=331
f(84)=349
f(85)=1381
f(86)=11279
f(87)=1439
f(88)=691
f(89)=1
f(90)=719
f(91)=1
f(92)=131
f(93)=1619
f(94)=1
f(95)=1
f(96)=1
f(97)=1
f(98)=14207
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+60x-1277 could be written as f(y)= y^2-2177 with x=y-30

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+30
f'(x)>2x+59

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

1277, 19, 1153, 17, 1021, 7, 881, 101, 733, 41, 577, 31, 59, 1, 241, 1, 61, 1, 127, 1, 1, 53, 1, 79, 739, 1, 137, 67, 1187, 163, 1423, 193, 1667, 1, 1, 1, 2179, 1, 2447, 1, 389, 179, 97, 197, 3299, 431, 1, 1, 3907, 1, 103, 1, 4547, 1, 1, 631, 307, 337, 293, 359, 5923, 109, 6287, 809, 6659, 107, 7039, 113, 1061, 953, 7823, 1, 433, 1, 1, 1, 9059, 1, 1, 1213, 9923, 317, 1481, 331, 349, 1381, 11279, 1439, 691, 1, 719, 1, 131, 1619, 1, 1, 1, 1, 14207, 1, 14723, 1873, 1, 277, 509, 1, 16319, 1, 167, 2143, 1, 2213, 17987, 571, 1, 1, 19139, 347, 19727, 2503, 20323, 1289, 1231, 1327, 181, 2731, 22159, 1, 22787, 1, 397, 1, 587, 3049, 1301, 1, 619, 1607, 1, 1, 26723, 199, 27407, 3469, 28099, 1, 929, 911, 1553, 3733, 30223, 3823, 4421, 1, 401, 2003, 1907, 4099, 1951, 599, 33923, 1, 34687, 1, 601, 4481, 1, 1, 607, 2339, 1, 2389, 1, 1, 39439, 1, 1, 1, 227, 1297, 1, 1, 42767, 5399, 823, 2753, 2341, 1, 1, 1, 2719, 1, 1, 743, 6857, 757, 457, 1, 49807, 1, 50723, 1, 51647, 3257, 52579, 1, 491, 1, 251, 1, 2917, 1747, 499, 7109, 1399, 1033, 257, 3677, 1447, 3739, 1, 7603, 8761, 1, 1, 1, 3727, 1, 1, 1, 65423, 8243, 66467, 1, 269, 4253, 1, 1, 271, 283, 1, 1, 71807, 1, 1, 9181, 73999, 9319, 1, 4729, 10889, 4799, 77347, 9739, 78479, 1, 773, 1, 4751, 1, 1, 10313, 4373, 10459, 12037, 5303, 85439, 1, 86627, 10903, 2833, 1579, 701, 2801, 90239, 1, 91459, 677, 13241, 1, 93923, 311, 1613, 5987, 96419, 1733, 1, 12289, 98947, 1, 1, 1, 853, 1, 6047, 1, 2539, 6547, 105407, 947, 1, 1, 1049, 1, 109379, 1, 15817, 1, 112067, 829, 1123, 839, 1453, 1031, 116159, 1, 117539, 14779, 118927, 787, 17189, 1, 121727, 1913, 6481, 1, 1, 2237, 7411, 7919, 127423, 8009, 4157, 1, 18617, 16381, 131779, 1, 7013, 1, 1, 1, 136207, 1, 137699, 1, 139199, 8747, 20101, 17683, 142223, 1, 4637, 1129, 1109, 1, 146819, 971, 148367, 1, 8819, 9419, 1, 1, 153059, 19231, 2621, 19429, 156227, 1, 157823, 4957, 1217, 20029, 161039, 20231, 1223, 1, 4007, 1, 1, 1097, 1, 1, 169219, 2657, 1597, 2683, 1583, 21673, 24889, 1, 3319, 11047, 1, 1, 1, 3217, 1, 1, 1, 5737, 184447, 5791, 26597, 1, 1663, 23599, 1, 11909, 1789, 1, 3167, 1427, 1, 24481, 196739, 1, 1, 1, 200323, 25153, 202127, 1, 1871, 1, 205759, 12917, 12211, 1, 1, 26293, 30181, 1, 213119, 6689, 1201, 1, 1, 3889, 218723, 13729, 1321, 1, 222499, 1, 32057, 1657, 226307, 1, 228223, 3581, 