Inhaltsverzeichnis

Development of
Algorithmic Constructions

18:39:36
Deutsch
28.Mar 2024

Polynom = x^2+292x-677

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) = 677 = 677
f(1) = 3 = 3
f(2) = 89 = 89
f(3) = 13 = 13
f(4) = 507 = 3*13*13
f(5) = 101 = 101
f(6) = 1111 = 11*101
f(7) = 177 = 3*59
f(8) = 1723 = 1723
f(9) = 127 = 127
f(10) = 2343 = 3*11*71
f(11) = 83 = 83
f(12) = 2971 = 2971
f(13) = 411 = 3*137
f(14) = 3607 = 3607
f(15) = 491 = 491
f(16) = 4251 = 3*13*109
f(17) = 143 = 11*13
f(18) = 4903 = 4903
f(19) = 327 = 3*109
f(20) = 5563 = 5563
f(21) = 737 = 11*67
f(22) = 6231 = 3*31*67
f(23) = 821 = 821
f(24) = 6907 = 6907
f(25) = 453 = 3*151
f(26) = 7591 = 7591
f(27) = 31 = 31
f(28) = 8283 = 3*11*251
f(29) = 1079 = 13*83
f(30) = 8983 = 13*691
f(31) = 1167 = 3*389
f(32) = 9691 = 11*881
f(33) = 157 = 157
f(34) = 10407 = 3*3469
f(35) = 673 = 673
f(36) = 11131 = 11131
f(37) = 1437 = 3*479
f(38) = 11863 = 11863
f(39) = 1529 = 11*139
f(40) = 12603 = 3*4201
f(41) = 811 = 811
f(42) = 13351 = 13*13*79
f(43) = 429 = 3*11*13
f(44) = 14107 = 14107
f(45) = 1811 = 1811
f(46) = 14871 = 3*4957
f(47) = 1907 = 1907
f(48) = 15643 = 15643
f(49) = 501 = 3*167
f(50) = 16423 = 11*1493
f(51) = 1051 = 1051
f(52) = 17211 = 3*5737
f(53) = 2201 = 31*71
f(54) = 18007 = 11*1637
f(55) = 2301 = 3*13*59
f(56) = 18811 = 13*1447
f(57) = 1201 = 1201
f(58) = 19623 = 3*31*211
f(59) = 313 = 313
f(60) = 20443 = 20443
f(61) = 2607 = 3*11*79
f(62) = 21271 = 89*239
f(63) = 2711 = 2711
f(64) = 22107 = 3*7369
f(65) = 11 = 11
f(66) = 22951 = 59*389
f(67) = 1461 = 3*487
f(68) = 23803 = 13*1831
f(69) = 3029 = 13*233
f(70) = 24663 = 3*8221
f(71) = 3137 = 3137
f(72) = 25531 = 11*11*211
f(73) = 1623 = 3*541
f(74) = 26407 = 26407
f(75) = 839 = 839
f(76) = 27291 = 3*11*827
f(77) = 3467 = 3467
f(78) = 28183 = 28183
f(79) = 3579 = 3*1193
f(80) = 29083 = 127*229
f(81) = 923 = 13*71
f(82) = 29991 = 3*13*769
f(83) = 1903 = 11*173
f(84) = 30907 = 31*997
f(85) = 3921 = 3*1307
f(86) = 31831 = 139*229
f(87) = 4037 = 11*367
f(88) = 32763 = 3*67*163
f(89) = 2077 = 31*67
f(90) = 33703 = 33703
f(91) = 267 = 3*89
f(92) = 34651 = 34651
f(93) = 4391 = 4391
f(94) = 35607 = 3*11*13*83
f(95) = 4511 = 13*347
f(96) = 36571 = 36571
f(97) = 579 = 3*193
f(98) = 37543 = 11*3413
f(99) = 2377 = 2377
f(100) = 38523 = 3*12841

