Inhaltsverzeichnis

Development of
Algorithmic Constructions

22:06:11
Deutsch
28.Mar 2024

Polynom = x^2+64x-449

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) = 449 = 449
f(1) = 3 = 3
f(2) = 317 = 317
f(3) = 31 = 31
f(4) = 177 = 3*59
f(5) = 13 = 13
f(6) = 29 = 29
f(7) = 3 = 3
f(8) = 127 = 127
f(9) = 13 = 13
f(10) = 291 = 3*97
f(11) = 47 = 47
f(12) = 463 = 463
f(13) = 69 = 3*23
f(14) = 643 = 643
f(15) = 23 = 23
f(16) = 831 = 3*277
f(17) = 29 = 29
f(18) = 1027 = 13*79
f(19) = 141 = 3*47
f(20) = 1231 = 1231
f(21) = 167 = 167
f(22) = 1443 = 3*13*37
f(23) = 97 = 97
f(24) = 1663 = 1663
f(25) = 111 = 3*37
f(26) = 1891 = 31*61
f(27) = 251 = 251
f(28) = 2127 = 3*709
f(29) = 281 = 281
f(30) = 2371 = 2371
f(31) = 39 = 3*13
f(32) = 2623 = 43*61
f(33) = 43 = 43
f(34) = 2883 = 3*31*31
f(35) = 377 = 13*29
f(36) = 3151 = 23*137
f(37) = 411 = 3*137
f(38) = 3427 = 23*149
f(39) = 223 = 223
f(40) = 3711 = 3*1237
f(41) = 241 = 241
f(42) = 4003 = 4003
f(43) = 519 = 3*173
f(44) = 4303 = 13*331
f(45) = 557 = 557
f(46) = 4611 = 3*29*53
f(47) = 149 = 149
f(48) = 4927 = 13*379
f(49) = 159 = 3*53
f(50) = 5251 = 59*89
f(51) = 677 = 677
f(52) = 5583 = 3*1861
f(53) = 719 = 719
f(54) = 5923 = 5923
f(55) = 381 = 3*127
f(56) = 6271 = 6271
f(57) = 403 = 13*31
f(58) = 6627 = 3*47*47
f(59) = 851 = 23*37
f(60) = 6991 = 6991
f(61) = 897 = 3*13*23
f(62) = 7363 = 37*199
f(63) = 59 = 59
f(64) = 7743 = 3*29*89
f(65) = 31 = 31
f(66) = 8131 = 47*173
f(67) = 1041 = 3*347
f(68) = 8527 = 8527
f(69) = 1091 = 1091
f(70) = 8931 = 3*13*229
f(71) = 571 = 571
f(72) = 9343 = 9343
f(73) = 597 = 3*199
f(74) = 9763 = 13*751
f(75) = 1247 = 29*43
f(76) = 10191 = 3*43*79
f(77) = 1301 = 1301
f(78) = 10627 = 10627
f(79) = 339 = 3*113
f(80) = 11071 = 11071
f(81) = 353 = 353
f(82) = 11523 = 3*23*167
f(83) = 1469 = 13*113
f(84) = 11983 = 23*521
f(85) = 1527 = 3*509
f(86) = 12451 = 12451
f(87) = 793 = 13*61
f(88) = 12927 = 3*31*139
f(89) = 823 = 823
f(90) = 13411 = 13411
f(91) = 1707 = 3*569
f(92) = 13903 = 13903
f(93) = 1769 = 29*61
f(94) = 14403 = 3*4801
f(95) = 229 = 229
f(96) = 14911 = 13*31*37
f(97) = 237 = 3*79
f(98) = 15427 = 15427
f(99) = 1961 = 37*53
f(100) = 15951 = 3*13*409

