Inhaltsverzeichnis

Development of
Algorithmic Constructions

13:04:34
Deutsch
29.Mar 2024

Polynom = x^2-44x+1451

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) = 1451 = 1451
f(1) = 11 = 11
f(2) = 1367 = 1367
f(3) = 83 = 83
f(4) = 1291 = 1291
f(5) = 157 = 157
f(6) = 1223 = 1223
f(7) = 149 = 149
f(8) = 1163 = 1163
f(9) = 71 = 71
f(10) = 1111 = 11*101
f(11) = 17 = 17
f(12) = 1067 = 11*97
f(13) = 131 = 131
f(14) = 1031 = 1031
f(15) = 127 = 127
f(16) = 1003 = 17*59
f(17) = 31 = 31
f(18) = 983 = 983
f(19) = 61 = 61
f(20) = 971 = 971
f(21) = 121 = 11*11
f(22) = 967 = 967
f(23) = 121 = 11*11
f(24) = 971 = 971
f(25) = 61 = 61
f(26) = 983 = 983
f(27) = 31 = 31
f(28) = 1003 = 17*59
f(29) = 127 = 127
f(30) = 1031 = 1031
f(31) = 131 = 131
f(32) = 1067 = 11*97
f(33) = 17 = 17
f(34) = 1111 = 11*101
f(35) = 71 = 71
f(36) = 1163 = 1163
f(37) = 149 = 149
f(38) = 1223 = 1223
f(39) = 157 = 157
f(40) = 1291 = 1291
f(41) = 83 = 83
f(42) = 1367 = 1367
f(43) = 11 = 11
f(44) = 1451 = 1451
f(45) = 187 = 11*17
f(46) = 1543 = 1543
f(47) = 199 = 199
f(48) = 1643 = 31*53
f(49) = 53 = 53
f(50) = 1751 = 17*103
f(51) = 113 = 113
f(52) = 1867 = 1867
f(53) = 241 = 241
f(54) = 1991 = 11*181
f(55) = 257 = 257
f(56) = 2123 = 11*193
f(57) = 137 = 137
f(58) = 2263 = 31*73
f(59) = 73 = 73
f(60) = 2411 = 2411
f(61) = 311 = 311
f(62) = 2567 = 17*151
f(63) = 331 = 331
f(64) = 2731 = 2731
f(65) = 11 = 11
f(66) = 2903 = 2903
f(67) = 187 = 11*17
f(68) = 3083 = 3083
f(69) = 397 = 397
f(70) = 3271 = 3271
f(71) = 421 = 421
f(72) = 3467 = 3467
f(73) = 223 = 223
f(74) = 3671 = 3671
f(75) = 59 = 59
f(76) = 3883 = 11*353
f(77) = 499 = 499
f(78) = 4103 = 11*373
f(79) = 527 = 17*31
f(80) = 4331 = 61*71
f(81) = 139 = 139
f(82) = 4567 = 4567
f(83) = 293 = 293
f(84) = 4811 = 17*283
f(85) = 617 = 617
f(86) = 5063 = 61*83
f(87) = 649 = 11*59
f(88) = 5323 = 5323
f(89) = 341 = 11*31
f(90) = 5591 = 5591
f(91) = 179 = 179
f(92) = 5867 = 5867
f(93) = 751 = 751
f(94) = 6151 = 6151
f(95) = 787 = 787
f(96) = 6443 = 17*379
f(97) = 103 = 103
f(98) = 6743 = 11*613
f(99) = 431 = 431
f(100) = 7051 = 11*641

