Inhaltsverzeichnis

Development of
Algorithmic Constructions

22:18:09
Deutsch
28.Mar 2024

Polynom = x^2-238x+557

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) = 557 = 557
f(1) = 5 = 5
f(2) = 85 = 5*17
f(3) = 37 = 37
f(4) = 379 = 379
f(5) = 19 = 19
f(6) = 835 = 5*167
f(7) = 265 = 5*53
f(8) = 1283 = 1283
f(9) = 47 = 47
f(10) = 1723 = 1723
f(11) = 485 = 5*97
f(12) = 2155 = 5*431
f(13) = 37 = 37
f(14) = 2579 = 2579
f(15) = 697 = 17*41
f(16) = 2995 = 5*599
f(17) = 25 = 5*5
f(18) = 3403 = 41*83
f(19) = 901 = 17*53
f(20) = 3803 = 3803
f(21) = 125 = 5*5*5
f(22) = 4195 = 5*839
f(23) = 1097 = 1097
f(24) = 4579 = 19*241
f(25) = 149 = 149
f(26) = 4955 = 5*991
f(27) = 1285 = 5*257
f(28) = 5323 = 5323
f(29) = 43 = 43
f(30) = 5683 = 5683
f(31) = 1465 = 5*293
f(32) = 6035 = 5*17*71
f(33) = 97 = 97
f(34) = 6379 = 6379
f(35) = 1637 = 1637
f(36) = 6715 = 5*17*79
f(37) = 215 = 5*43
f(38) = 7043 = 7043
f(39) = 1801 = 1801
f(40) = 7363 = 37*199
f(41) = 235 = 5*47
f(42) = 7675 = 5*5*307
f(43) = 1957 = 19*103
f(44) = 7979 = 79*101
f(45) = 127 = 127
f(46) = 8275 = 5*5*331
f(47) = 2105 = 5*421
f(48) = 8563 = 8563
f(49) = 17 = 17
f(50) = 8843 = 37*239
f(51) = 2245 = 5*449
f(52) = 9115 = 5*1823
f(53) = 289 = 17*17
f(54) = 9379 = 83*113
f(55) = 2377 = 2377
f(56) = 9635 = 5*41*47
f(57) = 305 = 5*61
f(58) = 9883 = 9883
f(59) = 2501 = 41*61
f(60) = 10123 = 53*191
f(61) = 5 = 5
f(62) = 10355 = 5*19*109
f(63) = 2617 = 2617
f(64) = 10579 = 71*149
f(65) = 167 = 167
f(66) = 10795 = 5*17*127
f(67) = 2725 = 5*5*109
f(68) = 11003 = 11003
f(69) = 347 = 347
f(70) = 11203 = 17*659
f(71) = 2825 = 5*5*113
f(72) = 11395 = 5*43*53
f(73) = 359 = 359
f(74) = 11579 = 11579
f(75) = 2917 = 2917
f(76) = 11755 = 5*2351
f(77) = 185 = 5*37
f(78) = 11923 = 11923
f(79) = 3001 = 3001
f(80) = 12083 = 43*281
f(81) = 95 = 5*19
f(82) = 12235 = 5*2447
f(83) = 3077 = 17*181
f(84) = 12379 = 12379
f(85) = 389 = 389
f(86) = 12515 = 5*2503
f(87) = 3145 = 5*17*37
f(88) = 12643 = 47*269
f(89) = 397 = 397
f(90) = 12763 = 12763
f(91) = 3205 = 5*641
f(92) = 12875 = 5*5*5*103
f(93) = 101 = 101
f(94) = 12979 = 12979
f(95) = 3257 = 3257
f(96) = 13075 = 5*5*523
f(97) = 205 = 5*41
f(98) = 13163 = 13163
f(99) = 3301 = 3301
f(100) = 13243 = 17*19*41