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+64x-449

f(0)=449
f(1)=3
f(2)=317
f(3)=31
f(4)=59
f(5)=13
f(6)=29
f(7)=1
f(8)=127
f(9)=1
f(10)=97
f(11)=47
f(12)=463
f(13)=23
f(14)=643
f(15)=1
f(16)=277
f(17)=1
f(18)=79
f(19)=1
f(20)=1231
f(21)=167
f(22)=37
f(23)=1
f(24)=1663
f(25)=1
f(26)=61
f(27)=251
f(28)=709
f(29)=281
f(30)=2371
f(31)=1
f(32)=43
f(33)=1
f(34)=1
f(35)=1
f(36)=137
f(37)=1
f(38)=149
f(39)=223
f(40)=1237
f(41)=241
f(42)=4003
f(43)=173
f(44)=331
f(45)=557
f(46)=53
f(47)=1
f(48)=379
f(49)=1
f(50)=89
f(51)=677
f(52)=1861
f(53)=719
f(54)=5923
f(55)=1
f(56)=6271
f(57)=1
f(58)=1
f(59)=1
f(60)=6991
f(61)=1
f(62)=199
f(63)=1
f(64)=1
f(65)=1
f(66)=1
f(67)=347
f(68)=8527
f(69)=1091
f(70)=229
f(71)=571
f(72)=9343
f(73)=1
f(74)=751
f(75)=1
f(76)=1
f(77)=1301
f(78)=10627
f(79)=113
f(80)=11071
f(81)=353
f(82)=1
f(83)=1
f(84)=521
f(85)=509
f(86)=12451
f(87)=1
f(88)=139
f(89)=823
f(90)=13411
f(91)=569
f(92)=13903
f(93)=1
f(94)=4801
f(95)=1
f(96)=1
f(97)=1
f(98)=15427
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+64x-449 could be written as f(y)= y^2-1473 with x=y-32

