Inhaltsverzeichnis

Development of
Algorithmic Constructions

22:54:48
Deutsch
28.Mar 2024

Polynom = x^2-92x+227

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) = 227 = 227
f(1) = 17 = 17
f(2) = 47 = 47
f(3) = 5 = 5
f(4) = 125 = 5*5*5
f(5) = 13 = 13
f(6) = 289 = 17*17
f(7) = 23 = 23
f(8) = 445 = 5*89
f(9) = 65 = 5*13
f(10) = 593 = 593
f(11) = 83 = 83
f(12) = 733 = 733
f(13) = 25 = 5*5
f(14) = 865 = 5*173
f(15) = 29 = 29
f(16) = 989 = 23*43
f(17) = 131 = 131
f(18) = 1105 = 5*13*17
f(19) = 145 = 5*29
f(20) = 1213 = 1213
f(21) = 79 = 79
f(22) = 1313 = 13*101
f(23) = 85 = 5*17
f(24) = 1405 = 5*281
f(25) = 181 = 181
f(26) = 1489 = 1489
f(27) = 191 = 191
f(28) = 1565 = 5*313
f(29) = 25 = 5*5
f(30) = 1633 = 23*71
f(31) = 13 = 13
f(32) = 1693 = 1693
f(33) = 215 = 5*43
f(34) = 1745 = 5*349
f(35) = 221 = 13*17
f(36) = 1789 = 1789
f(37) = 113 = 113
f(38) = 1825 = 5*5*73
f(39) = 115 = 5*23
f(40) = 1853 = 17*109
f(41) = 233 = 233
f(42) = 1873 = 1873
f(43) = 235 = 5*47
f(44) = 1885 = 5*13*29
f(45) = 59 = 59
f(46) = 1889 = 1889
f(47) = 59 = 59
f(48) = 1885 = 5*13*29
f(49) = 235 = 5*47
f(50) = 1873 = 1873
f(51) = 233 = 233
f(52) = 1853 = 17*109
f(53) = 115 = 5*23
f(54) = 1825 = 5*5*73
f(55) = 113 = 113
f(56) = 1789 = 1789
f(57) = 221 = 13*17
f(58) = 1745 = 5*349
f(59) = 215 = 5*43
f(60) = 1693 = 1693
f(61) = 13 = 13
f(62) = 1633 = 23*71
f(63) = 25 = 5*5
f(64) = 1565 = 5*313
f(65) = 191 = 191
f(66) = 1489 = 1489
f(67) = 181 = 181
f(68) = 1405 = 5*281
f(69) = 85 = 5*17
f(70) = 1313 = 13*101
f(71) = 79 = 79
f(72) = 1213 = 1213
f(73) = 145 = 5*29
f(74) = 1105 = 5*13*17
f(75) = 131 = 131
f(76) = 989 = 23*43
f(77) = 29 = 29
f(78) = 865 = 5*173
f(79) = 25 = 5*5
f(80) = 733 = 733
f(81) = 83 = 83
f(82) = 593 = 593
f(83) = 65 = 5*13
f(84) = 445 = 5*89
f(85) = 23 = 23
f(86) = 289 = 17*17
f(87) = 13 = 13
f(88) = 125 = 5*5*5
f(89) = 5 = 5
f(90) = 47 = 47
f(91) = 17 = 17
f(92) = 227 = 227
f(93) = 5 = 5
f(94) = 415 = 5*83
f(95) = 1 = 1
f(96) = 611 = 13*47
f(97) = 89 = 89
f(98) = 815 = 5*163
f(99) = 115 = 5*23
f(100) = 1027 = 13*79