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+292x-677

f(0)=677
f(1)=3
f(2)=89
f(3)=13
f(4)=1
f(5)=101
f(6)=11
f(7)=59
f(8)=1723
f(9)=127
f(10)=71
f(11)=83
f(12)=2971
f(13)=137
f(14)=3607
f(15)=491
f(16)=109
f(17)=1
f(18)=4903
f(19)=1
f(20)=5563
f(21)=67
f(22)=31
f(23)=821
f(24)=6907
f(25)=151
f(26)=7591
f(27)=1
f(28)=251
f(29)=1
f(30)=691
f(31)=389
f(32)=881
f(33)=157
f(34)=3469
f(35)=673
f(36)=11131
f(37)=479
f(38)=11863
f(39)=139
f(40)=4201
f(41)=811
f(42)=79
f(43)=1
f(44)=14107
f(45)=1811
f(46)=4957
f(47)=1907
f(48)=15643
f(49)=167
f(50)=1493
f(51)=1051
f(52)=5737
f(53)=1
f(54)=1637
f(55)=1
f(56)=1447
f(57)=1201
f(58)=211
f(59)=313
f(60)=20443
f(61)=1
f(62)=239
f(63)=2711
f(64)=7369
f(65)=1
f(66)=1
f(67)=487
f(68)=1831
f(69)=233
f(70)=8221
f(71)=3137
f(72)=1
f(73)=541
f(74)=26407
f(75)=839
f(76)=827
f(77)=3467
f(78)=28183
f(79)=1193
f(80)=229
f(81)=1
f(82)=769
f(83)=173
f(84)=997
f(85)=1307
f(86)=1
f(87)=367
f(88)=163
f(89)=1
f(90)=33703
f(91)=1
f(92)=34651
f(93)=4391
f(94)=1
f(95)=347
f(96)=36571
f(97)=193
f(98)=3413
f(99)=2377

b) Substitution of the polynom
The polynom f(x)=x^2+292x-677 could be written as f(y)= y^2-21993 with x=y-146