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^2-238x+557

f(0)=557
f(1)=5
f(2)=17
f(3)=37
f(4)=379
f(5)=19
f(6)=167
f(7)=53
f(8)=1283
f(9)=47
f(10)=1723
f(11)=97
f(12)=431
f(13)=1
f(14)=2579
f(15)=41
f(16)=599
f(17)=1
f(18)=83
f(19)=1
f(20)=3803
f(21)=1
f(22)=839
f(23)=1097
f(24)=241
f(25)=149
f(26)=991
f(27)=257
f(28)=5323
f(29)=43
f(30)=5683
f(31)=293
f(32)=71
f(33)=1
f(34)=6379
f(35)=1637
f(36)=79
f(37)=1
f(38)=7043
f(39)=1801
f(40)=199
f(41)=1
f(42)=307
f(43)=103
f(44)=101
f(45)=127
f(46)=331
f(47)=421
f(48)=8563
f(49)=1
f(50)=239
f(51)=449
f(52)=1823
f(53)=1
f(54)=113
f(55)=2377
f(56)=1
f(57)=61
f(58)=9883
f(59)=1
f(60)=191
f(61)=1
f(62)=109
f(63)=2617
f(64)=1
f(65)=1
f(66)=1
f(67)=1
f(68)=11003
f(69)=347
f(70)=659
f(71)=1
f(72)=1
f(73)=359
f(74)=11579
f(75)=2917
f(76)=2351
f(77)=1
f(78)=11923
f(79)=3001
f(80)=281
f(81)=1
f(82)=2447
f(83)=181
f(84)=12379
f(85)=389
f(86)=2503
f(87)=1
f(88)=269
f(89)=397
f(90)=12763
f(91)=641
f(92)=1
f(93)=1
f(94)=12979
f(95)=3257
f(96)=523
f(97)=1
f(98)=13163
f(99)=3301

b) Substitution of the polynom
The polynom f(x)=x^2-238x+557 could be written as f(y)= y^2-13604 with x=y+119

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-119
f'(x)>2x-239 with x > 117