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+32
f'(x)>2x+63

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

449, 3, 317, 31, 59, 13, 29, 1, 127, 1, 97, 47, 463, 23, 643, 1, 277, 1, 79, 1, 1231, 167, 37, 1, 1663, 1, 61, 251, 709, 281, 2371, 1, 43, 1, 1, 1, 137, 1, 149, 223, 1237, 241, 4003, 173, 331, 557, 53, 1, 379, 1, 89, 677, 1861, 719, 5923, 1, 6271, 1, 1, 1, 6991, 1, 199, 1, 1, 1, 1, 347, 8527, 1091, 229, 571, 9343, 1, 751, 1, 1, 1301, 10627, 113, 11071, 353, 1, 1, 521, 509, 12451, 1, 139, 823, 13411, 569, 13903, 1, 4801, 1, 1, 1, 15427, 1, 409, 2027, 311, 349, 587, 1, 5857, 1, 18127, 1, 18691, 593, 6421, 1, 19843, 839, 20431, 2591, 163, 1, 1, 457, 1, 2819, 7621, 2897, 1, 1, 1049, 191, 359, 3137, 25423, 1, 26083, 1, 1, 1693, 27427, 1, 28111, 3557, 9601, 911, 181, 1, 30211, 3821, 1, 3911, 1021, 1, 1, 1, 1, 1, 33871, 1427, 1117, 547, 1, 1, 1, 1523, 36943, 1, 12577, 2383, 653, 811, 1063, 4967, 13381, 1, 1, 431, 1, 1319, 1093, 5381, 1499, 1, 1, 2797, 15061, 2851, 46051, 1, 1, 1, 15937, 1, 1, 1, 49603, 6257, 1, 1, 51427, 1, 4027, 3301, 17761, 6719, 1, 1, 55171, 1, 1, 1, 1543, 2399, 1873, 563, 19681, 1, 1, 1, 61027, 7691, 1, 7817, 2741, 1, 64063, 1009, 1669, 1, 743, 2777, 5167, 4231, 22741, 4297, 877, 2909, 70351, 8861, 821, 1, 1229, 761, 827, 1, 673, 1, 1613, 1, 1789, 1, 26017, 1, 6091, 3323, 1709, 1, 2089, 641, 1559, 3467, 83791, 1, 1, 5347, 86143, 1, 3797, 1, 1283, 857, 1471, 941, 1, 2861, 991, 11597, 1531, 3917, 1, 5953, 31957, 1, 1, 4073, 2659, 12377, 1, 1567, 100927, 1, 2377, 1, 34501, 1, 104803, 1, 3659, 6673, 35809, 1, 108751, 1, 110083, 3461, 2857, 1, 112771, 1, 8779, 1, 1, 1, 5081, 1, 1, 14867, 39877, 1, 3271, 1, 1, 1, 41281, 15569, 1, 1, 126691, 7963, 42709, 8053, 9967, 1, 131023, 383, 1, 1, 1, 1, 135427, 17021, 971, 17207, 138403, 1, 4513, 1, 47137, 1367, 3041, 5987, 144451, 2269, 48661, 2293, 5087, 1, 11467, 18731, 1, 9463, 1, 3187, 153763, 19319, 977, 1, 1129, 1, 158527, 1, 53377, 20117, 161743, 1, 163363, 1, 1279, 1, 166627, 6977, 881, 919, 4357, 1, 171583, 1, 13327, 1, 58309, 1, 176611, 3697, 1, 11197, 2069, 1, 181711, 7607, 1, 443, 61717, 5813, 186883, 7823, 1, 23687, 1, 11953, 14779, 4021, 193891, 1, 1, 1, 1, 1033, 1193, 1, 1, 1, 1, 1, 1811, 1, 68821, 997, 208291, 1, 210127, 1, 70657, 6653, 5779, 2237, 1, 27077, 1, 1, 16879, 4591, 221311, 479, 1583, 28019, 1381, 9419, 227011, 1, 1, 1, 10037, 1, 1, 1, 78241, 14731, 1, 4951, 238627, 491, 1, 30197, 5641, 1, 1447, 7673, 2221, 30941, 8017, 1, 1, 1, 84181, 1, 1, 1, 4349, 2477, 86209, 4057, 260671, 1, 262723, 32969, 88261, 1, 1579, 5581, 268927, 1, 6949, 1097, 3457, 11423, 1847, 1, 4019, 8699, 12149, 1, 281551, 35327, 94561, 1, 285823, 1, 1, 36131, 1, 1, 10079, 1, 22651, 