Inhaltsverzeichnis

Development of
Algorithmic Constructions

18:31:38
Deutsch
28.Mar 2024

Polynom = x^2+223

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) = 223 = 223
f(1) = 7 = 7
f(2) = 227 = 227
f(3) = 29 = 29
f(4) = 239 = 239
f(5) = 31 = 31
f(6) = 259 = 7*37
f(7) = 17 = 17
f(8) = 287 = 7*41
f(9) = 19 = 19
f(10) = 323 = 17*19
f(11) = 43 = 43
f(12) = 367 = 367
f(13) = 49 = 7*7
f(14) = 419 = 419
f(15) = 7 = 7
f(16) = 479 = 479
f(17) = 1 = 1
f(18) = 547 = 547
f(19) = 73 = 73
f(20) = 623 = 7*89
f(21) = 83 = 83
f(22) = 707 = 7*101
f(23) = 47 = 47
f(24) = 799 = 17*47
f(25) = 53 = 53
f(26) = 899 = 29*31
f(27) = 119 = 7*17
f(28) = 1007 = 19*53
f(29) = 133 = 7*19
f(30) = 1123 = 1123
f(31) = 37 = 37
f(32) = 1247 = 29*43
f(33) = 41 = 41
f(34) = 1379 = 7*197
f(35) = 181 = 181
f(36) = 1519 = 7*7*31
f(37) = 199 = 199
f(38) = 1667 = 1667
f(39) = 109 = 109
f(40) = 1823 = 1823
f(41) = 119 = 7*17
f(42) = 1987 = 1987
f(43) = 259 = 7*37
f(44) = 2159 = 17*127
f(45) = 281 = 281
f(46) = 2339 = 2339
f(47) = 19 = 19
f(48) = 2527 = 7*19*19
f(49) = 41 = 41
f(50) = 2723 = 7*389
f(51) = 353 = 353
f(52) = 2927 = 2927
f(53) = 379 = 379
f(54) = 3139 = 43*73
f(55) = 203 = 7*29
f(56) = 3359 = 3359
f(57) = 217 = 7*31
f(58) = 3587 = 17*211
f(59) = 463 = 463
f(60) = 3823 = 3823
f(61) = 493 = 17*29
f(62) = 4067 = 7*7*83
f(63) = 131 = 131
f(64) = 4319 = 7*617
f(65) = 139 = 139
f(66) = 4579 = 19*241
f(67) = 589 = 19*31
f(68) = 4847 = 37*131
f(69) = 623 = 7*89
f(70) = 5123 = 47*109
f(71) = 329 = 7*47
f(72) = 5407 = 5407
f(73) = 347 = 347
f(74) = 5699 = 41*139
f(75) = 731 = 17*43
f(76) = 5999 = 7*857
f(77) = 769 = 769
f(78) = 6307 = 7*17*53
f(79) = 101 = 101
f(80) = 6623 = 37*179
f(81) = 53 = 53
f(82) = 6947 = 6947
f(83) = 889 = 7*127
f(84) = 7279 = 29*251
f(85) = 931 = 7*7*19
f(86) = 7619 = 19*401
f(87) = 487 = 487
f(88) = 7967 = 31*257
f(89) = 509 = 509
f(90) = 8323 = 7*29*41
f(91) = 1063 = 1063
f(92) = 8687 = 7*17*73
f(93) = 1109 = 1109
f(94) = 9059 = 9059
f(95) = 289 = 17*17
f(96) = 9439 = 9439
f(97) = 301 = 7*43
f(98) = 9827 = 31*317
f(99) = 1253 = 7*179
f(100) = 10223 = 10223