12113, 4127, 232079, 29131, 7549, 1, 235967, 1, 1, 29863, 1, 30109, 1, 7589, 243839, 1093, 1, 30853, 3137, 1637, 249827, 1, 35977, 15803, 1, 31859, 255887, 1889, 2659, 1, 259967, 2039, 4441, 1, 1459, 1069, 1, 16699, 1, 16829, 270307, 1, 272399, 1, 1, 1, 1, 8677, 1669, 1, 1, 1, 4639, 1, 285119, 1, 2789, 1, 289423, 36313, 2887, 1, 15461, 1, 1, 1, 1, 1, 300323, 18839, 1, 2711, 304739, 38231, 2417, 1, 16273, 9697, 2617, 9767, 18451, 1, 5179, 39631, 318179, 2851, 10337, 1, 1783, 40483, 7927, 1, 46757, 1283, 8039, 1, 331907, 1, 1181, 1, 17713, 21107, 1, 1, 341219, 1, 49081, 43093, 345923, 10847, 348287, 1, 20627, 1, 1, 44279, 355427, 1, 357823, 1, 51461, 45179, 2647, 45481, 4621, 1, 2053, 1, 11933, 2729, 372367, 1, 374819, 1237, 53897, 1, 1, 47623, 20117, 47933, 1, 1723, 387199, 1, 389699, 48869, 23071, 1, 1, 24749, 6733, 24907, 1, 50131, 3691, 7207, 7639, 1, 2111, 1, 409987, 1, 58937, 1, 3169, 1531, 417727, 26189, 420323, 7529, 422927, 2791, 1, 13339, 7019, 13421, 1, 54013, 6469, 1753, 1, 27337, 1, 3929, 7481, 55339, 444047, 55673, 4423, 7001, 1, 7043, 1543, 56681, 454799, 3001, 457507, 1, 6869, 1697, 462947, 1, 465679, 1, 1, 1, 1, 14767, 473923, 1, 15377, 8537, 4243, 1, 1493, 1, 1, 1483, 1, 61153, 490627, 1, 5087, 1933, 496259, 8887, 1, 62563, 26417, 1, 504767, 1861, 1, 1, 1, 1, 513347, 16087, 516223, 2311, 519107, 2099, 521999, 1, 524899, 1, 75401, 1741, 31219, 66523, 31391, 66889, 1, 1, 13159, 1, 542467, 67993, 1, 1, 78341, 34367, 1, 1, 1, 1, 29333, 1, 9497, 1, 563327, 1, 18269, 70981, 1, 71359, 572387, 35869, 2239, 1, 578467, 10357, 1, 2351, 1, 1, 2953, 4603, 84389, 1, 31253, 74419, 596899, 1, 599999, 1, 9887, 75583, 606223, 1, 2371, 1, 1, 1, 32401, 1, 1, 77543, 621923, 1, 625087, 1, 9377, 1, 20369, 1, 5333, 9941, 2207, 1, 2383, 80329, 644239, 1, 3877, 1, 10667, 1, 1, 81943, 1, 82349, 9857, 1, 1, 1, 16267, 11939, 670223, 83983, 16427, 42197, 1, 42403, 97157, 2749, 683407, 4507, 686723, 2689, 40591, 1, 40787, 1, 696719, 87299, 700067, 1, 1, 1, 706787, 1, 6637, 1, 13463, 1, 1, 1, 13591, 5309, 723727, 90679, 1, 45553, 730559, 45767, 23677, 2243, 737423, 1, 38993, 1, 4457, 11657, 43987, 4931, 1, 1, 754723, 47279, 7507, 47497, 4673, 13633, 765199, 95869, 1, 24077, 2333, 1, 110821, 5717, 1, 5743, 41201, 49037, 6959, 1, 7247, 98963, 793487, 99409, 2297, 1, 114377, 6269, 8291, 1901, 1, 101203, 1, 1, 1, 2687, 818659, 102559, 6277, 3323, 117989, 25867, 4211, 25981, 833219, 104381, 836879, 1, 840547, 1, 1, 52879, 10733, 106219, 1, 106681, 6529, 1, 12821, 13451, 14143, 15439, 4787, 1, 3011, 54503, 51407, 1, 6599, 109943, 28433, 2693, 885187, 1, 888959, 1, 8839, 111829, 13381, 112303, 8741, 1, 2437, 3331, 2287, 113731, 17203, 6011, 915587, 1, 919423, 1, 48593, 1, 3347, 116131, 1, 1, 1, 58549, 55219, 3793, 942607, 1, 11981, 1, 50021, 29761, 954307, 119533, 136889, 6317, 1, 60257, 1, 3559, 1, 1, 2713, 121993, 977923, 1, 981887, 15373, 140837, 1, 989839, 123979, 993827, 1, 52517, 1, 1, 1, 59167, 2377, 1009859, 1, 1, 1, 1017923, 127493, 1, 4129, 54001, 1, 1, 1, 1034147, 1, 17597, 7649, 2441, 1, 1046399, 8191, 5443, 3209, 1054607, 18869,