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^2-44x+1451

f(0)=1451
f(1)=11
f(2)=1367
f(3)=83
f(4)=1291
f(5)=157
f(6)=1223
f(7)=149
f(8)=1163
f(9)=71
f(10)=101
f(11)=17
f(12)=97
f(13)=131
f(14)=1031
f(15)=127
f(16)=59
f(17)=31
f(18)=983
f(19)=61
f(20)=971
f(21)=1
f(22)=967
f(23)=1
f(24)=1
f(25)=1
f(26)=1
f(27)=1
f(28)=1
f(29)=1
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=1
f(43)=1
f(44)=1
f(45)=1
f(46)=1543
f(47)=199
f(48)=53
f(49)=1
f(50)=103
f(51)=113
f(52)=1867
f(53)=241
f(54)=181
f(55)=257
f(56)=193
f(57)=137
f(58)=73
f(59)=1
f(60)=2411
f(61)=311
f(62)=151
f(63)=331
f(64)=2731
f(65)=1
f(66)=2903
f(67)=1
f(68)=3083
f(69)=397
f(70)=3271
f(71)=421
f(72)=3467
f(73)=223
f(74)=3671
f(75)=1
f(76)=353
f(77)=499
f(78)=373
f(79)=1
f(80)=1
f(81)=139
f(82)=4567
f(83)=293
f(84)=283
f(85)=617
f(86)=1
f(87)=1
f(88)=5323
f(89)=1
f(90)=5591
f(91)=179
f(92)=5867
f(93)=751
f(94)=6151
f(95)=787
f(96)=379
f(97)=1
f(98)=613
f(99)=431

b) Substitution of the polynom
The polynom f(x)=x^2-44x+1451 could be written as f(y)= y^2+967 with x=y+22