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

557, 5, 17, 37, 379, 19, 167, 53, 1283, 47, 1723, 97, 431, 1, 2579, 41, 599, 1, 83, 1, 3803, 1, 839, 1097, 241, 149, 991, 257, 5323, 43, 5683, 293, 71, 1, 6379, 1637, 79, 1, 7043, 1801, 199, 1, 307, 103, 101, 127, 331, 421, 8563, 1, 239, 449, 1823, 1, 113, 2377, 1, 61, 9883, 1, 191, 1, 109, 2617, 1, 1, 1, 1, 11003, 347, 659, 1, 1, 359, 11579, 2917, 2351, 1, 11923, 3001, 281, 1, 2447, 181, 12379, 389, 2503, 1, 269, 397, 12763, 641, 1, 1, 12979, 3257, 523, 1, 13163, 3301, 1, 1, 2663, 1, 787, 419, 2687, 673, 139, 211, 13523, 677, 2711, 1, 367, 1, 2719, 1, 223, 179, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 443, 1, 1, 1, 1, 3037, 1, 3557, 1, 1, 1, 4621, 1223, 1033, 1, 5717, 1499, 6277, 1, 1, 1783, 1, 1, 1601, 1, 8597, 1, 541, 1, 1, 1, 613, 2683, 1, 1, 11677, 2999, 1, 1, 2593, 3323, 1, 1, 2857, 1, 14957, 1, 823, 1, 653, 521, 17021, 1, 709, 1, 1, 1, 19157, 1, 1, 1, 1213, 1, 4273, 1087, 1301, 1, 22877, 1163, 4729, 751, 24421, 6203, 1, 1, 25997, 6599, 1, 1, 5521, 1, 1, 1, 5849, 1483, 1583, 953, 719, 1567, 6353, 503, 32621, 8263, 1, 1, 1, 8699, 1, 1, 1, 1, 37021, 1171, 1, 1, 547, 1, 479, 2011, 1, 643, 41621, 619, 8513, 1, 43517, 647, 563, 1, 1, 11483, 761, 733, 499, 1, 48397, 1, 1051, 1, 593, 1, 51421, 12983, 617, 1, 1009, 13499, 54517, 1, 11113, 1, 1381, 1, 1, 1, 58757, 1, 1129, 3019, 2437, 1, 569, 15643, 1, 1, 64237, 1, 65357, 1, 13297, 16763, 3559, 2131, 809, 3467, 1, 2203, 1, 3583, 14449, 1, 73421, 18503, 1, 1, 75797, 1, 2081, 1, 15641, 1, 1847, 1, 1, 1, 1997, 1289, 83117, 1, 1, 1, 85621, 21563, 17377, 1, 1, 1, 5261, 1, 1, 1, 5413, 1, 3733, 1, 937, 1489, 95957, 4831, 19457, 3061, 98621, 1, 19993, 1, 1427, 1, 102677, 1, 20809, 26183, 2243, 829, 1, 1, 1, 1, 109597, 1103, 1, 3491, 1, 28283, 22769, 1, 6781, 1, 6143, 1, 23633, 29723, 1, 3761, 1, 6091, 122557, 3853, 1747, 1, 5021, 1973, 3433, 1879, 1, 1, 1193, 1721, 1, 827, 1, 1, 1307, 4231, 1, 1, 8101, 1, 1, 1, 1657, 2213, 142421, 35803, 1, 1, 1, 36599, 147197, 1, 29761, 1, 1, 1, 1, 7643, 1, 1, 155317, 1, 31393, 4931, 158621, 39863, 32057, 1, 161957, 40699, 163637, 1, 1, 41543, 167021, 1, 1, 1, 4157, 1, 172157, 1, 1, 1, 175621, 44123, 1867, 1, 1, 2647, 947, 1, 36529, 2699, 1, 5791, 1, 1871, 5081, 5903, 189797, 1907, 38321, 1, 1523, 2557, 2297, 