2309, 1621, 37217, 1, 12497, 301027, 1, 1657, 1, 1, 1, 1, 2969, 1949, 9719, 1, 1, 10141, 1, 105541, 39719, 318883, 1, 6833, 20143, 8293, 1399, 8803, 13619, 1, 1, 4787, 5179, 332611, 13907, 7789, 1, 1423, 1627, 339583, 1, 1, 3299, 114757, 1489, 346627, 3623, 348991, 1, 1, 1, 27211, 1, 1597, 1, 1, 22483, 1303, 1, 363343, 45569, 121921, 1, 6947, 1, 1, 1, 3361, 1, 375523, 1, 1, 1, 4373, 47711, 16649, 16007, 1289, 1, 1, 12161, 1, 16319, 392911, 1699, 1481, 24793, 397951, 8317, 1, 3863, 134341, 1, 1, 1, 408127, 6397, 136897, 2239, 413263, 1, 11239, 1, 10729, 1, 1747, 17597, 32587, 53117, 3023, 1, 428863, 4481, 14879, 54101, 144709, 1, 9293, 9127, 8291, 1, 1, 1, 1, 18587, 447427, 1753, 2543, 3527, 1, 1, 1, 57107, 1, 28723, 4079, 9631, 5869, 1571, 155461, 1, 1, 1, 471871, 1, 158209, 1, 1, 19949, 1913, 30097, 160981, 30271, 485731, 20297, 37579, 1, 1, 7699, 38011, 1, 496963, 62297, 166597, 1, 11689, 10501, 21977, 2437, 1, 63719, 17627, 1, 514051, 1, 4657, 1, 4093, 1, 522703, 65519, 13477, 32941, 528511, 1, 40879, 1, 178117, 66977, 1, 1, 1, 1, 1, 5237, 1, 22817, 549091, 2647, 3119, 1, 555043, 23189, 1, 69941, 187009, 17579, 1, 1, 4139, 71069, 1, 1931, 24917, 11971, 1, 1, 193057, 2503, 582223, 1871, 585283, 1, 196117, 1, 11159, 797, 594511, 74507, 6869, 37447, 600703, 12547, 46447, 2441, 202309, 3307, 1, 1, 1601, 19211, 205441, 77237, 619471, 1, 1, 3001, 7193, 1, 1, 1, 1, 2141, 211777, 1, 638527, 1667, 1, 1, 1, 80819, 28181, 13537, 1, 40813, 7039, 82031, 2621, 27479, 2887, 1, 221461, 1, 6883, 1, 12659, 1, 224737, 1, 4547, 14149, 680803, 3709, 1, 3727, 52879, 1, 23819, 1, 1, 86969, 697423, 29129, 1, 1, 2971, 1, 15053, 2273, 710863, 89069, 238081, 1721, 717631, 1, 1, 1, 10499, 1, 727843, 15199, 4327, 1, 244897, 92051, 56779, 1, 12569, 5807, 8011, 2917, 1, 31259, 20323, 7247, 251809, 1, 24481, 1, 1, 4153, 4817, 95957, 769411, 1, 2153, 1, 1, 1, 779983, 32573, 60271, 49081, 1, 1, 1, 1, 794191, 99497, 265921, 1, 34841, 1, 1, 7757, 1, 1, 7187, 1, 2903, 1, 2417, 102647, 1, 34367, 4567, 25889, 1, 26003, 833923, 1, 837583, 1, 4597, 1, 844927, 1, 848611, 106307, 6607, 1, 1, 1, 8863, 1, 1, 108161, 867151, 36209, 2161, 54547, 6203, 1889, 67567, 36677, 5099, 110501, 1, 27743, 1, 1, 2531, 8609, 10313, 1, 2203, 1, 1, 56671, 2179, 113819, 912463, 1, 916291, 14347, 23593, 14407, 923971, 1, 1, 1, 10709, 1, 21757, 19531, 2029, 117671, 3533, 1, 947203, 9887, 7489, 1, 318337, 1, 16253, 40037, 1, 1, 322261, 1, 1, 40529, 1, 122081, 1, 1, 7069, 1, 986563, 4261, 10651, 124067, 12589, 1, 998527, 62533, 334177, 1, 1006543, 42023, 34847, 31643, 338197, 31769, 23689, 1, 1, 5569, 342241, 1, 2143, 21517, 4519, 1, 346309, 130121, 1043011, 5443, 36107, 1, 5939, 1, 7703, 3389, 22541, 66343,

