Inhaltsverzeichnis

Development of
Algorithmic Constructions

07:35:44
Deutsch
29.Mar 2024

Polynom = x^2+36x-7829

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) = 7829 = 7829
f(1) = 487 = 487
f(2) = 7753 = 7753
f(3) = 241 = 241
f(4) = 7669 = 7669
f(5) = 953 = 953
f(6) = 7577 = 7577
f(7) = 941 = 941
f(8) = 7477 = 7477
f(9) = 29 = 29
f(10) = 7369 = 7369
f(11) = 457 = 457
f(12) = 7253 = 7253
f(13) = 899 = 29*31
f(14) = 7129 = 7129
f(15) = 883 = 883
f(16) = 6997 = 6997
f(17) = 433 = 433
f(18) = 6857 = 6857
f(19) = 53 = 53
f(20) = 6709 = 6709
f(21) = 829 = 829
f(22) = 6553 = 6553
f(23) = 809 = 809
f(24) = 6389 = 6389
f(25) = 197 = 197
f(26) = 6217 = 6217
f(27) = 383 = 383
f(28) = 6037 = 6037
f(29) = 743 = 743
f(30) = 5849 = 5849
f(31) = 719 = 719
f(32) = 5653 = 5653
f(33) = 347 = 347
f(34) = 5449 = 5449
f(35) = 167 = 167
f(36) = 5237 = 5237
f(37) = 641 = 641
f(38) = 5017 = 29*173
f(39) = 613 = 613
f(40) = 4789 = 4789
f(41) = 73 = 73
f(42) = 4553 = 29*157
f(43) = 277 = 277
f(44) = 4309 = 31*139
f(45) = 523 = 523
f(46) = 4057 = 4057
f(47) = 491 = 491
f(48) = 3797 = 3797
f(49) = 229 = 229
f(50) = 3529 = 3529
f(51) = 53 = 53
f(52) = 3253 = 3253
f(53) = 389 = 389
f(54) = 2969 = 2969
f(55) = 353 = 353
f(56) = 2677 = 2677
f(57) = 79 = 79
f(58) = 2377 = 2377
f(59) = 139 = 139
f(60) = 2069 = 2069
f(61) = 239 = 239
f(62) = 1753 = 1753
f(63) = 199 = 199
f(64) = 1429 = 1429
f(65) = 79 = 79
f(66) = 1097 = 1097
f(67) = 29 = 29
f(68) = 757 = 757
f(69) = 73 = 73
f(70) = 409 = 409
f(71) = 29 = 29
f(72) = 53 = 53
f(73) = 1 = 1
f(74) = 311 = 311
f(75) = 31 = 31
f(76) = 683 = 683
f(77) = 109 = 109
f(78) = 1063 = 1063
f(79) = 157 = 157
f(80) = 1451 = 1451
f(81) = 103 = 103
f(82) = 1847 = 1847
f(83) = 1 = 1
f(84) = 2251 = 2251
f(85) = 307 = 307
f(86) = 2663 = 2663
f(87) = 359 = 359
f(88) = 3083 = 3083
f(89) = 103 = 103
f(90) = 3511 = 3511
f(91) = 233 = 233
f(92) = 3947 = 3947
f(93) = 521 = 521
f(94) = 4391 = 4391
f(95) = 577 = 577
f(96) = 4843 = 29*167
f(97) = 317 = 317
f(98) = 5303 = 5303
f(99) = 173 = 173
f(100) = 5771 = 29*199