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+223

f(0)=223
f(1)=7
f(2)=227
f(3)=29
f(4)=239
f(5)=31
f(6)=37
f(7)=17
f(8)=41
f(9)=19
f(10)=1
f(11)=43
f(12)=367
f(13)=1
f(14)=419
f(15)=1
f(16)=479
f(17)=1
f(18)=547
f(19)=73
f(20)=89
f(21)=83
f(22)=101
f(23)=47
f(24)=1
f(25)=53
f(26)=1
f(27)=1
f(28)=1
f(29)=1
f(30)=1123
f(31)=1
f(32)=1
f(33)=1
f(34)=197
f(35)=181
f(36)=1
f(37)=199
f(38)=1667
f(39)=109
f(40)=1823
f(41)=1
f(42)=1987
f(43)=1
f(44)=127
f(45)=281
f(46)=2339
f(47)=1
f(48)=1
f(49)=1
f(50)=389
f(51)=353
f(52)=2927
f(53)=379
f(54)=1
f(55)=1
f(56)=3359
f(57)=1
f(58)=211
f(59)=463
f(60)=3823
f(61)=1
f(62)=1
f(63)=131
f(64)=617
f(65)=139
f(66)=241
f(67)=1
f(68)=1
f(69)=1
f(70)=1
f(71)=1
f(72)=5407
f(73)=347
f(74)=1
f(75)=1
f(76)=857
f(77)=769
f(78)=1
f(79)=1
f(80)=179
f(81)=1
f(82)=6947
f(83)=1
f(84)=251
f(85)=1
f(86)=401
f(87)=487
f(88)=257
f(89)=509
f(90)=1
f(91)=1063
f(92)=1
f(93)=1109
f(94)=9059
f(95)=1
f(96)=9439
f(97)=1
f(98)=317
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+223 could be written as f(y)= y^2+223 with x=y+0

