Inhaltsverzeichnis

Development of
Algorithmic Constructions

06:50:07
Deutsch
29.Mar 2024

Polynom = x^2-212x+659

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) = 659 = 659
f(1) = 7 = 7
f(2) = 239 = 239
f(3) = 1 = 1
f(4) = 173 = 173
f(5) = 47 = 47
f(6) = 577 = 577
f(7) = 97 = 97
f(8) = 973 = 7*139
f(9) = 73 = 73
f(10) = 1361 = 1361
f(11) = 97 = 97
f(12) = 1741 = 1741
f(13) = 241 = 241
f(14) = 2113 = 2113
f(15) = 287 = 7*41
f(16) = 2477 = 2477
f(17) = 83 = 83
f(18) = 2833 = 2833
f(19) = 47 = 47
f(20) = 3181 = 3181
f(21) = 419 = 419
f(22) = 3521 = 7*503
f(23) = 461 = 461
f(24) = 3853 = 3853
f(25) = 251 = 251
f(26) = 4177 = 4177
f(27) = 271 = 271
f(28) = 4493 = 4493
f(29) = 581 = 7*83
f(30) = 4801 = 4801
f(31) = 619 = 619
f(32) = 5101 = 5101
f(33) = 41 = 41
f(34) = 5393 = 5393
f(35) = 173 = 173
f(36) = 5677 = 7*811
f(37) = 727 = 727
f(38) = 5953 = 5953
f(39) = 761 = 761
f(40) = 6221 = 6221
f(41) = 397 = 397
f(42) = 6481 = 6481
f(43) = 413 = 7*59
f(44) = 6733 = 6733
f(45) = 857 = 857
f(46) = 6977 = 6977
f(47) = 887 = 887
f(48) = 7213 = 7213
f(49) = 229 = 229
f(50) = 7441 = 7*1063
f(51) = 59 = 59
f(52) = 7661 = 47*163
f(53) = 971 = 971
f(54) = 7873 = 7873
f(55) = 997 = 997
f(56) = 8077 = 41*197
f(57) = 511 = 7*73
f(58) = 8273 = 8273
f(59) = 523 = 523
f(60) = 8461 = 8461
f(61) = 1069 = 1069
f(62) = 8641 = 8641
f(63) = 1091 = 1091
f(64) = 8813 = 7*1259
f(65) = 139 = 139
f(66) = 8977 = 47*191
f(67) = 283 = 283
f(68) = 9133 = 9133
f(69) = 1151 = 1151
f(70) = 9281 = 9281
f(71) = 1169 = 7*167
f(72) = 9421 = 9421
f(73) = 593 = 593
f(74) = 9553 = 41*233
f(75) = 601 = 601
f(76) = 9677 = 9677
f(77) = 1217 = 1217
f(78) = 9793 = 7*1399
f(79) = 1231 = 1231
f(80) = 9901 = 9901
f(81) = 311 = 311
f(82) = 10001 = 73*137
f(83) = 157 = 157
f(84) = 10093 = 10093
f(85) = 1267 = 7*181
f(86) = 10177 = 10177
f(87) = 1277 = 1277
f(88) = 10253 = 10253
f(89) = 643 = 643
f(90) = 10321 = 10321
f(91) = 647 = 647
f(92) = 10381 = 7*1483
f(93) = 1301 = 1301
f(94) = 10433 = 10433
f(95) = 1307 = 1307
f(96) = 10477 = 10477
f(97) = 41 = 41
f(98) = 10513 = 10513
f(99) = 329 = 7*47
f(100) = 10541 = 83*127