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+36x-7829

f(0)=7829
f(1)=487
f(2)=7753
f(3)=241
f(4)=7669
f(5)=953
f(6)=7577
f(7)=941
f(8)=7477
f(9)=29
f(10)=7369
f(11)=457
f(12)=7253
f(13)=31
f(14)=7129
f(15)=883
f(16)=6997
f(17)=433
f(18)=6857
f(19)=53
f(20)=6709
f(21)=829
f(22)=6553
f(23)=809
f(24)=6389
f(25)=197
f(26)=6217
f(27)=383
f(28)=6037
f(29)=743
f(30)=5849
f(31)=719
f(32)=5653
f(33)=347
f(34)=5449
f(35)=167
f(36)=5237
f(37)=641
f(38)=173
f(39)=613
f(40)=4789
f(41)=73
f(42)=157
f(43)=277
f(44)=139
f(45)=523
f(46)=4057
f(47)=491
f(48)=3797
f(49)=229
f(50)=3529
f(51)=1
f(52)=3253
f(53)=389
f(54)=2969
f(55)=353
f(56)=2677
f(57)=79
f(58)=2377
f(59)=1
f(60)=2069
f(61)=239
f(62)=1753
f(63)=199
f(64)=1429
f(65)=1
f(66)=1097
f(67)=1
f(68)=757
f(69)=1
f(70)=409
f(71)=1
f(72)=1
f(73)=1
f(74)=311
f(75)=1
f(76)=683
f(77)=109
f(78)=1063
f(79)=1
f(80)=1451
f(81)=103
f(82)=1847
f(83)=1
f(84)=2251
f(85)=307
f(86)=2663
f(87)=359
f(88)=3083
f(89)=1
f(90)=3511
f(91)=233
f(92)=3947
f(93)=521
f(94)=4391
f(95)=577
f(96)=1
f(97)=317
f(98)=5303
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+36x-7829 could be written as f(y)= y^2-8153 with x=y-18