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+0
f'(x)>2x-1 with x > 15

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

223, 7, 227, 29, 239, 31, 37, 17, 41, 19, 1, 43, 367, 1, 419, 1, 479, 1, 547, 73, 89, 83, 101, 47, 1, 53, 1, 1, 1, 1, 1123, 1, 1, 1, 197, 181, 1, 199, 1667, 109, 1823, 1, 1987, 1, 127, 281, 2339, 1, 1, 1, 389, 353, 2927, 379, 1, 1, 3359, 1, 211, 463, 3823, 1, 1, 131, 617, 139, 241, 1, 1, 1, 1, 1, 5407, 347, 1, 1, 857, 769, 1, 1, 179, 1, 6947, 1, 251, 1, 401, 487, 257, 509, 1, 1063, 1, 1109, 9059, 1, 9439, 1, 317, 1, 10223, 1303, 10627, 677, 1, 1, 1637, 1459, 11887, 1, 12323, 1, 751, 1, 13219, 1, 13679, 1, 1, 1, 2089, 929, 15107, 1, 821, 283, 947, 1, 16607, 1, 17123, 1, 2521, 2239, 1, 1153, 18719, 1187, 19267, 349, 461, 359, 1, 1, 20959, 1, 1, 2729, 1, 2803, 733, 1439, 23327, 1, 647, 433, 599, 3109, 1, 797, 1, 1, 1, 1, 577, 1, 27779, 1, 28447, 1, 29123, 1, 727, 3769, 4357, 1, 4457, 1, 31907, 1, 1, 1, 33347, 1, 643, 2153, 34819, 1, 5081, 4493, 5189, 1, 37087, 1171, 1, 683, 38639, 1, 443, 1, 1, 2539, 5861, 5179, 1, 5281, 1471, 673, 43487, 1, 1, 1, 1, 1, 2707, 2903, 1, 2957, 1, 1, 48623, 6133, 1597, 1, 499, 1, 1193, 6469, 1, 1, 7589, 1, 1103, 3407, 54979, 1, 1, 1, 1, 1, 1091, 911, 58787, 1, 8537, 1, 8677, 1, 3631, 3889, 62723, 1129, 63727, 1, 1579, 2039, 3461, 1, 1, 1, 9689, 8543, 4051, 4337, 2411, 1, 70979, 1277, 72047, 1, 881, 1151, 10601, 1, 1, 1, 4021, 9619, 1, 1, 78623, 1, 4691, 10039, 2609, 10181, 11717, 1, 1, 2617, 1, 10613, 85487, 1, 4561, 1, 5167, 5527, 701, 659, 12889, 11353, 1, 719, 92639, 1, 1997, 1, 95087, 1709, 96323, 1, 2269, 1, 743, 1, 1, 12589, 101347, 3187, 2503, 1, 3583, 1867, 105199, 1, 1, 1, 15401, 6779, 1, 13723, 5813, 1, 111779, 1, 1, 1, 114467, 1, 115823, 14563, 16741, 1, 16937, 1, 1, 887, 121327, 2179, 7219, 1, 1, 1, 125539, 1, 2591, 15959, 18341, 8069, 1, 1, 131267, 2357, 1, 2383, 1231, 1, 135647, 2131, 1031, 907, 19801, 17419, 4831, 8803, 1, 1, 143107, 1, 4987, 1069, 1447, 4591, 1, 4639, 21317, 18749, 150767, 997, 8017, 1367, 153887, 1381, 1873, 19531, 2963, 1, 1, 1, 1, 1, 3947, 1, 1, 1, 165059, 1481, 1, 1, 168323, 21143, 24281, 1, 24517, 1, 1, 5441, 1, 1, 1, 3169, 178307, 11197, 179999, 1, 1, 1201, 1, 1, 185123, 1453, 1, 1, 188579, 1, 2293, 23899, 1, 1, 27689, 1, 27941, 1, 197359, 24781, 1, 1, 200927, 1, 5479, 25453, 1, 25679, 4211, 12953, 1, 1, 209987, 1, 6833, 1, 4969, 1, 1697, 1, 1, 941, 31321, 1619, 1019, 13879, 223007, 13997, 1, 1, 226799, 1, 228707, 7177, 1093, 7237, 1, 1, 1, 1549, 13907, 1, 1, 2137, 8287, 1, 3319, 1, 5197, 1, 1213, 3863, 35461, 31153, 1, 1013, 4759, 1, 13381, 1, 2351, 32159, 1427, 32413, 37189, 8167, 1, 8231, 2971, 