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-22
f'(x)>2x-45 with x > 31

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

1451, 11, 1367, 83, 1291, 157, 1223, 149, 1163, 71, 101, 17, 97, 131, 1031, 127, 59, 31, 983, 61, 971, 1, 967, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1543, 199, 53, 1, 103, 113, 1867, 241, 181, 257, 193, 137, 73, 1, 2411, 311, 151, 331, 2731, 1, 2903, 1, 3083, 397, 3271, 421, 3467, 223, 3671, 1, 353, 499, 373, 1, 1, 139, 4567, 293, 283, 617, 1, 1, 5323, 1, 5591, 179, 5867, 751, 6151, 787, 379, 1, 613, 431, 641, 1, 1, 941, 7691, 491, 1, 1, 8363, 1, 281, 1, 9067, 1, 9431, 601, 9803, 1249, 599, 1297, 1, 673, 997, 349, 1, 1447, 11783, 1499, 12203, 1, 743, 1, 1, 1, 229, 1, 13963, 887, 14423, 1, 14891, 1, 1, 1951, 1, 503, 277, 1, 16843, 2137, 17351, 1, 1051, 1, 347, 1, 1, 2399, 19463, 2467, 20011, 317, 1, 1303, 1, 2677, 1973, 2749, 22283, 1, 22871, 1, 757, 2971, 24071, 1, 24683, 1, 25303, 1601, 25931, 1, 857, 3361, 27211, 1721, 1, 881, 2593, 3607, 29191, 3691, 29867, 1, 1, 1931, 1, 359, 1879, 367, 1, 2063, 547, 1, 467, 1, 1, 1, 1, 1123, 3301, 2293, 1, 1, 37831, 1, 38603, 2437, 39383, 1, 1, 461, 577, 5171, 41771, 659, 439, 2687, 43403, 5477, 4021, 5581, 1, 2843, 1481, 1, 463, 1, 47623, 6007, 48491, 1, 49367, 1, 1621, 6337, 1, 6449, 733, 1, 52951, 1669, 1, 6791, 1, 6907, 541, 1, 56663, 3571, 1087, 1, 58567, 1, 1009, 1, 3559, 953, 61483, 1, 1, 1, 63467, 1999, 5861, 1, 5953, 1, 911, 8377, 67531, 4253, 68567, 1, 1, 797, 70663, 809, 4219, 1129, 877, 4583, 73867, 1, 1, 9437, 1, 4787, 7013, 607, 4603, 9851, 79367, 1, 80491, 1, 2633, 1, 82763, 947, 83911, 1, 1, 1, 1627, 2713, 87403, 647, 8053, 1, 8161, 1, 5351, 1, 1511, 11597, 93383, 1, 94603, 1, 1571, 1, 859, 12211, 5783, 1, 99563, 1, 100823, 1, 9281, 12841, 9397, 13001, 104651, 6581, 105943, 3331, 107243, 13487, 1487, 1, 1, 1, 111191, 6991, 6619, 14149, 3673, 1, 115211, 7243, 10597, 1, 1, 14827, 2251, 1, 1, 3793, 2069, 7673, 1, 1, 1, 1427, 1, 7937, 127703, 4013, 929, 16231, 2213, 16411, 1091, 1, 1103, 8387, 134923, 1, 1, 1, 137867, 8663, 1, 1, 1109, 1609, 1039, 1, 1483, 4519, 1, 9133, 146891, 18457, 1, 1097, 13633, 9421, 1193, 4759, 153067, 19231, 1531, 19427, 863, 1, 1901, 1, 1, 20021, 160967, 1, 1, 10211, 164183, 1289, 15073, 1, 1, 21031, 169067, 5309, 170711, 1, 10139, 21649, 1723, 1987, 175691, 1, 919, 5569, 179051, 1, 2963, 22699, 182443, 1, 16741, 1, 1, 1373, 1259, 23557, 2281, 11887, 11239, 2999, 192811, 1, 1889, 2221, 196331, 6163, 1, 12437, 1459, 25097, 11863, 25321, 1, 1, 18661, 1, 1319, 25999, 1, 26227, 210731, 3307, 6857, 1213, 1439, 2447, 216263, 1597, 3697, 13691, 1229, 1, 1, 27851, 20341, 28087, 1, 1, 1637, 14281, 3889, 1, 231367, 1, 13723, 1, 3313, 1, 237163, 1, 1523, 30011, 241067, 