1, 197077, 49499, 11701, 1249, 40153, 50423, 202621, 6361, 40897, 10271, 4799, 1, 3929, 10459, 1, 1, 11159, 1439, 1, 1, 2137, 1, 3067, 1367, 1, 55163, 221621, 1, 1, 1, 3697, 3539, 13381, 11423, 1, 1, 13613, 1, 46681, 1, 235397, 1, 5051, 1, 47881, 60103, 241421, 1, 1, 1, 245477, 7703, 6037, 1, 1, 1, 251621, 1, 1, 1, 255757, 1493, 257837, 1, 1, 1231, 15413, 8221, 2113, 13259, 15661, 8353, 2657, 1, 1151, 4243, 272621, 1291, 1, 1, 276917, 69499, 1873, 1, 56249, 70583, 2039, 1, 1, 1, 3643, 1, 15263, 1, 58441, 4583, 1, 73883, 1, 1861, 17581, 2027, 2371, 1889, 1, 76123, 305621, 4793, 1, 15451, 1153, 1, 3221, 15679, 12589, 9871, 317021, 4679, 1, 2003, 1777, 1, 323957, 1, 65257, 1, 328621, 5153, 1789, 16607, 1, 10453, 3259, 16843, 1, 10601, 7243, 2083, 1, 1, 4159, 86599, 1237, 1, 70001, 87803, 352421, 1, 70969, 1, 5857, 1, 5897, 18047, 1, 1, 364621, 91463, 73417, 1, 369557, 92699, 372037, 2333, 1, 2539, 377021, 11821, 1, 1, 382037, 1, 22621, 1, 77417, 1, 389621, 1, 1913, 2459, 394717, 98999, 9239, 1, 1, 1, 3907, 1, 81001, 1, 9479, 6389, 2753, 1, 82561, 12941, 5851, 104183, 83609, 2621, 420677, 105499, 1, 1327, 85193, 106823, 1, 3359, 86257, 1, 1, 1, 11801, 1, 17573, 1, 10781, 1, 17789, 1, 1741, 112199, 4457, 1, 1709, 1, 1, 14281, 91673, 1, 9811, 1, 1511, 1, 93329, 1, 1, 117703, 94441, 1, 27941, 119099, 477797, 1, 5059, 120503, 2531, 1, 1, 1, 3851, 1, 4513, 1451, 98953, 7753, 6299, 1, 1, 3137, 12277, 126199, 10771, 1, 4073, 127643, 512021, 1, 1, 25819, 1, 4057, 8537, 26111, 1, 16411, 1, 1, 105913, 3319, 1, 133499, 28183, 1, 1, 1, 14633, 1, 108881, 1, 547397, 1, 1, 5519, 110681, 17341, 1, 139483, 2729, 1753, 562477, 1, 565517, 1, 6689, 1, 1, 17911, 6761, 1, 577757, 1, 580837, 1, 23357, 2287, 587021, 1, 4721, 1, 31223, 8747, 3571, 1, 2551, 8839, 602621, 1, 1, 30367, 11489, 9539, 2089, 1, 123049, 1, 618421, 155003, 1, 1, 624797, 1, 1, 1, 126241, 1, 634421, 4969, 127529, 31963, 17321, 10039, 644117, 1, 3011, 1, 1, 163063, 6883, 1, 17761, 1, 15359, 2069, 1, 1, 667021, 1, 26813, 33599, 673637, 1, 39821, 33931, 136057, 1, 40213, 1, 137393, 1, 1, 172999, 693677, 1, 139409, 1, 5039, 1, 140761, 1, 707197, 22153, 6899, 1, 2341, 1, 1, 1, 144169, 1, 724277, 181499, 727717, 1, 1, 4261, 1, 23011, 147617, 1, 1, 5807, 745037, 37339, 1, 1, 3779, 188443, 1, 4733, 759037, 4639,