33181, 266479, 1, 268547, 1, 15919, 16979, 1, 1801, 1061, 1, 5651, 1, 279007, 1, 3851, 5039, 9137, 5077, 16787, 17903, 1, 1, 41381, 36343, 1, 1, 15473, 9221, 296159, 1327, 298339, 5347, 1, 1, 302723, 1117, 1, 1, 1, 1, 309359, 1, 5879, 1, 1, 1, 316067, 1279, 318319, 1, 1, 20107, 2713, 20249, 7561, 2399, 327407, 5867, 1, 1, 331999, 1, 1847, 2207, 2531, 42223, 1181, 1, 1, 1259, 1, 1, 1, 6199, 348323, 1, 2677, 2749, 1627, 44281, 50777, 44579, 1, 1, 360223, 1, 1, 1, 365039, 2693, 12671, 1, 1, 11597, 53189, 1, 12923, 1, 377219, 1, 379679, 1, 20113, 2819, 384623, 1, 3253, 1, 7951, 1, 392099, 49169, 3907, 7069, 2857, 3557, 399647, 25057, 1, 1, 1, 2671, 1, 1, 409823, 1, 412387, 1, 414959, 7433, 13469, 26177, 10247, 26339, 8627, 53003, 60761, 3137, 427939, 1, 1, 1, 433187, 7759, 11779, 1, 438467, 27487, 1, 27653, 63397, 55639, 446447, 1, 1, 2011, 12211, 1, 1, 2999, 6263, 1, 65701, 1, 66089, 1, 9901, 1, 468079, 1, 470819, 1, 1, 1, 28019, 1, 1, 1, 3623, 30203, 484639, 1, 487427, 1, 490223, 8779, 493027, 15451, 29167, 1, 71237, 3677, 2311, 1699, 504323, 1, 26693, 1, 9623, 9133, 1, 1, 1, 8081, 1723, 1, 1, 1, 1, 65731, 6353, 4721, 1, 1, 13003, 3517, 1, 67189, 2081, 16889, 77417, 16981, 32051, 2203, 547823, 1, 12809, 4931, 553759, 34703, 1, 69779, 11423, 1, 4231, 1, 565727, 8863, 2887, 1, 571759, 1, 33811, 1, 4157, 36209, 82981, 1693, 83417, 73181, 586979, 1, 7109, 1, 1, 1, 2063, 74719, 1, 1, 1831, 37747, 86501, 75883, 1, 1, 1, 1, 8423, 1, 1, 1889, 32693, 1, 89189, 39119, 5273, 39317, 630659, 79031, 3541, 1621, 2267, 2851, 640223, 1, 5903, 80629, 92377, 81031, 1, 2143, 1, 1, 656323, 1, 16087, 11807, 662819, 1, 666079, 10433, 95621, 83873, 1, 1, 1, 1, 1, 6079, 2113, 1, 1, 85933, 6823, 21587, 3191, 1, 99397, 87181, 699119, 1, 24223, 6287, 41519, 6317, 22877, 5227, 24571, 1, 1, 11213, 102761, 1, 722723, 1, 1733, 1, 5569, 6529, 732959, 1, 736387, 1, 6217, 92693, 106181, 1, 39301, 1, 750179, 1, 753647, 1, 1, 1, 760607, 1, 6421, 95731, 2069, 5657, 1, 3019, 1, 1, 21031, 1, 41141, 97931, 785219, 1, 1, 49409, 113189, 5839, 1, 99709, 1, 1, 27691, 3593, 3347, 1, 810223, 101503, 1, 2683, 3767, 51203, 3617, 102859, 1, 14759, 1, 1, 831967, 3257, 19433, 104681, 2551, 2237, 17203, 1, 3373, 2791, 44753, 15217, 853999, 1, 16183, 26861, 50671, 1, 1, 1, 124121, 1, 10513, 1, 2441, 7841, 880067, 15749, 1, 5827, 1, 6949, 18191, 1, 127877, 112129, 898927, 112603, 902723, 1, 3793, 8111, 31391, 114031, 1, 2663, 1, 1, 1, 28867, 3203, 2467, 19777, 16633, 30109, 1, 1, 58699, 941123, 117883, 1, 118369, 1, 1, 1, 7459, 1, 1, 7333, 17189, 18199, 2083, 968479, 1, 138917, 3929, 1, 4217, 1, 30697, 24007, 1, 5521, 17683, 58367, 1, 52433, 1,