1, 243031, 1, 22273, 1, 22453, 1, 248971, 1, 250967, 1, 252971, 1, 1, 2909, 257003, 1, 1, 16253, 3677, 1, 263111, 1, 265163, 1, 1429, 1, 24481, 33791, 271367, 2003, 8821, 4289, 275543, 1, 277643, 3167, 279751, 3191, 281867, 17683, 9161, 1, 1, 35899, 2551, 1, 1553, 9109, 26597, 18353, 294731, 1, 1, 1, 4903, 1, 4127, 1, 17851, 3461, 5011, 1237, 1, 1, 5851, 1, 312331, 39181, 28597, 39461, 1, 1, 2113, 5003, 321323, 2371, 323591, 40591, 1423, 1, 1, 1871, 1, 1, 332743, 41737, 335051, 21013, 337367, 1, 30881, 1, 1, 42899, 344363, 5399, 346711, 1279, 3389, 43781, 3623, 4007, 1, 2017, 6037, 1, 358571, 44971, 4349, 2663, 363371, 11393, 3023, 22937, 1, 1, 370631, 1, 7039, 1, 375511, 1, 377963, 1, 380423, 4337, 1, 3001, 5279, 1, 1949, 2861, 390343, 1579, 1, 24631, 1, 6197, 397867, 49891, 400391, 50207, 402923, 1, 2917, 2311, 408011, 4651, 24151, 51481, 3253, 1, 2753, 1, 1, 1, 38261, 1, 1, 6637, 1, 26711, 428683, 1, 13913, 3181, 433931, 2473, 1, 1, 8287, 1, 3373, 55399, 444523, 13933, 7331, 1, 40897, 56401, 41141, 56737, 1, 28537, 457943, 1, 1663, 57751, 1, 5281, 466091, 1, 1, 29387, 27739, 1907, 474311, 1, 1, 1759, 1, 1, 1, 1, 485383, 1, 488171, 15299, 490967, 30773, 8369, 1, 496583, 5659, 499403, 1, 1, 15739, 505067, 63311, 1, 63667, 1, 1, 1, 32191, 16661, 1, 1, 65101, 522251, 1, 525143, 1, 528043, 1, 6397, 1, 3583, 16729, 536791, 33641, 539723, 1, 49333, 4001, 1, 1, 548567, 17189, 32443, 69127, 554503, 69499, 557483, 1, 560471, 1, 563467, 70621, 1, 70997, 1, 1, 3163, 8969, 52321, 4243, 1, 1, 5147, 18223, 584663, 36637, 1, 1, 1, 1, 19157, 1, 1, 1, 600043, 1, 2087, 1, 606251, 9497, 1787, 38183, 55681, 76757, 615623, 1, 6379, 1, 36583, 4871, 10247, 7121, 628231, 1, 8893, 1, 1, 39761, 637771, 1, 640967, 2591, 1, 40361, 1, 1, 3371, 81527, 653831, 81931, 38651, 1, 2083, 3761, 663563, 7559, 666823, 83557, 2287, 41983, 1, 1, 1, 84787, 61813, 85199, 1, 1, 686551, 43013, 689867, 86441, 1, 86857, 696523, 3967, 699863, 1993, 703211, 1, 9679, 1, 22901, 11119, 41959, 44687, 5923, 89797, 1, 1, 723467, 1, 7057, 1, 730283, 91499, 43159, 1, 737131, 2099, 1, 2729, 1, 1, 747463, 1, 4973, 47041, 68581, 23629, 68897, 94951, 761351, 1, 14431, 1, 768343, 48131, 45403, 1, 775367, 8831, 6133, 1, 1, 12253, 785963, 98467, 789511, 98911, 4241, 1, 72421, 1, 1, 5897, 6329, 1, 807371, 1, 810967, 2309, 2269, 9277, 26393, 102499, 821803, 1, 6301, 1, 11677, 1, 1, 1, 1, 52387, 840023, 6577, 13831, 105691, 847367, 1, 851051, 2423, 1, 1, 6553, 2029, 16267, 6353, 865867, 1, 10477, 1, 79393, 3529, 1, 1, 14929, 1, 3019, 3259, 1, 1, 892103, 10159, 1, 5101, 899671, 14087, 15313, 113171, 2741, 113647, 1, 1, 2683, 57301, 1, 1, 9511, 115561, 926411, 3413, 5197, 29131, 30133, 1, 937991, 1, 4733, 14747, 3041, 1, 949643, 6997, 953543, 1,