6. Sequence of the polynom (only primes)

557, 5, 17, 37, 379, 19, 167, 53, 1283, 47, 1723, 97, 431, 2579, 41, 599, 83, 3803, 839, 1097, 241, 149, 991, 257, 5323, 43, 5683, 293, 71, 6379, 1637, 79, 7043, 1801, 199, 307, 103, 101, 127, 331, 421, 8563, 239, 449, 1823, 113, 2377, 61, 9883, 191, 109, 2617, 11003, 347, 659, 359, 11579, 2917, 2351, 11923, 3001, 281, 2447, 181, 12379, 389, 2503, 269, 397, 12763, 641, 12979, 3257, 523, 13163, 3301, 2663, 787, 419, 2687, 673, 139, 211, 13523, 677, 2711, 367, 2719, 223, 179, 443, 3037, 3557, 4621, 1223, 1033, 5717, 1499, 6277, 1783, 1601, 8597, 541, 613, 2683, 11677, 2999, 2593, 3323, 2857, 14957, 823, 653, 521, 17021, 709, 19157, 1213, 4273, 1087, 1301, 22877, 1163, 4729, 751, 24421, 6203, 25997, 6599, 5521, 5849, 1483, 1583, 953, 719, 1567, 6353, 503, 32621, 8263, 8699, 37021, 1171, 547, 479, 2011, 643, 41621, 619, 8513, 43517, 647, 563, 11483, 761, 733, 499, 48397, 1051, 593, 51421, 12983, 617, 1009, 13499, 54517, 11113, 1381, 58757, 1129, 3019, 2437, 569, 15643, 64237, 65357, 13297, 16763, 3559, 2131, 809, 3467, 2203, 3583, 14449, 73421, 18503, 75797, 2081, 15641, 1847, 1997, 1289, 83117, 85621, 21563, 17377, 5261, 5413, 3733, 937, 1489, 95957, 4831, 19457, 3061, 98621, 19993, 1427, 102677, 20809, 26183, 2243, 829, 109597, 1103, 3491, 28283, 22769, 6781, 6143, 23633, 29723, 3761, 6091, 122557, 3853, 1747, 5021, 1973, 3433, 1879, 1193, 1721, 827, 1307, 4231, 8101, 1657, 2213, 142421, 35803, 36599, 147197, 29761, 7643, 155317, 31393, 4931, 158621, 39863, 32057, 161957, 40699, 163637, 41543, 167021, 4157, 172157, 175621, 44123, 1867, 2647, 947, 36529, 2699, 5791, 1871, 5081, 5903, 189797, 1907, 38321, 1523, 2557, 2297, 197077, 49499, 11701, 1249, 40153, 50423, 202621, 6361, 40897, 10271, 4799, 3929, 10459, 11159, 1439, 2137, 3067, 1367, 55163, 221621, 3697, 3539, 13381, 11423, 13613, 46681, 235397, 5051, 47881, 60103, 241421, 245477, 7703, 6037, 251621, 255757, 1493, 257837, 1231, 15413, 8221, 2113, 13259, 15661, 8353, 2657, 1151, 4243, 272621, 1291, 276917, 69499, 1873, 56249, 70583, 2039, 3643, 15263, 58441, 4583, 73883, 1861, 17581, 2027, 2371, 1889, 76123, 305621, 4793, 15451, 1153, 3221, 15679, 12589, 9871, 317021, 4679, 2003, 1777, 323957, 65257, 328621, 5153, 1789, 16607, 10453, 3259, 16843, 10601, 7243, 2083, 4159, 86599, 1237, 70001, 87803, 352421, 70969, 5857, 5897, 18047, 364621, 91463, 73417, 369557, 92699, 372037, 2333, 2539, 377021, 11821, 382037, 22621, 77417, 389621, 1913, 2459, 394717, 98999, 9239, 3907, 81001, 9479, 6389, 2753, 82561, 12941, 5851, 104183, 83609, 2621, 420677, 105499, 1327, 85193, 106823, 3359, 86257, 11801, 17573, 10781, 17789, 1741, 112199, 4457, 1709, 14281, 91673, 9811, 1511, 93329, 117703, 94441, 27941, 119099, 477797, 5059, 120503, 2531, 3851, 4513, 1451, 98953, 7753, 6299, 3137, 12277, 126199, 10771, 4073, 127643, 512021, 25819, 4057, 8537, 26111, 16411, 105913, 3319, 133499, 28183, 14633, 108881, 547397, 5519, 110681, 17341, 139483, 2729, 1753, 562477, 565517, 6689, 17911, 6761, 577757, 580837, 23357, 2287, 587021, 4721, 31223, 8747, 3571, 2551, 8839, 602621, 30367, 11489, 9539, 2089, 123049, 618421, 155003, 624797, 126241, 634421, 4969, 127529, 31963, 17321, 10039, 644117, 3011, 163063, 6883, 17761, 15359, 2069, 667021, 26813, 33599, 673637, 39821, 33931, 136057, 40213, 137393, 172999, 693677, 139409, 5039, 140761, 707197, 22153, 6899, 2341, 144169, 724277, 181499, 727717, 4261, 23011, 147617, 5807, 745037, 37339, 3779, 188443, 4733, 759037, 4639,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-238x+557 and
the reducible primes which appear as divisor for the first time
p | x^2-238x+557 and p < x^2-238x+557

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 : 557, 5, 17, 37, 379, 19, 167, 53, 1283, 47, 1723, 97, 431, 1, 2579, 41, 599, 1, 83, 1,
Found in Database : 557, 5, 17, 37, 379, 19, 167, 53, 1283, 47, 1723, 97, 431, 2579, 41, 599, 83, 3803, 839, 1097, 241, 149, 991, 257, 5323, 43, 5683, 293, 71, 6379, 1637, 79, 7043, 1801,
Found in Database : 5, 17, 19, 37, 41, 43, 47, 53, 61, 71, 79, 83, 97, 101, 103, 109, 113, 127, 139, 149,