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+18
f'(x)>2x+35

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

7829, 487, 7753, 241, 7669, 953, 7577, 941, 7477, 29, 7369, 457, 7253, 31, 7129, 883, 6997, 433, 6857, 53, 6709, 829, 6553, 809, 6389, 197, 6217, 383, 6037, 743, 5849, 719, 5653, 347, 5449, 167, 5237, 641, 173, 613, 4789, 73, 157, 277, 139, 523, 4057, 491, 3797, 229, 3529, 1, 3253, 389, 2969, 353, 2677, 79, 2377, 1, 2069, 239, 1753, 199, 1429, 1, 1097, 1, 757, 1, 409, 1, 1, 1, 311, 1, 683, 109, 1063, 1, 1451, 103, 1847, 1, 2251, 307, 2663, 359, 3083, 1, 3511, 233, 3947, 521, 4391, 577, 1, 317, 5303, 1, 1, 751, 6247, 811, 127, 1, 1, 467, 7723, 997, 8231, 1061, 8747, 563, 1, 149, 9803, 1259, 10343, 1327, 10891, 349, 11447, 733, 12011, 1, 12583, 1609, 13163, 1, 13751, 439, 14347, 1831, 14951, 1907, 1, 1, 16183, 1031, 16811, 2141, 1, 2221, 1, 1151, 18743, 1, 19403, 2467, 20071, 2551, 20747, 659, 739, 1361, 22123, 1, 787, 2897, 23531, 1493, 24247, 769, 24971, 3167, 25703, 3259, 853, 419, 27191, 1723, 27947, 3541, 28711, 3637, 29483, 1867, 571, 479, 31051, 3931, 31847, 1, 1, 1033, 1, 1, 34283, 4337, 35111, 4441, 1, 2273, 36791, 1163, 37643, 4759, 1, 1, 39371, 1, 1, 2543, 41131, 5197, 42023, 5309, 42923, 2711, 827, 1, 1543, 5651, 1, 1, 1607, 1471, 47543, 3001, 48491, 6121, 251, 1, 50411, 3181, 51383, 1621, 52363, 6607, 1721, 1, 54347, 857, 55351, 3491, 1, 7109, 57383, 7237, 58411, 1, 59447, 937, 1, 263, 61543, 7759, 62603, 1973, 63671, 4013, 64747, 8161, 65831, 8297, 66923, 4217, 68023, 2143, 947, 281, 1, 1, 1, 1, 72503, 4567, 73643, 9277, 2579, 9421, 1, 4783, 2659, 607, 78283, 9859, 1, 10007, 80651, 2539, 81847, 5153, 1567, 10457, 84263, 1, 1171, 5381, 86711, 2729, 2837, 11071, 1129, 1, 1, 1423, 91703, 1, 1, 11701, 1193, 1, 95531, 6011, 96823, 1523, 98123, 12347, 99431, 12511, 100747, 3169, 102071, 6421, 1951, 13009, 104743, 13177, 1, 6673, 773, 1, 1, 13687, 110183, 13859, 3847, 877, 112951, 7103, 3943, 1, 115751, 14557, 117163, 1, 118583, 1, 120011, 15091, 121447, 15271, 122891, 3863, 124343, 7817, 125803, 15817, 127271, 16001, 128747, 8093, 4201, 4093, 839, 1, 1049, 16747, 134731, 1, 136247, 8563, 137771, 17317, 139303, 17509, 1109, 1, 142391, 2237, 143947, 1, 145511, 18287, 147083, 4621, 148663, 9341, 1, 1, 151847, 19081, 887, 1, 5347, 4871, 156683, 19687, 1, 19891, 1, 1, 161591, 10151, 163243, 20509, 1601, 20717, 166571, 10463, 168247, 1321, 1559, 21347, 2351, 21559, 1, 5443, 1, 10993, 176747, 1, 1, 1, 5813, 11317, 181943, 1, 183691, 23071, 3499, 23291, 187211, 2939, 188983, 11867, 190763, 23957, 1153, 24181, 1783, 12203, 2687, 3079, 197963, 24859, 199783, 25087, 201611, 6329, 1, 1, 7079, 1, 1, 1, 7207, 13121, 1, 6619, 2693, 26711, 214631, 26947, 216523, 1699, 218423, 13711, 2789, 1, 222247, 27901, 224171, 14071, 226103, 1, 499, 28627, 1, 28871, 231947, 1, 233911, 1, 1697, 1021, 7673, 1, 239851, 15053, 241847, 7589, 243851, 1, 245863, 30859, 509, 3889, 249911, 15683, 251947, 1, 1, 1, 4831, 16067, 1297, 4049, 260171, 1, 9043, 32911, 264331, 8293, 9187, 1, 1, 1087, 270631, 33961, 