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^2-92x+227

f(0)=227
f(1)=17
f(2)=47
f(3)=5
f(4)=1
f(5)=13
f(6)=1
f(7)=23
f(8)=89
f(9)=1
f(10)=593
f(11)=83
f(12)=733
f(13)=1
f(14)=173
f(15)=29
f(16)=43
f(17)=131
f(18)=1
f(19)=1
f(20)=1213
f(21)=79
f(22)=101
f(23)=1
f(24)=281
f(25)=181
f(26)=1489
f(27)=191
f(28)=313
f(29)=1
f(30)=71
f(31)=1
f(32)=1693
f(33)=1
f(34)=349
f(35)=1
f(36)=1789
f(37)=113
f(38)=73
f(39)=1
f(40)=109
f(41)=233
f(42)=1873
f(43)=1
f(44)=1
f(45)=59
f(46)=1889
f(47)=1
f(48)=1
f(49)=1
f(50)=1
f(51)=1
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=1
f(57)=1
f(58)=1
f(59)=1
f(60)=1
f(61)=1
f(62)=1
f(63)=1
f(64)=1
f(65)=1
f(66)=1
f(67)=1
f(68)=1
f(69)=1
f(70)=1
f(71)=1
f(72)=1
f(73)=1
f(74)=1
f(75)=1
f(76)=1
f(77)=1
f(78)=1
f(79)=1
f(80)=1
f(81)=1
f(82)=1
f(83)=1
f(84)=1
f(85)=1
f(86)=1
f(87)=1
f(88)=1
f(89)=1
f(90)=1
f(91)=1
f(92)=1
f(93)=1
f(94)=1
f(95)=1
f(96)=1
f(97)=1
f(98)=163
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-92x+227 could be written as f(y)= y^2-1889 with x=y+46