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+146
f'(x)>2x+291

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

677, 3, 89, 13, 1, 101, 11, 59, 1723, 127, 71, 83, 2971, 137, 3607, 491, 109, 1, 4903, 1, 5563, 67, 31, 821, 6907, 151, 7591, 1, 251, 1, 691, 389, 881, 157, 3469, 673, 11131, 479, 11863, 139, 4201, 811, 79, 1, 14107, 1811, 4957, 1907, 15643, 167, 1493, 1051, 5737, 1, 1637, 1, 1447, 1201, 211, 313, 20443, 1, 239, 2711, 7369, 1, 1, 487, 1831, 233, 8221, 3137, 1, 541, 26407, 839, 827, 3467, 28183, 1193, 229, 1, 769, 173, 997, 1307, 1, 367, 163, 1, 33703, 1, 34651, 4391, 1, 347, 36571, 193, 3413, 2377, 12841, 4877, 39511, 1667, 40507, 1, 1, 1, 3271, 1, 43543, 5507, 179, 1409, 773, 1, 4241, 5897, 15901, 6029, 1, 1, 49831, 787, 1, 1, 51991, 199, 1, 419, 18061, 311, 55291, 1, 4339, 7121, 1, 3631, 5333, 617, 757, 7547, 1847, 7691, 359, 653, 1, 307, 21481, 739, 1, 1, 941, 383, 22669, 1, 1033, 2909, 70423, 683, 1, 1129, 72871, 1531, 6737, 9341, 25117, 9497, 76603, 1609, 77863, 223, 2029, 1, 80407, 1, 1, 1, 1, 5227, 947, 3539, 1, 10781, 1, 421, 1, 463, 89563, 11279, 1, 11447, 92251, 1, 93607, 1, 31657, 1087, 7411, 1, 1237, 6151, 33037, 3119, 9137, 4217, 457, 1, 1, 3251, 104743, 1, 8167, 431, 35869, 1231, 1, 2287, 110503, 1, 37321, 14087, 1693, 1, 8839, 1, 3527, 7321, 117883, 4943, 10853, 15017, 1, 7603, 1123, 1283, 123931, 1, 3217, 509, 1427, 1, 128551, 1, 1399, 16361, 131671, 5519, 12113, 8377, 3457, 1, 12401, 5717, 138007, 17351, 269, 1097, 141223, 1, 1721, 17957, 48157, 1, 11239, 3061, 2081, 4643, 49801, 1, 443, 6329, 937, 4799, 4679, 1, 12007, 503, 1901, 1, 53161, 911, 449, 1, 162907, 1861, 54877, 1, 166363, 1, 1, 1, 5147, 21341, 2417, 7187, 15761, 10891, 1, 5501, 176923, 1, 1, 1, 60169, 5669, 5881, 1, 1823, 1, 61981, 23357, 1049, 3931, 1567, 1, 4909, 1, 17573, 8093, 195163, 1, 65677, 12373, 1, 1, 1009, 1, 5197, 1, 204583, 2141, 6661, 25931, 547, 26171, 19121, 1, 1, 13327, 6491, 2069, 1279, 1, 218107, 13693, 1, 1, 222043, 9293, 3797, 2557, 75337, 3547, 17539, 1, 3433, 1, 1, 29129, 234043, 1, 1951, 1, 571, 1, 240151, 1, 601, 1, 81421, 15331, 1, 1, 248407, 31181, 83497, 1, 252583, 1321, 1, 2459, 85597, 1, 23537, 1, 261031, 16381, 87721, 33029, 2089, 1, 267451, 1291, 1, 1, 271771, 11369, 273943, 1109, 92041, 8663, 25301, 5821, 280507, 35201, 659, 2729, 4013, 1, 2843, 563, 96457, 3301, 1789, 12197, 643, 1, 1, 1429, 1, 12479, 300631, 1, 1, 613, 3677, 3191, 27953, 1, 3331, 38867, 24007, 1, 314407, 1, 1487, 1, 2113, 1213, 321403, 20161, 1, 5077, 3929, 1, 2297, 1, 110281, 1297, 30293, 6967, 3079, 42101, 1, 1, 1, 647, 26371, 1, 733, 1, 347671, 14537, 350107, 10979, 117517, 22111, 1, 14843, 1997, 3449, 1, 22573, 362407, 1, 364891, 45767, 1, 1, 1753, 1933, 1, 1, 9613, 3617, 661, 1, 379963, 23827, 1, 1, 12421, 16097, 1, 48611, 130057, 1, 30211, 8209, 395323, 4507, 132637, 1, 2551, 