6. Sequence of the polynom (only primes)

1277, 19, 1153, 17, 1021, 7, 881, 101, 733, 41, 577, 31, 59, 241, 61, 127, 53, 79, 739, 137, 67, 1187, 163, 1423, 193, 1667, 2179, 2447, 389, 179, 97, 197, 3299, 431, 3907, 103, 4547, 631, 307, 337, 293, 359, 5923, 109, 6287, 809, 6659, 107, 7039, 113, 1061, 953, 7823, 433, 9059, 1213, 9923, 317, 1481, 331, 349, 1381, 11279, 1439, 691, 719, 131, 1619, 14207, 14723, 1873, 277, 509, 16319, 167, 2143, 2213, 17987, 571, 19139, 347, 19727, 2503, 20323, 1289, 1231, 1327, 181, 2731, 22159, 22787, 397, 587, 3049, 1301, 619, 1607, 26723, 199, 27407, 3469, 28099, 929, 911, 1553, 3733, 30223, 3823, 4421, 401, 2003, 1907, 4099, 1951, 599, 33923, 34687, 601, 4481, 607, 2339, 2389, 39439, 227, 1297, 42767, 5399, 823, 2753, 2341, 2719, 743, 6857, 757, 457, 49807, 50723, 51647, 3257, 52579, 491, 251, 2917, 1747, 499, 7109, 1399, 1033, 257, 3677, 1447, 3739, 7603, 8761, 3727, 65423, 8243, 66467, 269, 4253, 271, 283, 71807, 9181, 73999, 9319, 4729, 10889, 4799, 77347, 9739, 78479, 773, 4751, 10313, 4373, 10459, 12037, 5303, 85439, 86627, 10903, 2833, 1579, 701, 2801, 90239, 91459, 677, 13241, 93923, 311, 1613, 5987, 96419, 1733, 12289, 98947, 853, 6047, 2539, 6547, 105407, 947, 1049, 109379, 15817, 112067, 829, 1123, 839, 1453, 1031, 116159, 117539, 14779, 118927, 787, 17189, 121727, 1913, 6481, 2237, 7411, 7919, 127423, 8009, 4157, 18617, 16381, 131779, 7013, 136207, 137699, 139199, 8747, 20101, 17683, 142223, 4637, 1129, 1109, 146819, 971, 148367, 8819, 9419, 153059, 19231, 2621, 19429, 156227, 157823, 4957, 1217, 20029, 161039, 20231, 1223, 4007, 1097, 169219, 2657, 1597, 2683, 1583, 21673, 24889, 3319, 11047, 3217, 5737, 184447, 5791, 26597, 1663, 23599, 11909, 1789, 3167, 1427, 24481, 196739, 200323, 25153, 202127, 1871, 205759, 12917, 12211, 26293, 30181, 213119, 6689, 1201, 3889, 218723, 13729, 1321, 222499, 32057, 1657, 226307, 228223, 3581, 12113, 4127, 232079, 29131, 7549, 235967, 29863, 30109, 7589, 243839, 1093, 30853, 3137, 1637, 249827, 35977, 15803, 31859, 255887, 1889, 2659, 259967, 2039, 4441, 1459, 1069, 16699, 16829, 270307, 272399, 8677, 1669, 4639, 285119, 2789, 289423, 36313, 2887, 15461, 300323, 18839, 2711, 304739, 38231, 2417, 16273, 9697, 2617, 9767, 18451, 5179, 39631, 318179, 2851, 10337, 1783, 40483, 7927, 46757, 