1, 1, 274871, 8623, 277003, 34759, 279143, 35027, 281291, 1103, 283447, 17783, 285611, 35837, 287783, 36109, 5471, 18191, 1483, 1, 294347, 36931, 296551, 1283, 1789, 9371, 1, 1, 9781, 1, 1, 38321, 307691, 19301, 1, 9721, 312203, 1, 2003, 39451, 4339, 4967, 319031, 20011, 321323, 1, 323623, 40597, 11239, 20443, 2203, 5147, 11399, 41467, 332903, 41759, 1, 10513, 1, 1, 1459, 42641, 1, 42937, 344683, 21617, 347063, 10883, 4787, 1, 351847, 44131, 354251, 2777, 356663, 22367, 1, 1553, 361511, 45341, 363947, 1, 2459, 1, 3581, 1, 371303, 46567, 2689, 11719, 1, 23593, 3677, 47497, 381223, 47809, 383723, 24061, 4889, 12109, 3061, 48751, 1, 1, 1999, 6173, 1, 24851, 1669, 50021, 1, 50341, 404011, 1, 406583, 6373, 409163, 51307, 411751, 51631, 414347, 1, 7867, 26141, 419563, 52609, 422183, 52937, 424811, 26633, 427447, 13399, 430091, 53927, 432743, 1871, 435403, 1, 4019, 1, 1, 1, 443431, 1, 446123, 27967, 1, 3517, 2269, 56611, 454247, 56951, 14741, 14323, 459703, 28817, 462443, 57977, 2963, 58321, 1, 29333, 470711, 14753, 1, 59359, 1, 59707, 16519, 7507, 481847, 30203, 2011, 60757, 487463, 1, 490283, 1, 493111, 7727, 495947, 62171, 498791, 2017, 1613, 1, 6911, 1, 507371, 63601, 510247, 1, 513131, 1, 516023, 1, 9791, 2243, 521831, 1, 1, 4111, 527671, 33071, 530603, 66509, 533543, 66877, 536491, 33623, 539447, 2113, 2161, 67987, 1, 1, 548363, 17183, 3187, 1, 554347, 69481, 19219, 69857, 2447, 35117, 19427, 1, 10687, 70991, 569447, 1, 572491, 8969, 2293, 36067, 578603, 1, 1, 72901, 2111, 36643, 4229, 9209, 590923, 2389, 594023, 1, 597131, 1, 600247, 1, 3613, 75617, 606503, 2621, 7717, 38201, 612791, 1, 1, 77191, 619111, 77587, 7877, 2437, 2617, 39191, 628651, 78781, 631847, 79181, 635051, 39791, 638263, 4999, 20693, 80387, 2767, 1, 22343, 1, 651191, 40801, 22567, 82009, 657703, 1, 2153, 41413, 6449, 20809, 667531, 1, 670823, 84059, 674123, 10559, 6577, 42443, 1, 1, 12907, 85717, 687403, 43067, 2179, 1, 694091, 2999, 697447, 1, 700811, 1, 704183, 1, 707563, 88657, 710951, 1, 1, 44753, 717751, 22483, 721163, 90359, 724583, 90787, 6679, 5701, 731447, 45823, 10067, 92077, 13931, 1, 741803, 46471, 1, 1, 1, 93811, 25939, 1, 1, 23671, 759223, 1, 5119, 95561, 766247, 96001, 4903, 48221, 6089, 1, 7127, 97327, 4673, 97771, 10739, 12277, 2843, 49331, 6229, 99109, 5717, 3433, 798251, 1, 1, 1, 805451, 100907, 809063, 101359, 1987, 25453, 2339, 51133, 5503, 1, 823591, 103177, 4157, 51817, 1, 1, 834571, 104551, 838247, 1, 841931, 1, 845623, 52967, 29287, 106397, 853031, 106861, 1, 1, 1, 6737, 864203, 1, 867943, 108727, 16447, 27299, 875447, 54833, 4463, 1, 11177, 110609, 1, 55541, 890551, 1, 1, 112031, 11369, 112507, 901963, 1, 905783, 56731, 909611, 3929, 913447, 3691, 917291, 1, 921143, 14423, 983, 1, 928871, 116351, 17599, 29209, 2609, 58661, 940523, 117809, 991, 118297, 948331, 59393, 952247, 29819, 5527, 1, 33107, 1, 2731, 7547, 1, 60623, 2801, 2297, 31481, 1, 979883, 1, 983863, 3851, 987851, 123731, 1009, 124231, 6343, 31183, 999863, 62617, 1013, 125737, 13807, 126241, 1011947, 1, 1015991, 1, 1020043, 1, 1024103, 4423,