6. Sequence of the polynom (only primes)

1451, 11, 1367, 83, 1291, 157, 1223, 149, 1163, 71, 101, 17, 97, 131, 1031, 127, 59, 31, 983, 61, 971, 967, 1543, 199, 53, 103, 113, 1867, 241, 181, 257, 193, 137, 73, 2411, 311, 151, 331, 2731, 2903, 3083, 397, 3271, 421, 3467, 223, 3671, 353, 499, 373, 139, 4567, 293, 283, 617, 5323, 5591, 179, 5867, 751, 6151, 787, 379, 613, 431, 641, 941, 7691, 491, 8363, 281, 9067, 9431, 601, 9803, 1249, 599, 1297, 673, 997, 349, 1447, 11783, 1499, 12203, 743, 229, 13963, 887, 14423, 14891, 1951, 503, 277, 16843, 2137, 17351, 1051, 347, 2399, 19463, 2467, 20011, 317, 1303, 2677, 1973, 2749, 22283, 22871, 757, 2971, 24071, 24683, 25303, 1601, 25931, 857, 3361, 27211, 1721, 881, 2593, 3607, 29191, 3691, 29867, 1931, 359, 1879, 367, 2063, 547, 467, 1123, 3301, 2293, 37831, 38603, 2437, 39383, 461, 577, 5171, 41771, 659, 439, 2687, 43403, 5477, 4021, 5581, 2843, 1481, 463, 47623, 6007, 48491, 49367, 1621, 6337, 6449, 733, 52951, 1669, 6791, 6907, 541, 56663, 3571, 1087, 58567, 1009, 3559, 953, 61483, 63467, 1999, 5861, 5953, 911, 8377, 67531, 4253, 68567, 797, 70663, 809, 4219, 1129, 877, 4583, 73867, 9437, 4787, 7013, 607, 4603, 9851, 79367, 80491, 2633, 82763, 947, 83911, 1627, 2713, 87403, 647, 8053, 8161, 5351, 1511, 11597, 93383, 94603, 1571, 859, 12211, 5783, 99563, 100823, 9281, 12841, 9397, 13001, 104651, 6581, 105943, 3331, 107243, 13487, 1487, 111191, 6991, 6619, 14149, 3673, 115211, 7243, 10597, 14827, 2251, 3793, 2069, 7673, 1427, 7937, 127703, 4013, 929, 16231, 2213, 16411, 1091, 1103, 8387, 134923, 137867, 8663, 1109, 1609, 1039, 1483, 4519, 9133, 146891, 18457, 1097, 13633, 9421, 1193, 4759, 153067, 19231, 1531, 19427, 863, 1901, 20021, 160967, 10211, 164183, 1289, 15073, 21031, 169067, 5309, 170711, 10139, 21649, 1723, 1987, 175691, 919, 5569, 179051, 2963, 22699, 182443, 16741, 1373, 1259, 23557, 2281, 11887, 11239, 2999, 192811, 1889, 2221, 196331, 6163, 12437, 1459, 25097, 11863, 25321, 18661, 1319, 25999, 26227, 210731, 3307, 6857, 1213, 1439, 2447, 216263, 1597, 3697, 13691, 1229, 27851, 20341, 28087, 1637, 14281, 3889, 231367, 13723, 3313, 237163, 1523, 30011, 241067, 243031, 22273, 22453, 248971, 250967, 252971, 2909, 257003, 16253, 3677, 263111, 265163, 1429, 24481, 33791, 271367, 2003, 8821, 4289, 275543, 277643, 3167, 279751, 3191, 281867, 17683, 9161, 35899, 2551, 1553, 9109, 26597, 18353, 294731, 4903, 4127, 17851, 3461, 5011, 1237, 5851, 312331, 39181, 28597, 39461, 2113, 5003, 321323, 2371, 323591, 40591, 1423, 1871, 332743, 41737, 335051, 21013, 337367, 30881, 42899, 344363, 5399, 346711, 1279, 3389, 43781, 3623, 4007, 2017, 6037, 358571, 44971, 4349, 2663, 363371, 11393, 3023, 22937, 370631, 7039, 375511, 377963, 380423, 4337, 3001, 5279, 1949, 2861, 390343, 1579, 24631, 6197, 397867, 49891, 400391, 50207, 402923, 2917, 2311, 408011, 4651, 24151, 51481, 3253, 2753, 38261, 6637, 26711, 428683, 13913, 3181, 433931, 2473, 8287, 3373, 55399, 444523, 13933, 7331, 40897, 56401, 41141, 56737, 28537, 457943, 1663, 57751, 5281, 466091, 29387, 27739, 1907, 474311, 1759, 485383, 488171, 15299, 490967, 30773, 8369, 496583, 5659, 499403, 15739, 505067, 63311, 63667, 32191, 16661, 65101, 522251, 525143, 528043, 6397, 3583, 16729, 536791, 33641, 539723, 49333, 4001, 548567, 17189, 32443, 69127, 554503, 69499, 557483, 560471, 563467, 70621, 70997, 3163, 8969, 52321, 4243, 5147, 18223, 584663, 36637, 19157, 600043, 2087, 606251, 9497, 1787, 38183, 55681, 76757, 615623, 6379, 36583, 4871, 10247, 7121, 628231, 8893, 39761, 637771, 640967, 2591, 40361, 3371, 81527, 653831, 81931, 38651, 2083, 3761, 663563, 7559, 666823, 83557, 2287, 41983, 84787, 61813, 85199, 686551, 43013, 689867, 86441, 86857, 696523, 3967, 699863, 1993, 703211, 9679, 22901, 11119, 41959, 44687, 5923, 89797, 723467, 7057, 730283, 91499, 43159, 737131, 2099, 2729, 747463, 4973, 47041, 68581, 23629, 68897, 94951, 761351, 14431, 768343, 48131, 45403, 775367, 8831, 6133, 12253, 785963, 98467, 789511, 98911, 4241, 72421, 5897, 6329, 807371, 810967, 2309, 2269, 9277, 26393, 102499, 821803, 6301, 11677, 52387, 840023, 6577, 13831, 105691, 847367, 851051, 2423, 6553, 2029, 16267, 6353, 865867, 10477, 79393, 3529, 14929, 3019, 3259, 892103, 10159, 5101, 899671, 14087, 15313, 113171, 2741, 113647, 2683, 57301, 9511, 115561, 926411, 3413, 5197, 29131, 30133, 937991, 4733, 14747, 3041, 949643, 6997, 953543,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-44x+1451 and
the reducible primes which appear as divisor for the first time
p | x^2-44x+1451 and p < x^2-44x+1451