761, 1, 1, 135241, 1, 1, 1, 37361, 3221, 137869, 25933, 1, 1, 13513, 52529, 140521, 26431, 424231, 1, 32839, 1, 143197, 1, 1607, 4517, 1733, 27271, 145897, 54881, 40037, 1, 1, 2137, 1, 1747, 1877, 18749, 751, 56591, 1, 1, 456871, 9547, 1, 1, 1, 57977, 465211, 9721, 1, 14669, 1, 1, 7069, 19793, 43313, 1, 12289, 2311, 4423, 20147, 484951, 5527, 162601, 30577, 490663, 1, 493531, 61871, 165469, 4787, 1, 1, 1, 31477, 5431, 63317, 46181, 21227, 510907, 1, 2903, 16103, 1, 1, 519703, 65147, 2099, 1489, 525607, 1, 5939, 1, 1, 66629, 48593, 859, 1, 4211, 1, 67751, 1, 22709, 546523, 8563, 2207, 1, 5471, 23087, 1, 1, 6007, 1, 561703, 5867, 857, 1, 17207, 71171, 8521, 1, 52181, 2767, 14797, 1, 580183, 24239, 583291, 3323, 195469, 9187, 589531, 2239, 592663, 74279, 15277, 1, 1, 12511, 1, 1, 201757, 2447, 55313, 1, 1, 19163, 919, 5927, 1, 2347, 10529, 19463, 208141, 3557, 627643, 1, 630871, 1, 211369, 39733, 4457, 1, 853, 80279, 1, 80687, 647131, 1, 8233, 1, 1, 1, 21193, 2111, 3907, 3761, 221197, 20789, 2657, 1, 670231, 83987, 20411, 21101, 676903, 1, 1, 1, 227869, 85661, 686971, 14347, 5039, 983, 231241, 86927, 697111, 2647, 1, 1, 18049, 44101, 1, 953, 64613, 89057, 2357, 1, 1, 1, 721051, 1, 1, 6983, 10253, 1, 1993, 45823, 1, 1, 738391, 30839, 741883, 46477, 248461, 1, 5237, 31277, 5413, 3041, 22907, 11839, 759463, 15859, 1, 95597, 255517, 8731, 59239, 1, 907, 2203, 1, 97379, 780823, 32609, 784411, 1, 23879, 49363, 1, 2543, 1, 1, 266281, 1, 802471, 1, 806107, 9181, 8707, 1, 3491, 1, 62851, 1, 3463, 1, 1, 1, 1, 51871, 277261, 1, 1, 1, 7699, 8087, 21613, 26399, 2011, 1, 1973, 1, 284701, 1, 857851, 17911, 3203, 3373, 22189, 1, 1, 36293, 872923, 13669, 1, 1, 3779, 36767, 884311, 110777, 296041, 1, 68611, 9311, 895771, 1, 2969, 112691, 903451, 9431, 5779, 56827, 27611, 1, 1, 2939, 83537, 1, 4591, 14449, 1, 38693, 930583, 10597, 3943, 1, 13217, 1, 1, 118037, 315421, 118529, 4261, 1, 1, 29879, 1, 120011, 7951, 40169, 74311, 1, 323341, 1, 1, 3697, 978007, 122501, 327337, 5591, 986023, 1, 990043, 9539, 1, 1, 2927, 5209, 1002151, 62761, 30491, 1, 1010263, 42179, 1014331, 1, 26113, 1, 1022491, 42689, 1026583, 11689, 343561, 1, 1034791, 21601, 6221, 130121, 31607, 1, 1, 21859, 3083, 1, 1, 1, 1153, 1, 1063771, 1, 2803, 66877, 82471, 1, 5101, 1, 360169, 1, 1084711, 11321, 98993, 136379, 1, 1, 99761, 1, 1, 1, 1, 138497, 1110103, 1, 8017, 2251, 372877, 1, 1, 46877, 2797, 10859, 2753, 17713, 1, 1, 1140091, 1, 34679, 1, 1148731, 23977, 1153063, 2777, 1, 13177, 13997, 48497, 1166107, 3319, 390157, 73291, 6791, 49043, 1179223, 2503, 1, 5701, 1188007, 6199, 108401, 4817, 398941, 2237, 17929, 1, 6247, 6863, 403369, 11657, 93427, 1, 1219003, 1, 407821, 38303, 8963, 51257, 1, 1, 5807, 38723, 8681, 1, 1245883, 1, 416797, 5051, 1254907, 2381, 1, 9857, 13591, 14389, 16057, 4073, 97927, 19927, 425869, 1, 1, 1, 4111, 2731,