6. Sequence of the polynom (only primes)

449, 3, 317, 31, 59, 13, 29, 127, 97, 47, 463, 23, 643, 277, 79, 1231, 167, 37, 1663, 61, 251, 709, 281, 2371, 43, 137, 149, 223, 1237, 241, 4003, 173, 331, 557, 53, 379, 89, 677, 1861, 719, 5923, 6271, 6991, 199, 347, 8527, 1091, 229, 571, 9343, 751, 1301, 10627, 113, 11071, 353, 521, 509, 12451, 139, 823, 13411, 569, 13903, 4801, 15427, 409, 2027, 311, 349, 587, 5857, 18127, 18691, 593, 6421, 19843, 839, 20431, 2591, 163, 457, 2819, 7621, 2897, 1049, 191, 359, 3137, 25423, 26083, 1693, 27427, 28111, 3557, 9601, 911, 181, 30211, 3821, 3911, 1021, 33871, 1427, 1117, 547, 1523, 36943, 12577, 2383, 653, 811, 1063, 4967, 13381, 431, 1319, 1093, 5381, 1499, 2797, 15061, 2851, 46051, 15937, 49603, 6257, 51427, 4027, 3301, 17761, 6719, 55171, 1543, 2399, 1873, 563, 19681, 61027, 7691, 7817, 2741, 64063, 1009, 1669, 743, 2777, 5167, 4231, 22741, 4297, 877, 2909, 70351, 8861, 821, 1229, 761, 827, 673, 1613, 1789, 26017, 6091, 3323, 1709, 2089, 641, 1559, 3467, 83791, 5347, 86143, 3797, 1283, 857, 1471, 941, 2861, 991, 11597, 1531, 3917, 5953, 31957, 4073, 2659, 12377, 1567, 100927, 2377, 34501, 104803, 3659, 6673, 35809, 108751, 110083, 3461, 2857, 112771, 8779, 5081, 14867, 39877, 3271, 41281, 15569, 126691, 7963, 42709, 8053, 9967, 131023, 383, 135427, 17021, 971, 17207, 138403, 4513, 47137, 1367, 3041, 5987, 144451, 2269, 48661, 2293, 5087, 11467, 18731, 9463, 3187, 153763, 19319, 977, 1129, 158527, 53377, 20117, 161743, 163363, 1279, 166627, 6977, 881, 919, 4357, 171583, 13327, 58309, 176611, 3697, 11197, 2069, 181711, 7607, 443, 61717, 5813, 186883, 7823, 23687, 11953, 14779, 4021, 193891, 1033, 1193, 1811, 68821, 997, 208291, 210127, 70657, 6653, 5779, 2237, 27077, 16879, 4591, 221311, 479, 1583, 28019, 1381, 9419, 227011, 10037, 78241, 14731, 4951, 238627, 491, 30197, 5641, 1447, 7673, 2221, 30941, 8017, 84181, 4349, 2477, 86209, 4057, 260671, 262723, 32969, 88261, 1579, 5581, 268927, 6949, 1097, 3457, 11423, 1847, 4019, 8699, 12149, 281551, 35327, 94561, 285823, 36131, 10079, 22651, 2309, 1621, 37217, 12497, 301027, 1657, 2969, 1949, 9719, 10141, 105541, 39719, 318883, 6833, 20143, 8293, 1399, 8803, 13619, 4787, 5179, 332611, 13907, 7789, 1423, 1627, 339583, 3299, 114757, 1489, 346627, 3623, 348991, 27211, 1597, 22483, 1303, 363343, 45569, 121921, 6947, 3361, 375523, 4373, 47711, 16649, 16007, 1289, 12161, 16319, 392911, 1699, 1481, 24793, 397951, 8317, 3863, 134341, 408127, 6397, 136897, 2239, 413263, 11239, 10729, 1747, 17597, 32587, 53117, 3023, 428863, 4481, 14879, 54101, 144709, 9293, 9127, 8291, 18587, 447427, 1753, 2543, 3527, 57107, 28723, 4079, 9631, 5869, 1571, 155461, 471871, 158209, 19949, 1913, 30097, 160981, 30271, 485731, 20297, 37579, 7699, 38011, 496963, 62297, 166597, 11689, 10501, 21977, 2437, 63719, 17627, 514051, 4657, 4093, 522703, 65519, 13477, 32941, 528511, 40879, 178117, 66977, 5237, 22817, 549091, 2647, 3119, 555043, 23189, 69941, 187009, 17579, 4139, 71069, 1931, 24917, 11971, 193057, 2503, 582223, 1871, 585283, 196117, 11159, 797, 594511, 74507, 6869, 37447, 600703, 12547, 46447, 2441, 202309, 3307, 1601, 19211, 205441, 77237, 619471, 3001, 7193, 2141, 211777, 638527, 1667, 80819, 28181, 13537, 40813, 7039, 82031, 2621, 27479, 2887, 221461, 6883, 12659, 224737, 4547, 14149, 680803, 3709, 3727, 52879, 23819, 86969, 697423, 29129, 2971, 15053, 2273, 710863, 89069, 238081, 1721, 717631, 10499, 727843, 15199, 4327, 244897, 92051, 56779, 12569, 5807, 8011, 2917, 31259, 20323, 7247, 251809, 24481, 4153, 4817, 95957, 769411, 2153, 779983, 32573, 60271, 49081, 794191, 99497, 265921, 34841, 7757, 7187, 2903, 2417, 102647, 34367, 4567, 25889, 26003, 833923, 837583, 4597, 844927, 848611, 106307, 6607, 8863, 108161, 867151, 36209, 2161, 54547, 6203, 1889, 67567, 36677, 5099, 110501, 27743, 2531, 8609, 10313, 2203, 56671, 2179, 113819, 912463, 916291, 14347, 23593, 14407, 923971, 10709, 21757, 19531, 2029, 117671, 3533, 947203, 9887, 7489, 318337, 16253, 40037, 322261, 40529, 122081, 7069, 986563, 4261, 10651, 124067, 12589, 998527, 62533, 334177, 1006543, 42023, 34847, 31643, 338197, 31769, 23689, 5569, 342241, 2143, 21517, 4519, 346309, 130121, 1043011, 5443, 36107, 5939, 7703, 3389, 22541, 66343,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+64x-449 and
the reducible primes which appear as divisor for the first time
p | x^2+64x-449 and p < x^2+64x-449

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 : 449, 3, 317, 31, 59, 13, 29, 1, 127, 1, 97, 47, 463, 23, 643, 1, 277, 1, 79, 1,
Found in Database : 449, 3, 317, 31, 59, 13, 29, 127, 97, 47, 463, 23, 643, 277, 79, 1231, 167, 37, 1663, 61, 251, 709, 281, 2371, 43, 137, 149, 223,
Found in Database : 3, 13, 23, 29, 31, 37, 43, 47, 53, 59, 61, 79, 89, 97, 113, 127, 137, 139, 149,