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^2-212x+659

f(0)=659
f(1)=7
f(2)=239
f(3)=1
f(4)=173
f(5)=47
f(6)=577
f(7)=97
f(8)=139
f(9)=73
f(10)=1361
f(11)=1
f(12)=1741
f(13)=241
f(14)=2113
f(15)=41
f(16)=2477
f(17)=83
f(18)=2833
f(19)=1
f(20)=3181
f(21)=419
f(22)=503
f(23)=461
f(24)=3853
f(25)=251
f(26)=4177
f(27)=271
f(28)=4493
f(29)=1
f(30)=4801
f(31)=619
f(32)=5101
f(33)=1
f(34)=5393
f(35)=1
f(36)=811
f(37)=727
f(38)=5953
f(39)=761
f(40)=6221
f(41)=397
f(42)=6481
f(43)=59
f(44)=6733
f(45)=857
f(46)=6977
f(47)=887
f(48)=7213
f(49)=229
f(50)=1063
f(51)=1
f(52)=163
f(53)=971
f(54)=7873
f(55)=997
f(56)=197
f(57)=1
f(58)=8273
f(59)=523
f(60)=8461
f(61)=1069
f(62)=8641
f(63)=1091
f(64)=1259
f(65)=1
f(66)=191
f(67)=283
f(68)=9133
f(69)=1151
f(70)=9281
f(71)=167
f(72)=9421
f(73)=593
f(74)=233
f(75)=601
f(76)=9677
f(77)=1217
f(78)=1399
f(79)=1231
f(80)=9901
f(81)=311
f(82)=137
f(83)=157
f(84)=10093
f(85)=181
f(86)=10177
f(87)=1277
f(88)=10253
f(89)=643
f(90)=10321
f(91)=647
f(92)=1483
f(93)=1301
f(94)=10433
f(95)=1307
f(96)=10477
f(97)=1
f(98)=10513
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-212x+659 could be written as f(y)= y^2-10577 with x=y+106

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-106
f'(x)>2x-213 with x > 103