6. Sequence of the polynom (only primes)

223, 7, 227, 29, 239, 31, 37, 17, 41, 19, 43, 367, 419, 479, 547, 73, 89, 83, 101, 47, 53, 1123, 197, 181, 199, 1667, 109, 1823, 1987, 127, 281, 2339, 389, 353, 2927, 379, 3359, 211, 463, 3823, 131, 617, 139, 241, 5407, 347, 857, 769, 179, 6947, 251, 401, 487, 257, 509, 1063, 1109, 9059, 9439, 317, 10223, 1303, 10627, 677, 1637, 1459, 11887, 12323, 751, 13219, 13679, 2089, 929, 15107, 821, 283, 947, 16607, 17123, 2521, 2239, 1153, 18719, 1187, 19267, 349, 461, 359, 20959, 2729, 2803, 733, 1439, 23327, 647, 433, 599, 3109, 797, 577, 27779, 28447, 29123, 727, 3769, 4357, 4457, 31907, 33347, 643, 2153, 34819, 5081, 4493, 5189, 37087, 1171, 683, 38639, 443, 2539, 5861, 5179, 5281, 1471, 673, 43487, 2707, 2903, 2957, 48623, 6133, 1597, 499, 1193, 6469, 7589, 1103, 3407, 54979, 1091, 911, 58787, 8537, 8677, 3631, 3889, 62723, 1129, 63727, 1579, 2039, 3461, 9689, 8543, 4051, 4337, 2411, 70979, 1277, 72047, 881, 1151, 10601, 4021, 9619, 78623, 4691, 10039, 2609, 10181, 11717, 2617, 10613, 85487, 4561, 5167, 5527, 701, 659, 12889, 11353, 719, 92639, 1997, 95087, 1709, 96323, 2269, 743, 12589, 101347, 3187, 2503, 3583, 1867, 105199, 15401, 6779, 13723, 5813, 111779, 114467, 115823, 14563, 16741, 16937, 887, 121327, 2179, 7219, 125539, 2591, 15959, 18341, 8069, 131267, 2357, 2383, 1231, 135647, 2131, 1031, 907, 19801, 17419, 4831, 8803, 143107, 4987, 1069, 1447, 4591, 4639, 21317, 18749, 150767, 997, 8017, 1367, 153887, 1381, 1873, 19531, 2963, 3947, 165059, 1481, 168323, 21143, 24281, 24517, 5441, 3169, 178307, 11197, 179999, 1201, 185123, 1453, 188579, 2293, 23899, 27689, 27941, 197359, 24781, 200927, 5479, 25453, 25679, 4211, 12953, 209987, 6833, 4969, 1697, 941, 31321, 1619, 1019, 13879, 223007, 13997, 226799, 228707, 7177, 1093, 7237, 1549, 13907, 2137, 8287, 3319, 5197, 1213, 3863, 35461, 31153, 1013, 4759, 13381, 2351, 32159, 1427, 32413, 37189, 8167, 8231, 2971, 33181, 266479, 268547, 15919, 16979, 1801, 1061, 5651, 279007, 3851, 5039, 9137, 5077, 16787, 17903, 41381, 36343, 15473, 9221, 296159, 1327, 298339, 5347, 302723, 1117, 309359, 5879, 316067, 1279, 318319, 20107, 2713, 20249, 7561, 2399, 327407, 5867, 331999, 1847, 2207, 2531, 42223, 1181, 1259, 6199, 348323, 2677, 2749, 1627, 44281, 50777, 44579, 360223, 365039, 2693, 12671, 11597, 53189, 12923, 377219, 379679, 20113, 2819, 384623, 3253, 7951, 392099, 49169, 3907, 7069, 2857, 3557, 399647, 25057, 2671, 409823, 412387, 414959, 7433, 13469, 26177, 10247, 26339, 8627, 53003, 60761, 3137, 427939, 433187, 7759, 11779, 438467, 27487, 27653, 63397, 55639, 446447, 2011, 12211, 2999, 6263, 65701, 66089, 9901, 468079, 470819, 28019, 3623, 30203, 484639, 487427, 490223, 8779, 493027, 15451, 29167, 71237, 3677, 2311, 1699, 504323, 26693, 9623, 9133, 8081, 1723, 65731, 6353, 4721, 13003, 3517, 67189, 2081, 16889, 77417, 16981, 32051, 2203, 547823, 12809, 4931, 553759, 34703, 69779, 11423, 4231, 565727, 8863, 2887, 571759, 33811, 4157, 36209, 82981, 1693, 83417, 73181, 586979, 7109, 2063, 74719, 1831, 37747, 86501, 75883, 8423, 1889, 32693, 89189, 39119, 5273, 39317, 630659, 79031, 3541, 1621, 2267, 2851, 640223, 5903, 80629, 92377, 81031, 2143, 656323, 16087, 11807, 662819, 666079, 10433, 95621, 83873, 6079, 2113, 85933, 6823, 21587, 3191, 99397, 87181, 699119, 24223, 6287, 41519, 6317, 22877, 5227, 24571, 11213, 102761, 722723, 1733, 5569, 6529, 732959, 736387, 6217, 92693, 106181, 39301, 750179, 753647, 760607, 6421, 95731, 2069, 5657, 3019, 21031, 41141, 97931, 785219, 49409, 113189, 5839, 99709, 27691, 3593, 3347, 810223, 101503, 2683, 3767, 51203, 3617, 102859, 14759, 831967, 3257, 19433, 104681, 2551, 2237, 17203, 3373, 2791, 44753, 15217, 853999, 16183, 26861, 50671, 124121, 10513, 2441, 7841, 880067, 15749, 5827, 6949, 18191, 127877, 112129, 898927, 112603, 902723, 3793, 8111, 31391, 114031, 2663, 28867, 3203, 2467, 19777, 16633, 30109, 58699, 941123, 117883, 118369, 7459, 7333, 17189, 18199, 2083, 968479, 138917, 3929, 4217, 30697, 24007, 5521, 17683, 58367, 52433,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2x+223 and
the reducible primes which appear as divisor for the first time
p | x^2x+223 and p < x^2x+223

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 : 223, 7, 227, 29, 239, 31, 37, 17, 41, 19, 1, 43, 367, 1, 419, 1, 479, 1, 547, 73,
Found in Database : 223, 7, 227, 29, 239, 31, 37, 17, 41, 19, 43, 367, 419, 479, 547, 73, 89, 83, 101, 47, 53, 1123, 197, 181, 199, 1667, 109,
Found in Database : 7, 17, 19, 29, 31, 37, 41, 43, 47, 53, 73, 83, 89, 101, 109, 127, 131, 139,