6. Sequence of the polynom (only primes)

677, 3, 89, 13, 101, 11, 59, 1723, 127, 71, 83, 2971, 137, 3607, 491, 109, 4903, 5563, 67, 31, 821, 6907, 151, 7591, 251, 691, 389, 881, 157, 3469, 673, 11131, 479, 11863, 139, 4201, 811, 79, 14107, 1811, 4957, 1907, 15643, 167, 1493, 1051, 5737, 1637, 1447, 1201, 211, 313, 20443, 239, 2711, 7369, 487, 1831, 233, 8221, 3137, 541, 26407, 839, 827, 3467, 28183, 1193, 229, 769, 173, 997, 1307, 367, 163, 33703, 34651, 4391, 347, 36571, 193, 3413, 2377, 12841, 4877, 39511, 1667, 40507, 3271, 43543, 5507, 179, 1409, 773, 4241, 5897, 15901, 6029, 49831, 787, 51991, 199, 419, 18061, 311, 55291, 4339, 7121, 3631, 5333, 617, 757, 7547, 1847, 7691, 359, 653, 307, 21481, 739, 941, 383, 22669, 1033, 2909, 70423, 683, 1129, 72871, 1531, 6737, 9341, 25117, 9497, 76603, 1609, 77863, 223, 2029, 80407, 5227, 947, 3539, 10781, 421, 463, 89563, 11279, 11447, 92251, 93607, 31657, 1087, 7411, 1237, 6151, 33037, 3119, 9137, 4217, 457, 3251, 104743, 8167, 431, 35869, 1231, 2287, 110503, 37321, 14087, 1693, 8839, 3527, 7321, 117883, 4943, 10853, 15017, 7603, 1123, 1283, 123931, 3217, 509, 1427, 128551, 1399, 16361, 131671, 5519, 12113, 8377, 3457, 12401, 5717, 138007, 17351, 269, 1097, 141223, 1721, 17957, 48157, 11239, 3061, 2081, 4643, 49801, 443, 6329, 937, 4799, 4679, 12007, 503, 1901, 53161, 911, 449, 162907, 1861, 54877, 166363, 5147, 21341, 2417, 7187, 15761, 10891, 5501, 176923, 60169, 5669, 5881, 1823, 61981, 23357, 1049, 3931, 1567, 4909, 17573, 8093, 195163, 65677, 12373, 1009, 5197, 204583, 2141, 6661, 25931, 547, 26171, 19121, 13327, 6491, 2069, 1279, 218107, 13693, 222043, 9293, 3797, 2557, 75337, 3547, 17539, 3433, 29129, 234043, 1951, 571, 240151, 601, 81421, 15331, 248407, 31181, 83497, 252583, 1321, 2459, 85597, 23537, 261031, 16381, 87721, 33029, 2089, 267451, 1291, 271771, 11369, 273943, 1109, 92041, 8663, 25301, 5821, 280507, 35201, 659, 2729, 4013, 2843, 563, 96457, 3301, 1789, 12197, 643, 1429, 12479, 300631, 613, 3677, 3191, 27953, 3331, 38867, 24007, 314407, 1487, 2113, 1213, 321403, 20161, 5077, 3929, 2297, 110281, 1297, 30293, 6967, 3079, 42101, 647, 26371, 733, 347671, 14537, 350107, 10979, 117517, 22111, 14843, 1997, 3449, 22573, 362407, 364891, 45767, 1753, 1933, 9613, 3617, 661, 379963, 23827, 12421, 16097, 48611, 130057, 30211, 8209, 395323, 4507, 132637, 2551, 761, 135241, 37361, 3221, 137869, 25933, 13513, 52529, 140521, 26431, 424231, 32839, 143197, 1607, 4517, 1733, 27271, 145897, 54881, 40037, 2137, 1747, 1877, 18749, 751, 56591, 456871, 9547, 57977, 465211, 9721, 14669, 7069, 19793, 43313, 12289, 2311, 4423, 20147, 484951, 5527, 162601, 30577, 490663, 493531, 61871, 165469, 4787, 31477, 5431, 63317, 46181, 21227, 510907, 2903, 16103, 519703, 65147, 2099, 1489, 525607, 5939, 66629, 48593, 859, 4211, 67751, 22709, 546523, 8563, 2207, 5471, 23087, 6007, 561703, 5867, 857, 17207, 71171, 8521, 52181, 2767, 14797, 580183, 24239, 583291, 3323, 195469, 9187, 589531, 2239, 592663, 74279, 15277, 12511, 201757, 2447, 55313, 19163, 919, 5927, 2347, 10529, 19463, 208141, 3557, 627643, 630871, 211369, 39733, 4457, 853, 80279, 80687, 647131, 8233, 21193, 2111, 3907, 3761, 221197, 20789, 2657, 670231, 83987, 20411, 21101, 676903, 227869, 85661, 686971, 14347, 5039, 983, 231241, 86927, 697111, 2647, 18049, 44101, 953, 64613, 89057, 2357, 721051, 6983, 10253, 1993, 45823, 738391, 30839, 741883, 46477, 248461, 5237, 31277, 5413, 3041, 22907, 11839, 759463, 15859, 95597, 255517, 8731, 59239, 907, 2203, 97379, 780823, 32609, 784411, 23879, 49363, 2543, 266281, 802471, 806107, 9181, 8707, 3491, 62851, 3463, 51871, 277261, 7699, 8087, 21613, 26399, 2011, 1973, 284701, 857851, 17911, 3203, 3373, 22189, 36293, 872923, 13669, 3779, 36767, 884311, 110777, 296041, 68611, 9311, 895771, 2969, 112691, 903451, 9431, 5779, 56827, 27611, 2939, 83537, 4591, 14449, 38693, 930583, 10597, 3943, 13217, 118037, 315421, 118529, 4261, 29879, 120011, 7951, 40169, 74311, 323341, 3697, 978007, 122501, 327337, 5591, 986023, 990043, 9539, 2927, 5209, 1002151, 62761, 30491, 1010263, 42179, 1014331, 26113, 1022491, 42689, 1026583, 11689, 343561, 1034791, 21601, 6221, 130121, 31607, 21859, 3083, 1153, 1063771, 2803, 66877, 82471, 5101, 360169, 1084711, 11321, 98993, 136379, 99761, 138497, 1110103, 8017, 2251, 372877, 46877, 2797, 10859, 2753, 17713, 1140091, 34679, 1148731, 23977, 1153063, 2777, 13177, 13997, 48497, 1166107, 3319, 390157, 73291, 6791, 49043, 1179223, 2503, 5701, 1188007, 6199, 108401, 4817, 398941, 2237, 17929, 6247, 6863, 403369, 11657, 93427, 1219003, 407821, 38303, 8963, 51257, 5807, 38723, 8681, 1245883, 416797, 5051, 1254907, 2381, 9857, 13591, 14389, 16057, 4073, 97927, 19927, 425869, 4111, 2731,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+292x-677 and
the reducible primes which appear as divisor for the first time
p | x^2+292x-677 and p < x^2+292x-677

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 : 677, 3, 89, 13, 1, 101, 11, 59, 1723, 127, 71, 83, 2971, 137, 3607, 491, 109, 1, 4903, 1,
Found in Database : 677, 3, 89, 13, 101, 11, 59, 1723, 127, 71, 83, 2971, 137, 3607, 491, 109, 4903, 5563, 67, 31, 821, 6907, 151, 7591, 251, 691, 389, 881, 157, 3469, 673, 11131, 479, 11863, 139,
Found in Database : 3, 11, 13, 31, 59, 67, 71, 79, 83, 89, 101, 109, 127, 137, 139,