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-46
f'(x)>2x-93 with x > 43

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

227, 17, 47, 5, 1, 13, 1, 23, 89, 1, 593, 83, 733, 1, 173, 29, 43, 131, 1, 1, 1213, 79, 101, 1, 281, 181, 1489, 191, 313, 1, 71, 1, 1693, 1, 349, 1, 1789, 113, 73, 1, 109, 233, 1873, 1, 1, 59, 1889, 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, 163, 1, 1, 1, 1, 1, 1, 199, 1, 229, 1, 1, 2207, 1, 2467, 1, 547, 359, 3011, 197, 659, 1, 211, 467, 1, 1, 839, 1, 347, 1, 967, 1, 5167, 1, 5507, 1, 1171, 1, 6211, 1, 263, 1, 6947, 223, 431, 1, 1543, 1, 8111, 1039, 1, 1, 1, 571, 719, 239, 1, 1249, 10211, 1, 2131, 1, 383, 1, 269, 1, 1, 1, 12511, 797, 1, 331, 13487, 1, 1, 1, 1, 461, 883, 1, 1, 1, 16067, 1021, 16607, 1, 1, 2179, 1, 1, 1, 1, 401, 1, 19427, 1, 4003, 2539, 20611, 1307, 4243, 1, 1, 2767, 22447, 569, 1, 1, 1, 751, 4871, 617, 1471, 3167, 25667, 1, 1, 1667, 27011, 1, 1, 701, 28387, 449, 1, 1, 1, 3769, 2347, 1, 6247, 1, 2459, 1, 32707, 827, 6691, 4229, 34211, 1, 1399, 1, 35747, 4517, 36527, 1, 439, 2357, 1657, 1, 1, 983, 39727, 1, 3119, 1, 1, 653, 1, 1, 1, 1087, 1, 1, 503, 1, 9127, 443, 46511, 5869, 9479, 1, 1, 1523, 487, 1, 1, 1, 1759, 3217, 1, 1, 1823, 1, 4139, 1, 1, 863, 55711, 1, 2267, 1429, 3391, 1, 2549, 739, 11923, 1, 60611, 7639, 12323, 1553, 62627, 1973, 63647, 1, 1, 1, 2857, 1, 1, 1, 67807, 4271, 4051, 1, 1, 1, 71011, 1, 14419, 1, 1, 709, 74287, 1871, 887, 1, 76511, 4817, 15527, 1, 1, 1, 79907, 1, 1, 2551, 1, 1, 1, 2099, 1433, 1, 1453, 1, 17383, 10939, 1, 853, 17863, 1, 90527, 1, 3989, 2309, 3719, 11699, 7247, 5927, 1123, 1201, 1, 1, 97967, 1, 1, 3121, 100511, 1, 20359, 1, 103087, 12967, 2221, 1, 21139, 1, 947, 1, 1, 1, 6451, 1, 8539, 1, 977, 1, 8747, 1, 4603, 1447, 116447, 7321, 1, 2963, 1, 1153, 4159, 1, 1061, 1, 123427, 1, 124847, 1, 5051, 7937, 127711, 1, 1987, 1, 997, 16417, 10159, 1, 1571, 1049, 1709, 1, 27299, 1, 3209, 1, 139487, 1753, 1, 1, 1, 17909, 28807, 1, 6329, 1, 147107, 3697, 2287, 18679, 150211, 9437, 1, 1907, 1723, 19267, 154927, 1, 1361, 1229, 3677, 1, 1879, 4013, 5563, 1559, 162947, 1, 1, 10337, 2341, 20879, 1, 4217, 1, 5323, 171167, 1, 2659, 1277, 1, 953, 1, 2213, 177887, 11171, 3821, 1, 1, 22769, 1, 1, 36947, 1, 1, 23417, 11071, 1, 37991, 11927, 14747, 12037, 1, 1, 1, 24517, 2699, 1237, 2339, 1, 2417, 25189, 40483, 1, 1, 12821, 206047, 1, 1663, 26099, 4877, 1, 1459, 1, 213407, 1, 1, 1, 43427, 27259, 991, 1, 44179, 1, 222787, 27967, 9769, 5641, 45319, 1, 228511, 1, 2711, 1, 232367, 29167, 5449, 1, 47251, 14827, 10357, 1031, 1, 6029, 242147, 1, 1, 1, 49223, 1, 1093, 31139, 50023, 1, 14831, 1217, 254147, 6379, 10247, 2473, 258211, 8101, 52051, 1, 5581, 32917, 15551, 1327, 4099, 1, 1, 1, 1, 6791, 9403, 34217, 274787, 1, 1, 1, 4729, 2693, 1, 1, 283267, 1367, 12409, 3581, 1, 1, 1, 1, 1, 1831, 22619, 1, 1, 7433, 4591, 1291, 17683, 1, 1, 1, 1597, 2251, 2719, 1, 12379, 1, 311711, 1, 62791, 7877, 1747, 39667, 318467, 1, 2789, 20117, 24847, 40519, 1, 8161, 1, 10273, 4519, 2069, 1, 41669, 334511, 41959, 1, 1, 1, 1, 341507, 1, 68771, 1, 1, 1, 1, 1, 350947, 44017, 27179, 8863, 71143, 22307, 1, 1321, 72103, 9043, 362927, 1979, 21491, 1, 1, 1, 1549, 1601, 74531, 1, 1, 1, 4549, 1, 1, 47659, 382511, 47969, 5923, 1, 1, 3037, 1, 9781, 3413, 1, 3911, 24767, 1, 1, 400067, 1, 13883, 1, 81031, 1, 1, 12781, 16411, 10289, 1319, 51767, 415427, 5209, 1, 1, 7129, 2293, 1, 10613, 1583, 1, 2153, 1, 1, 54049, 1, 1, 1, 5471, 439007, 1, 441667, 1, 88867, 1, 447011, 14011, 89939, 2819, 1, 1, 1, 11411, 7043, 28697, 1, 28867, 92647, 1, 3557, 58417, 1, 1, 94291, 1847, 474211, 1, 19079, 11959, 1, 30071, 482527, 1, 3347, 1, 37547, 1423, 2089, 1, 1, 15473, 21589, 1, 1, 1, 1439, 31477, 1, 1, 507907, 63667, 510767, 1, 2389, 1, 1, 8093, 1, 1, 3019, 3851, 1, 1, 105619, 1, 40847, 1, 106787, 2677, 536867, 16823, 6833, 1, 1487, 5233, 545711, 68399, 1, 1, 12829, 1, 554627, 13903, 1, 1, 6299, 8783, 1, 1, 33331, 1511, 1, 1, 22907, 2111, 575711, 1, 115751, 1, 2609, 1, 1, 1, 1993, 1, 2801, 4357, 118819, 14891, 1721, 1627, 35311, 7523, 9283, 75619, 1, 2621, 1, 1, 6067, 1, 615907, 1, 7283, 3373, 1, 2999, 5003, 1, 628547, 1, 13441, 1, 5521, 19891, 5647, 19991, 2729, 16073, 1, 4751, 647747, 8117, 2003, 40787, 1, 1, 5717, 16477, 22783, 1, 663967, 1, 133447, 83609, 670511, 1, 7927, 8443, 2833, 1, 680387, 1, 136739, 85669, 1, 21521, 1, 1, 53359, 3779, 9817, 17467, 140071, 1, 703711, 44087, 28283, 1, 1, 89017, 713827, 1, 143443, 1, 720611, 90289, 1, 18143, 727427, 1, 3307, 9157, 1, 1, 56747, 5437, 1, 4643, 744607, 1, 6863, 1, 1, 7243, 2281, 47297, 5231, 1, 1, 95467, 765487, 19181, 1, 12043, 2017, 1, 11939, 3889, 1, 1, 1, 9811, 1, 49277, 1, 98999, 31751, 19889, 10093, 1, 17041, 1, 160903, 7753, 808111, 3491, 162343, 10169, 35449, 51071, 818947, 1, 2531, 103049, 826211, 6469, 1, 1, 4231, 104417, 19469, 1, 1, 52667, 844511, 1, 169639, 1, 50111, 8209, 1, 1, 1931, 1, 29759, 108109, 173347, 1, 66959, 54521, 1, 1, 1, 1, 881711, 1, 35419, 1, 889247, 1, 1, 1, 2161, 112339, 39157, 4339, 180883, 11329,