6. Sequence of the polynom (only primes)

7829, 487, 7753, 241, 7669, 953, 7577, 941, 7477, 29, 7369, 457, 7253, 31, 7129, 883, 6997, 433, 6857, 53, 6709, 829, 6553, 809, 6389, 197, 6217, 383, 6037, 743, 5849, 719, 5653, 347, 5449, 167, 5237, 641, 173, 613, 4789, 73, 157, 277, 139, 523, 4057, 491, 3797, 229, 3529, 3253, 389, 2969, 353, 2677, 79, 2377, 2069, 239, 1753, 199, 1429, 1097, 757, 409, 311, 683, 109, 1063, 1451, 103, 1847, 2251, 307, 2663, 359, 3083, 3511, 233, 3947, 521, 4391, 577, 317, 5303, 751, 6247, 811, 127, 467, 7723, 997, 8231, 1061, 8747, 563, 149, 9803, 1259, 10343, 1327, 10891, 349, 11447, 733, 12011, 12583, 1609, 13163, 13751, 439, 14347, 1831, 14951, 1907, 16183, 1031, 16811, 2141, 2221, 1151, 18743, 19403, 2467, 20071, 2551, 20747, 659, 739, 1361, 22123, 787, 2897, 23531, 1493, 24247, 769, 24971, 3167, 25703, 3259, 853, 419, 27191, 1723, 27947, 3541, 28711, 3637, 29483, 1867, 571, 479, 31051, 3931, 31847, 1033, 34283, 4337, 35111, 4441, 2273, 36791, 1163, 37643, 4759, 39371, 2543, 41131, 5197, 42023, 5309, 42923, 2711, 827, 1543, 5651, 1607, 1471, 47543, 3001, 48491, 6121, 251, 50411, 3181, 51383, 1621, 52363, 6607, 1721, 54347, 857, 55351, 3491, 7109, 57383, 7237, 58411, 59447, 937, 263, 61543, 7759, 62603, 1973, 63671, 4013, 64747, 8161, 65831, 8297, 66923, 4217, 68023, 2143, 947, 281, 72503, 4567, 73643, 9277, 2579, 9421, 4783, 2659, 607, 78283, 9859, 10007, 80651, 2539, 81847, 5153, 1567, 10457, 84263, 1171, 5381, 86711, 2729, 2837, 11071, 1129, 1423, 91703, 11701, 1193, 95531, 6011, 96823, 1523, 98123, 12347, 99431, 12511, 100747, 3169, 102071, 6421, 1951, 13009, 104743, 13177, 6673, 773, 13687, 110183, 13859, 3847, 877, 112951, 7103, 3943, 115751, 14557, 117163, 118583, 120011, 15091, 121447, 15271, 122891, 3863, 124343, 7817, 125803, 15817, 127271, 16001, 128747, 8093, 4201, 4093, 839, 1049, 16747, 134731, 136247, 8563, 137771, 17317, 139303, 17509, 1109, 142391, 2237, 143947, 145511, 18287, 147083, 4621, 148663, 9341, 151847, 19081, 887, 5347, 4871, 156683, 19687, 19891, 161591, 10151, 163243, 20509, 1601, 20717, 166571, 10463, 168247, 1321, 1559, 21347, 2351, 21559, 5443, 10993, 176747, 5813, 11317, 181943, 183691, 23071, 3499, 23291, 187211, 2939, 188983, 11867, 190763, 23957, 1153, 24181, 1783, 12203, 2687, 3079, 197963, 24859, 199783, 25087, 201611, 6329, 7079, 7207, 13121, 6619, 2693, 26711, 214631, 26947, 216523, 1699, 218423, 13711, 2789, 222247, 27901, 224171, 14071, 226103, 499, 28627, 28871, 231947, 233911, 1697, 1021, 7673, 239851, 15053, 241847, 7589, 243851, 245863, 30859, 509, 3889, 249911, 15683, 251947, 4831, 16067, 1297, 4049, 260171, 9043, 32911, 264331, 8293, 9187, 1087, 270631, 33961, 274871, 8623, 277003, 34759, 279143, 35027, 281291, 1103, 283447, 17783, 285611, 35837, 287783, 36109, 5471, 18191, 1483, 294347, 36931, 296551, 1283, 1789, 9371, 9781, 38321, 307691, 19301, 9721, 312203, 2003, 39451, 4339, 4967, 319031, 20011, 321323, 323623, 40597, 11239, 20443, 2203, 5147, 11399, 41467, 332903, 41759, 10513, 1459, 42641, 