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

659, 7, 239, 1, 173, 47, 577, 97, 139, 73, 1361, 1, 1741, 241, 2113, 41, 2477, 83, 2833, 1, 3181, 419, 503, 461, 3853, 251, 4177, 271, 4493, 1, 4801, 619, 5101, 1, 5393, 1, 811, 727, 5953, 761, 6221, 397, 6481, 59, 6733, 857, 6977, 887, 7213, 229, 1063, 1, 163, 971, 7873, 997, 197, 1, 8273, 523, 8461, 1069, 8641, 1091, 1259, 1, 191, 283, 9133, 1151, 9281, 167, 9421, 593, 233, 601, 9677, 1217, 1399, 1231, 9901, 311, 137, 157, 10093, 181, 10177, 1277, 10253, 643, 10321, 647, 1483, 1301, 10433, 1307, 10477, 1, 10513, 1, 127, 1319, 179, 1321, 109, 661, 1511, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1087, 1, 1523, 1, 281, 1, 1, 331, 2879, 389, 3347, 1, 3823, 1, 1, 569, 4799, 631, 757, 347, 5807, 379, 6323, 823, 1, 1, 1, 1, 7919, 1, 8467, 1093, 1289, 1163, 9587, 617, 10159, 653, 10739, 1, 1, 1453, 11923, 1, 12527, 401, 1877, 1, 13759, 1759, 14387, 919, 1, 1, 15667, 1999, 16319, 2081, 16979, 541, 2521, 1, 1, 2333, 1, 1, 19699, 1, 20399, 1297, 21107, 2683, 1, 1, 3221, 1, 23279, 739, 24019, 3049, 24767, 449, 25523, 1619, 1, 1667, 27059, 1, 1, 3529, 28627, 907, 29423, 1, 1, 547, 31039, 3931, 31859, 2017, 32687, 2069, 4789, 4243, 34367, 4349, 859, 557, 1, 1, 36947, 4673, 1, 4783, 38707, 2447, 5657, 2503, 40499, 5119, 881, 5233, 42323, 1, 733, 683, 44179, 5581, 45119, 1, 6581, 2909, 797, 2969, 1021, 1, 1, 883, 49939, 1, 1, 1607, 1, 6553, 7561, 6679, 53939, 1, 54959, 3467, 55987, 1009, 1, 7193, 58067, 1831, 59119, 1, 8597, 7589, 839, 7723, 62323, 3929, 1, 571, 64499, 1, 65599, 8269, 1627, 1051, 9689, 2137, 68947, 8689, 70079, 8831, 1, 641, 72367, 1, 73523, 1, 74687, 1, 10837, 2389, 1879, 1213, 1, 1, 79423, 1429, 80627, 5077, 81839, 5153, 83059, 10459, 12041, 10613, 85523, 673, 86767, 2731, 88019, 1583, 1223, 11239, 90547, 1, 91823, 5779, 1, 11719, 94399, 1, 1153, 3011, 97007, 1, 98323, 12373, 1, 12539, 1, 6353, 1, 1, 1249, 13043, 105023, 1, 1, 1, 1, 3389, 109139, 13729, 110527, 13903, 1, 7039, 113327, 7127, 1, 14431, 116159, 2087, 1993, 3697, 2903, 1871, 1, 15149, 17417, 15331, 123379, 7757, 124847, 1, 126323, 2269, 127807, 16069, 1, 1, 941, 4111, 1, 16633, 821, 16823, 135347, 1, 136879, 1229, 1427, 1, 139967, 1, 1459, 4447, 20441, 1, 144659, 18181, 146239, 18379, 147827, 1327, 149423, 1, 151027, 18979, 152639, 19181, 22037, 2423, 155887, 1, 157523, 19793, 159167, 2857, 2203, 10103, 3457, 1, 164147, 1, 23689, 1, 877, 5261, 169199, 2657, 170899, 3067, 172607, 21683, 3709, 10949, 176047, 11057, 1, 1, 179519, 22549, 1663, 1423, 183023, 1, 4507, 23209, 947, 23431, 1, 11827, 1, 11939, 1, 24103, 193727, 24329, 195539, 1, 197359, 1549, 1433, 