6. Sequence of the polynom (only primes)

227, 17, 47, 5, 13, 23, 89, 593, 83, 733, 173, 29, 43, 131, 1213, 79, 101, 281, 181, 1489, 191, 313, 71, 1693, 349, 1789, 113, 73, 109, 233, 1873, 59, 1889, 163, 199, 229, 2207, 2467, 547, 359, 3011, 197, 659, 211, 467, 839, 347, 967, 5167, 5507, 1171, 6211, 263, 6947, 223, 431, 1543, 8111, 1039, 571, 719, 239, 1249, 10211, 2131, 383, 269, 12511, 797, 331, 13487, 461, 883, 16067, 1021, 16607, 2179, 401, 19427, 4003, 2539, 20611, 1307, 4243, 2767, 22447, 569, 751, 4871, 617, 1471, 3167, 25667, 1667, 27011, 701, 28387, 449, 3769, 2347, 6247, 2459, 32707, 827, 6691, 4229, 34211, 1399, 35747, 4517, 36527, 439, 2357, 1657, 983, 39727, 3119, 653, 1087, 503, 9127, 443, 46511, 5869, 9479, 1523, 487, 1759, 3217, 1823, 4139, 863, 55711, 2267, 1429, 3391, 2549, 739, 11923, 60611, 7639, 12323, 1553, 62627, 1973, 63647, 2857, 67807, 4271, 4051, 71011, 14419, 709, 74287, 1871, 887, 76511, 4817, 15527, 79907, 2551, 2099, 1433, 1453, 17383, 10939, 853, 17863, 90527, 3989, 2309, 3719, 11699, 7247, 5927, 1123, 1201, 97967, 3121, 100511, 20359, 103087, 12967, 2221, 21139, 947, 6451, 8539, 977, 8747, 4603, 1447, 116447, 7321, 2963, 1153, 4159, 1061, 123427, 124847, 5051, 7937, 127711, 1987, 997, 16417, 10159, 1571, 1049, 1709, 27299, 3209, 139487, 1753, 17909, 28807, 6329, 147107, 3697, 2287, 18679, 150211, 9437, 1907, 1723, 19267, 154927, 1361, 1229, 3677, 1879, 4013, 5563, 1559, 162947, 10337, 2341, 20879, 4217, 5323, 171167, 2659, 1277, 953, 2213, 177887, 11171, 3821, 22769, 36947, 23417, 11071, 37991, 11927, 14747, 12037, 24517, 2699, 1237, 2339, 2417, 25189, 40483, 12821, 206047, 1663, 26099, 4877, 1459, 213407, 43427, 27259, 991, 44179, 222787, 27967, 9769, 5641, 45319, 228511, 2711, 232367, 29167, 5449, 47251, 14827, 10357, 1031, 6029, 242147, 49223, 1093, 31139, 50023, 14831, 1217, 254147, 6379, 10247, 2473, 258211, 8101, 52051, 5581, 32917, 15551, 1327, 4099, 6791, 9403, 34217, 274787, 4729, 2693, 283267, 1367, 12409, 3581, 1831, 22619, 7433, 4591, 1291, 17683, 1597, 2251, 2719, 12379, 311711, 62791, 7877, 1747, 39667, 318467, 2789, 20117, 24847, 40519, 8161, 10273, 4519, 2069, 41669, 334511, 41959, 341507, 68771, 350947, 44017, 27179, 8863, 71143, 22307, 1321, 72103, 9043, 362927, 1979, 21491, 1549, 1601, 74531, 4549, 47659, 382511, 47969, 5923, 3037, 9781, 3413, 3911, 24767, 400067, 13883, 81031, 12781, 16411, 10289, 1319, 51767, 415427, 5209, 7129, 2293, 10613, 1583, 2153, 54049, 5471, 439007, 441667, 88867, 447011, 14011, 89939, 2819, 11411, 7043, 28697, 28867, 92647, 3557, 58417, 94291, 1847, 474211, 19079, 11959, 30071, 482527, 3347, 37547, 1423, 2089, 15473, 21589, 1439, 31477, 507907, 63667, 510767, 2389, 8093, 3019, 3851, 105619, 40847, 106787, 2677, 536867, 16823, 6833, 1487, 5233, 545711, 68399, 12829, 554627, 13903, 6299, 8783, 33331, 1511, 22907, 2111, 575711, 115751, 2609, 1993, 2801, 4357, 118819, 14891, 1721, 1627, 35311, 7523, 9283, 75619, 2621, 6067, 615907, 7283, 3373, 2999, 5003, 628547, 13441, 5521, 19891, 5647, 19991, 2729, 16073, 4751, 647747, 8117, 2003, 40787, 5717, 16477, 22783, 663967, 133447, 83609, 670511, 7927, 8443, 2833, 680387, 136739, 85669, 21521, 53359, 3779, 9817, 17467, 140071, 703711, 44087, 28283, 89017, 713827, 143443, 720611, 90289, 18143, 727427, 3307, 9157, 56747, 5437, 4643, 744607, 6863, 7243, 2281, 47297, 5231, 95467, 765487, 19181, 12043, 2017, 11939, 3889, 9811, 49277, 98999, 31751, 19889, 10093, 17041, 160903, 7753, 808111, 3491, 162343, 10169, 35449, 51071, 818947, 2531, 103049, 826211, 6469, 4231, 104417, 19469, 52667, 844511, 169639, 50111, 8209, 1931, 29759, 108109, 173347, 66959, 54521, 881711, 35419, 889247, 2161, 112339, 39157, 4339, 180883, 11329,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-92x+227 and
the reducible primes which appear as divisor for the first time
p | x^2-92x+227 and p < x^2-92x+227