1283, 8039, 331907, 1181, 17713, 21107, 341219, 49081, 43093, 345923, 10847, 348287, 20627, 44279, 355427, 357823, 51461, 45179, 2647, 45481, 4621, 2053, 11933, 2729, 372367, 374819, 1237, 53897, 47623, 20117, 47933, 1723, 387199, 389699, 48869, 23071, 24749, 6733, 24907, 50131, 3691, 7207, 7639, 2111, 409987, 58937, 3169, 1531, 417727, 26189, 420323, 7529, 422927, 2791, 13339, 7019, 13421, 54013, 6469, 1753, 27337, 3929, 7481, 55339, 444047, 55673, 4423, 7001, 7043, 1543, 56681, 454799, 3001, 457507, 6869, 1697, 462947, 465679, 14767, 473923, 15377, 8537, 4243, 1493, 1483, 61153, 490627, 5087, 1933, 496259, 8887, 62563, 26417, 504767, 1861, 513347, 16087, 516223, 2311, 519107, 2099, 521999, 524899, 75401, 1741, 31219, 66523, 31391, 66889, 13159, 542467, 67993, 78341, 34367, 29333, 9497, 563327, 18269, 70981, 71359, 572387, 35869, 2239, 578467, 10357, 2351, 2953, 4603, 84389, 31253, 74419, 596899, 599999, 9887, 75583, 606223, 2371, 32401, 77543, 621923, 625087, 9377, 20369, 5333, 9941, 2207, 2383, 80329, 644239, 3877, 10667, 81943, 82349, 9857, 16267, 11939, 670223, 83983, 16427, 42197, 42403, 97157, 2749, 683407, 4507, 686723, 2689, 40591, 40787, 696719, 87299, 700067, 706787, 6637, 13463, 13591, 5309, 723727, 90679, 45553, 730559, 45767, 23677, 2243, 737423, 38993, 4457, 11657, 43987, 4931, 754723, 47279, 7507, 47497, 4673, 13633, 765199, 95869, 24077, 2333, 110821, 5717, 5743, 41201, 49037, 6959, 7247, 98963, 793487, 99409, 2297, 114377, 6269, 8291, 1901, 101203, 2687, 818659, 102559, 6277, 3323, 117989, 25867, 4211, 25981, 833219, 104381, 836879, 840547, 52879, 10733, 106219, 106681, 6529, 12821, 13451, 14143, 15439, 4787, 3011, 54503, 51407, 6599, 109943, 28433, 2693, 885187, 888959, 8839, 111829, 13381, 112303, 8741, 2437, 3331, 2287, 113731, 17203, 6011, 915587, 919423, 48593, 3347, 116131, 58549, 55219, 3793, 942607, 11981, 50021, 29761, 954307, 119533, 136889, 6317, 60257, 3559, 2713, 121993, 977923, 981887, 15373, 140837, 989839, 123979, 993827, 52517, 59167, 2377, 1009859, 1017923, 127493, 4129, 54001, 1034147, 17597, 7649, 2441, 1046399, 8191, 5443, 3209, 1054607, 18869,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+60x-1277 and
the reducible primes which appear as divisor for the first time
p | x^2+60x-1277 and p < x^2+60x-1277

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 : 1277, 19, 1153, 17, 1021, 7, 881, 101, 733, 41, 577, 31, 59, 1, 241, 1, 61, 1, 127, 1,
Found in Database : 1277, 19, 1153, 17, 1021, 7, 881, 101, 733, 41, 577, 31, 59, 241, 61, 127, 53, 79, 739, 137, 67, 1187, 163, 1423, 193, 1667, 2179, 2447,
Found in Database : 7, 17, 19, 31, 41, 53, 59, 61, 67, 79, 97, 101, 103, 107, 109, 113, 127, 131, 137,