25013, 4903, 25243, 1, 1, 204719, 12853, 1237, 25939, 3533, 3739, 210323, 3301, 212207, 6661, 1, 26881, 1, 1, 217907, 13679, 219823, 13799, 221747, 1, 223679, 28081, 225619, 1, 227567, 3571, 32789, 28813, 2789, 29059, 1487, 14653, 235439, 2111, 1019, 29803, 239423, 1, 1901, 1, 1, 7639, 1, 30809, 1, 31063, 5309, 2237, 251567, 15787, 1997, 1, 255679, 32089, 36821, 8087, 259823, 1, 1447, 32869, 1, 4733, 266099, 1, 268207, 16829, 270323, 33923, 38921, 1, 274579, 1, 1657, 8681, 278867, 4999, 281023, 1, 6907, 1, 1, 17903, 41077, 1, 289727, 36353, 291923, 9157, 294127, 1, 1, 37181, 298559, 1, 300787, 18869, 1, 19009, 305267, 38299, 1699, 1, 309779, 1, 312047, 9787, 1, 1, 2311, 39719, 45557, 1, 321199, 20147, 323507, 40583, 3359, 5839, 1373, 1, 3407, 2591, 5641, 1, 47881, 42043, 1951, 21169, 339887, 21317, 5801, 6133, 344639, 1, 2129, 5441, 349423, 10957, 50261, 44129, 7537, 1, 8699, 22367, 4919, 3217, 361523, 45343, 363967, 1, 366419, 11489, 52697, 5783, 7901, 46573, 373823, 1, 376307, 3371, 9239, 23753, 1, 47819, 383807, 1, 1, 1, 1, 1, 391379, 49081, 393919, 7057, 1, 24859, 399023, 1, 401587, 50359, 57737, 1, 406739, 1, 409327, 1, 411923, 1, 3803, 1, 3001, 26153, 3851, 26317, 1, 52963, 1, 1, 1, 6703, 430319, 1, 432979, 1, 435647, 54623, 5281, 27479, 1, 27647, 1579, 55631, 446399, 1, 449107, 2011, 451823, 1, 454547, 56989, 457279, 57331, 65717, 28837, 11287, 29009, 465523, 58363, 1, 8387, 1, 3691, 473839, 14851, 10141, 59753, 68489, 60103, 482227, 1, 8221, 30403, 1, 8737, 1, 1, 493523, 15467, 1, 3889, 71317, 62581, 502079, 62939, 504947, 31649, 2837, 4547, 510707, 64019, 2689, 64381, 516499, 8093, 74201, 1, 522323, 1, 525247, 65839, 1949, 4729, 3821, 33287, 2711, 66943, 537023, 1, 77141, 16921, 2371, 1, 545939, 1669, 548927, 9829, 1, 34589, 554927, 1, 7643, 69931, 1, 70309, 563987, 1, 1, 1, 3631, 1, 573119, 1, 576179, 36107, 4561, 36299, 2029, 1, 1, 73369, 6067, 18439, 591599, 1, 6131, 1, 597823, 74923, 600947, 37657, 86297, 37853, 607219, 76099, 14887, 76493, 613523, 1, 13121, 1, 3583, 77681, 1, 78079, 1, 39239, 3769, 39439, 632627, 79279, 3329, 11383, 13597, 20021, 4091, 10061, 2281, 1973, 92681, 81299, 652019, 40853, 2719, 41057, 658547, 11789, 661823, 1, 11273, 5209, 8053, 20939, 95957, 1, 1, 2063, 11497, 42499, 681647, 6101, 2729, 85831, 8293, 86249, 3511, 1, 99289, 5443, 698387, 87509, 1, 87931, 5147, 6311, 708527, 44389, 711923, 89203, 1, 1907, 102677, 11257, 722159, 22621, 725587, 1, 729023, 13049, 732467, 45887, 735919, 46103, 739379, 92639, 106121, 1, 1, 1, 749807, 11743, 6911, 1, 756799, 94819, 2297, 47629, 763823, 1, 109621, 1, 4259, 96581, 16477, 1, 2749, 1, 781523, 1, 2897, 2399,