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 4 2 2 1 0.5 0.5
3 8 7 2 5 0.875 0.25 0.625
4 16 13 4 9 0.8125 0.25 0.5625
5 32 24 7 17 0.75 0.21875 0.53125
6 64 33 10 23 0.515625 0.15625 0.359375
7 128 48 13 35 0.375 0.1015625 0.2734375
8 256 118 34 84 0.4609375 0.1328125 0.328125
9 512 270 61 209 0.52734375 0.11914063 0.40820313
10 1024 585 110 475 0.57128906 0.10742188 0.46386719
11 2048 1231 187 1044 0.60107422 0.09130859 0.50976563
12 4096 2531 371 2160 0.61791992 0.09057617 0.52734375
13 8192 5160 665 4495 0.62988281 0.08117676 0.54870605
14 16384 10445 1229 9216 0.63751221 0.07501221 0.5625
15 32768 21011 2231 18780 0.64120483 0.06808472 0.57312012
16 65536 42248 4220 38028 0.64465332 0.06439209 0.58026123
17 131072 84894 7876 77018 0.64768982 0.06008911 0.58760071
18 262144 170665 14798 155867 0.65103531 0.05644989 0.59458542
19 524288 342840 27673 315167 0.65391541 0.05278206 0.60113335
20 1048576 687722 52265 635457 0.65586281 0.04984379 0.60601902
21 2097152 1379257 99224 1280033 0.65768099 0.04731369 0.6103673
22 4194304 2765776 188228 2577548 0.65941238 0.04487705 0.61453533
23 8388608 5544300 359042 5185258 0.66093206 0.04280114 0.61813092
24 16777216 11110836 685752 10425084 0.66225743 0.040874 0.62138343


8. Check for existing Integer Sequences by OEIS

Found in Database : 227, 17, 47, 5, 1, 13, 1, 23, 89, 1, 593, 83, 733, 1, 173, 29, 43, 131, 1, 1,
Found in Database : 227, 17, 47, 5, 13, 23, 89, 593, 83, 733, 173, 29, 43, 131, 1213, 79, 101, 281, 181, 1489, 191, 313, 71, 1693, 349, 1789, 113, 73,
Found in Database : 5, 13, 17, 23, 29, 43, 47, 59, 71, 73, 79, 83, 89, 101, 109, 113, 131,