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

A B C D E F G H
exponent
=log2 (x)
<=x number
of all primes
number of primes
p = f(x)
number of primes
p | f(x)
C / x D / x E / x
1 2 3 2 1 1.5 1 0.5
2 4 5 3 2 1.25 0.75 0.5
3 8 9 5 4 1.125 0.625 0.5
4 16 17 6 11 1.0625 0.375 0.6875
5 32 22 9 13 0.6875 0.28125 0.40625
6 64 39 13 26 0.609375 0.203125 0.40625
7 128 85 30 55 0.6640625 0.234375 0.4296875
8 256 168 55 113 0.65625 0.21484375 0.44140625
9 512 346 96 250 0.67578125 0.1875 0.48828125
10 1024 683 190 493 0.66699219 0.18554688 0.48144531
11 2048 1367 326 1041 0.66748047 0.15917969 0.50830078
12 4096 2746 607 2139 0.67041016 0.14819336 0.5222168
13 8192 5485 1132 4353 0.66955566 0.13818359 0.53137207
14 16384 11062 2076 8986 0.6751709 0.12670898 0.54846191
15 32768 22185 3788 18397 0.67703247 0.11560059 0.56143188
16 65536 44532 6984 37548 0.67950439 0.10656738 0.57293701
17 131072 89127 13128 75999 0.67998505 0.10015869 0.57982635
18 262144 178536 24794 153742 0.68106079 0.0945816 0.58647919
19 524288 357537 46700 310837 0.68194771 0.08907318 0.59287453
20 1048576 715789 88142 627647 0.68262959 0.08405876 0.59857082
21 2097152 1432787 167145 1265642 0.68320608 0.07970095 0.60350513
22 4194304 2867476 317768 2549708 0.68365955 0.0757618 0.60789776
23 8388608 5738409 606049 5132360 0.68407166 0.07224667 0.61182499
24 16777216 11483050 1158039 10325011 0.68444312 0.0690245 0.61541861


8. Check for existing Integer Sequences by OEIS

Found in Database : 1451, 11, 1367, 83, 1291, 157, 1223, 149, 1163, 71, 101, 17, 97, 131, 1031, 127, 59, 31, 983, 61,
Found in Database : 1451, 11, 1367, 83, 1291, 157, 1223, 149, 1163, 71, 101, 17, 97, 131, 1031, 127, 59, 31, 983, 61, 971, 967,
Found in Database : 11, 17, 31, 53, 59, 61, 71, 73, 83, 97, 101, 103, 113, 127, 131, 137, 139, 149,