6. Sequence of the polynom (only primes)

659, 7, 239, 173, 47, 577, 97, 139, 73, 1361, 1741, 241, 2113, 41, 2477, 83, 2833, 3181, 419, 503, 461, 3853, 251, 4177, 271, 4493, 4801, 619, 5101, 5393, 811, 727, 5953, 761, 6221, 397, 6481, 59, 6733, 857, 6977, 887, 7213, 229, 1063, 163, 971, 7873, 997, 197, 8273, 523, 8461, 1069, 8641, 1091, 1259, 191, 283, 9133, 1151, 9281, 167, 9421, 593, 233, 601, 9677, 1217, 1399, 1231, 9901, 311, 137, 157, 10093, 181, 10177, 1277, 10253, 643, 10321, 647, 1483, 1301, 10433, 1307, 10477, 10513, 127, 1319, 179, 1321, 109, 661, 1511, 1087, 1523, 281, 331, 2879, 389, 3347, 3823, 569, 4799, 631, 757, 347, 5807, 379, 6323, 823, 7919, 8467, 1093, 1289, 1163, 9587, 617, 10159, 653, 10739, 1453, 11923, 12527, 401, 1877, 13759, 1759, 14387, 919, 15667, 1999, 16319, 2081, 16979, 541, 2521, 2333, 19699, 20399, 1297, 21107, 2683, 3221, 23279, 739, 24019, 3049, 24767, 449, 25523, 1619, 1667, 27059, 3529, 28627, 907, 29423, 547, 31039, 3931, 31859, 2017, 32687, 2069, 4789, 4243, 34367, 4349, 859, 557, 36947, 4673, 4783, 38707, 2447, 5657, 2503, 40499, 5119, 881, 5233, 42323, 733, 683, 44179, 5581, 45119, 6581, 2909, 797, 2969, 1021, 883, 49939, 1607, 6553, 7561, 6679, 53939, 54959, 3467, 55987, 1009, 7193, 58067, 1831, 59119, 8597, 7589, 839, 7723, 62323, 3929, 571, 64499, 65599, 8269, 1627, 1051, 9689, 2137, 68947, 8689, 70079, 8831, 641, 72367, 73523, 74687, 10837, 2389, 1879, 1213, 79423, 1429, 80627, 5077, 81839, 5153, 83059, 10459, 12041, 10613, 85523, 673, 86767, 2731, 88019, 1583, 1223, 11239, 90547, 91823, 5779, 11719, 94399, 1153, 3011, 97007, 98323, 12373, 12539, 6353, 1249, 13043, 105023, 3389, 109139, 13729, 110527, 13903, 7039, 113327, 7127, 14431, 116159, 2087, 1993, 3697, 2903, 1871, 15149, 17417, 15331, 123379, 7757, 124847, 126323, 2269, 127807, 16069, 941, 4111, 16633, 821, 16823, 135347, 136879, 1229, 1427, 139967, 1459, 4447, 20441, 144659, 18181, 146239, 18379, 147827, 1327, 149423, 151027, 18979, 152639, 19181, 22037, 2423, 155887, 157523, 19793, 159167, 2857, 2203, 10103, 3457, 164147, 23689, 877, 5261, 169199, 2657, 170899, 3067, 172607, 21683, 3709, 10949, 176047, 11057, 179519, 22549, 1663, 1423, 183023, 4507, 23209, 947, 23431, 11827, 11939, 24103, 193727, 24329, 195539, 197359, 1549, 1433, 25013, 4903, 25243, 204719, 12853, 1237, 25939, 3533, 3739, 210323, 3301, 212207, 6661, 26881, 217907, 13679, 219823, 13799, 221747, 223679, 28081, 225619, 227567, 3571, 32789, 28813, 2789, 29059, 1487, 14653, 235439, 2111, 1019, 29803, 239423, 1901, 7639, 30809, 31063, 5309, 2237, 251567, 15787, 1997, 255679, 32089, 36821, 8087, 259823, 1447, 32869, 4733, 266099, 268207, 16829, 270323, 33923, 38921, 274579, 1657, 8681, 278867, 4999, 281023, 6907, 17903, 41077, 289727, 36353, 291923, 9157, 294127, 37181, 298559, 300787, 18869, 19009, 305267, 38299, 1699, 309779, 312047, 9787, 2311, 39719, 45557, 321199, 20147, 323507, 40583, 3359, 5839, 1373, 3407, 2591, 5641, 47881, 42043, 1951, 21169, 339887, 21317, 5801, 6133, 344639, 2129, 5441, 349423, 10957, 50261, 44129, 7537, 8699, 22367, 4919, 3217, 361523, 45343, 363967, 366419, 11489, 52697, 5783, 7901, 46573, 373823, 376307, 3371, 9239, 23753, 47819, 383807, 391379, 49081, 393919, 7057, 24859, 399023, 401587, 50359, 57737, 406739, 409327, 411923, 3803, 3001, 26153, 3851, 26317, 52963, 6703, 430319, 432979, 435647, 54623, 5281, 27479, 27647, 1579, 55631, 446399, 449107, 2011, 451823, 454547, 56989, 457279, 57331, 65717, 28837, 11287, 29009, 465523, 58363, 8387, 3691, 473839, 14851, 10141, 59753, 68489, 60103, 482227, 8221, 30403, 8737, 493523, 15467, 3889, 71317, 62581, 502079, 62939, 504947, 31649, 2837, 4547, 510707, 64019, 2689, 64381, 516499, 8093, 74201, 522323, 525247, 65839, 1949, 4729, 3821, 33287, 2711, 66943, 537023, 77141, 16921, 2371, 545939, 1669, 548927, 9829, 34589, 554927, 7643, 69931, 70309, 563987, 3631, 573119, 576179, 36107, 4561, 36299, 2029, 73369, 6067, 18439, 591599, 6131, 597823, 74923, 600947, 37657, 86297, 37853, 607219, 76099, 14887, 76493, 613523, 13121, 3583, 77681, 78079, 39239, 3769, 39439, 632627, 79279, 3329, 11383, 13597, 20021, 4091, 10061, 2281, 1973, 92681, 81299, 652019, 40853, 2719, 41057, 658547, 11789, 661823, 11273, 5209, 8053, 20939, 95957, 2063, 11497, 42499, 681647, 6101, 2729, 85831, 8293, 86249, 3511, 99289, 5443, 698387, 87509, 87931, 5147, 6311, 708527, 44389, 711923, 89203, 1907, 102677, 11257, 722159, 22621, 725587, 729023, 13049, 732467, 45887, 735919, 46103, 739379, 92639, 106121, 749807, 11743, 6911, 756799, 94819, 2297, 47629, 763823, 109621, 4259, 96581, 16477, 2749, 781523, 2897, 2399,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-212x+659 and
the reducible primes which appear as divisor for the first time
p | x^2-212x+659 and p < x^2-212x+659

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 : 659, 7, 239, 1, 173, 47, 577, 97, 139, 73, 1361, 1, 1741, 241, 2113, 41, 2477, 83, 2833, 1,
Found in Database : 659, 7, 239, 173, 47, 577, 97, 139, 73, 1361, 1741, 241, 2113, 41, 2477, 83, 2833, 3181, 419, 503, 461, 3853, 251, 4177, 271, 4493, 4801, 619, 5101, 5393, 811, 727, 5953, 761,
Found in Database : 7, 41, 47, 59, 73, 83, 97, 109, 127, 137, 139,