42937, 344683, 21617, 347063, 10883, 4787, 351847, 44131, 354251, 2777, 356663, 22367, 1553, 361511, 45341, 363947, 2459, 3581, 371303, 46567, 2689, 11719, 23593, 3677, 47497, 381223, 47809, 383723, 24061, 4889, 12109, 3061, 48751, 1999, 6173, 24851, 1669, 50021, 50341, 404011, 406583, 6373, 409163, 51307, 411751, 51631, 414347, 7867, 26141, 419563, 52609, 422183, 52937, 424811, 26633, 427447, 13399, 430091, 53927, 432743, 1871, 435403, 4019, 443431, 446123, 27967, 3517, 2269, 56611, 454247, 56951, 14741, 14323, 459703, 28817, 462443, 57977, 2963, 58321, 29333, 470711, 14753, 59359, 59707, 16519, 7507, 481847, 30203, 2011, 60757, 487463, 490283, 493111, 7727, 495947, 62171, 498791, 2017, 1613, 6911, 507371, 63601, 510247, 513131, 516023, 9791, 2243, 521831, 4111, 527671, 33071, 530603, 66509, 533543, 66877, 536491, 33623, 539447, 2113, 2161, 67987, 548363, 17183, 3187, 554347, 69481, 19219, 69857, 2447, 35117, 19427, 10687, 70991, 569447, 572491, 8969, 2293, 36067, 578603, 72901, 2111, 36643, 4229, 9209, 590923, 2389, 594023, 597131, 600247, 3613, 75617, 606503, 2621, 7717, 38201, 612791, 77191, 619111, 77587, 7877, 2437, 2617, 39191, 628651, 78781, 631847, 79181, 635051, 39791, 638263, 4999, 20693, 80387, 2767, 22343, 651191, 40801, 22567, 82009, 657703, 2153, 41413, 6449, 20809, 667531, 670823, 84059, 674123, 10559, 6577, 42443, 12907, 85717, 687403, 43067, 2179, 694091, 2999, 697447, 700811, 704183, 707563, 88657, 710951, 44753, 717751, 22483, 721163, 90359, 724583, 90787, 6679, 5701, 731447, 45823, 10067, 92077, 13931, 741803, 46471, 93811, 25939, 23671, 759223, 5119, 95561, 766247, 96001, 4903, 48221, 6089, 7127, 97327, 4673, 97771, 10739, 12277, 2843, 49331, 6229, 99109, 5717, 3433, 798251, 805451, 100907, 809063, 101359, 1987, 25453, 2339, 51133, 5503, 823591, 103177, 4157, 51817, 834571, 104551, 838247, 841931, 845623, 52967, 29287, 106397, 853031, 106861, 6737, 864203, 867943, 108727, 16447, 27299, 875447, 54833, 4463, 11177, 110609, 55541, 890551, 112031, 11369, 112507, 901963, 905783, 56731, 909611, 3929, 913447, 3691, 917291, 921143, 14423, 983, 928871, 116351, 17599, 29209, 2609, 58661, 940523, 117809, 991, 118297, 948331, 59393, 952247, 29819, 5527, 33107, 2731, 7547, 60623, 2801, 2297, 31481, 979883, 983863, 3851, 987851, 123731, 1009, 124231, 6343, 31183, 999863, 62617, 1013, 125737, 13807, 126241, 1011947, 1015991, 1020043, 1024103, 4423,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+36x-7829 and
the reducible primes which appear as divisor for the first time
p | x^2+36x-7829 and p < x^2+36x-7829

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 : 7829, 487, 7753, 241, 7669, 953, 7577, 941, 7477, 29, 7369, 457, 7253, 31, 7129, 883, 6997, 433, 6857, 53,
Found in Database : 7829, 487, 7753, 241, 7669, 953, 7577, 941, 7477, 29, 7369, 457, 7253, 31, 7129, 883, 6997, 433, 6857, 53, 6709, 829, 6553, 809, 6389, 197, 6217, 383, 6037, 743, 5849, 719, 5653, 347, 5449, 167, 5237, 641, 173, 613,
Found in Database : 29, 31, 53, 73, 79, 103, 109, 127, 139, 149,