Inhaltsverzeichnis

Development of
Algorithmic Constructions

17:30:01
Deutsch
28.Mar 2024

Polynom = 2x^2+1

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) = 1 = 1
f(1) = 3 = 3
f(2) = 9 = 3*3
f(3) = 19 = 19
f(4) = 33 = 3*11
f(5) = 51 = 3*17
f(6) = 73 = 73
f(7) = 99 = 3*3*11
f(8) = 129 = 3*43
f(9) = 163 = 163
f(10) = 201 = 3*67
f(11) = 243 = 3*3*3*3*3
f(12) = 289 = 17*17
f(13) = 339 = 3*113
f(14) = 393 = 3*131
f(15) = 451 = 11*41
f(16) = 513 = 3*3*3*19
f(17) = 579 = 3*193
f(18) = 649 = 11*59
f(19) = 723 = 3*241
f(20) = 801 = 3*3*89
f(21) = 883 = 883
f(22) = 969 = 3*17*19
f(23) = 1059 = 3*353
f(24) = 1153 = 1153
f(25) = 1251 = 3*3*139
f(26) = 1353 = 3*11*41
f(27) = 1459 = 1459
f(28) = 1569 = 3*523
f(29) = 1683 = 3*3*11*17
f(30) = 1801 = 1801
f(31) = 1923 = 3*641
f(32) = 2049 = 3*683
f(33) = 2179 = 2179
f(34) = 2313 = 3*3*257
f(35) = 2451 = 3*19*43
f(36) = 2593 = 2593
f(37) = 2739 = 3*11*83
f(38) = 2889 = 3*3*3*107
f(39) = 3043 = 17*179
f(40) = 3201 = 3*11*97
f(41) = 3363 = 3*19*59
f(42) = 3529 = 3529
f(43) = 3699 = 3*3*3*137
f(44) = 3873 = 3*1291
f(45) = 4051 = 4051
f(46) = 4233 = 3*17*83
f(47) = 4419 = 3*3*491
f(48) = 4609 = 11*419
f(49) = 4803 = 3*1601
f(50) = 5001 = 3*1667
f(51) = 5203 = 11*11*43
f(52) = 5409 = 3*3*601
f(53) = 5619 = 3*1873
f(54) = 5833 = 19*307
f(55) = 6051 = 3*2017
f(56) = 6273 = 3*3*17*41
f(57) = 6499 = 67*97
f(58) = 6729 = 3*2243
f(59) = 6963 = 3*11*211
f(60) = 7201 = 19*379
f(61) = 7443 = 3*3*827
f(62) = 7689 = 3*11*233
f(63) = 7939 = 17*467
f(64) = 8193 = 3*2731
f(65) = 8451 = 3*3*3*313
f(66) = 8713 = 8713
f(67) = 8979 = 3*41*73
f(68) = 9249 = 3*3083
f(69) = 9523 = 89*107
f(70) = 9801 = 3*3*3*3*11*11
f(71) = 10083 = 3*3361
f(72) = 10369 = 10369
f(73) = 10659 = 3*11*17*19
f(74) = 10953 = 3*3*1217
f(75) = 11251 = 11251
f(76) = 11553 = 3*3851
f(77) = 11859 = 3*59*67
f(78) = 12169 = 43*283
f(79) = 12483 = 3*3*19*73
f(80) = 12801 = 3*17*251
f(81) = 13123 = 11*1193
f(82) = 13449 = 3*4483
f(83) = 13779 = 3*3*1531
f(84) = 14113 = 11*1283
f(85) = 14451 = 3*4817
f(86) = 14793 = 3*4931
f(87) = 15139 = 15139
f(88) = 15489 = 3*3*1721
f(89) = 15843 = 3*5281
f(90) = 16201 = 17*953
f(91) = 16563 = 3*5521
f(92) = 16929 = 3*3*3*3*11*19
f(93) = 17299 = 17299
f(94) = 17673 = 3*43*137
f(95) = 18051 = 3*11*547
f(96) = 18433 = 18433
f(97) = 18819 = 3*3*3*17*41
f(98) = 19209 = 3*19*337
f(99) = 19603 = 19603
f(100) = 20001 = 3*59*113

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) = 2x^2+1

f(0)=1
f(1)=3
f(2)=1
f(3)=19
f(4)=11
f(5)=17
f(6)=73
f(7)=1
f(8)=43
f(9)=163
f(10)=67
f(11)=1
f(12)=1
f(13)=113
f(14)=131
f(15)=41
f(16)=1
f(17)=193
f(18)=59
f(19)=241
f(20)=89
f(21)=883
f(22)=1
f(23)=353
f(24)=1153
f(25)=139
f(26)=1
f(27)=1459
f(28)=523
f(29)=1
f(30)=1801
f(31)=641
f(32)=683
f(33)=2179
f(34)=257
f(35)=1
f(36)=2593
f(37)=83
f(38)=107
f(39)=179
f(40)=97
f(41)=1
f(42)=3529
f(43)=137
f(44)=1291
f(45)=4051
f(46)=1
f(47)=491
f(48)=419
f(49)=1601
f(50)=1667
f(51)=1
f(52)=601
f(53)=1873
f(54)=307
f(55)=2017
f(56)=1
f(57)=1
f(58)=2243
f(59)=211
f(60)=379
f(61)=827
f(62)=233
f(63)=467
f(64)=2731
f(65)=313
f(66)=8713
f(67)=1
f(68)=3083
f(69)=1
f(70)=1
f(71)=3361
f(72)=10369
f(73)=1
f(74)=1217
f(75)=11251
f(76)=3851
f(77)=1
f(78)=283
f(79)=1
f(80)=251
f(81)=1193
f(82)=4483
f(83)=1531
f(84)=1283
f(85)=4817
f(86)=4931
f(87)=15139
f(88)=1721
f(89)=5281
f(90)=953
f(91)=5521
f(92)=1
f(93)=17299
f(94)=1
f(95)=547
f(96)=18433
f(97)=1
f(98)=337
f(99)=19603

b) Substitution of the polynom
The polynom f(x)=2x^2+1 could be written as f(y)= 2y^2+1 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 > 1

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

1, 3, 1, 19, 11, 17, 73, 1, 43, 163, 67, 1, 1, 113, 131, 41, 1, 193, 59, 241, 89, 883, 1, 353, 1153, 139, 1, 1459, 523, 1, 1801, 641, 683, 2179, 257, 1, 2593, 83, 107, 179, 97, 1, 3529, 137, 1291, 4051, 1, 491, 419, 1601, 1667, 1, 601, 1873, 307, 2017, 1, 1, 2243, 211, 379, 827, 233, 467, 2731, 313, 8713, 1, 3083, 1, 1, 3361, 10369, 1, 1217, 11251, 3851, 1, 283, 1, 251, 1193, 4483, 1531, 1283, 4817, 4931, 15139, 1721, 5281, 953, 5521, 1, 17299, 1, 547, 18433, 1, 337, 19603, 1, 2267, 20809, 643, 7211, 22051, 227, 449, 569, 1, 2689, 1297, 8363, 8513, 1, 2939, 8971, 1, 9283, 1049, 347, 1, 9923, 30259, 1, 947, 281, 10753, 331, 401, 593, 673, 34849, 3931, 11971, 36451, 1, 1, 929, 1171, 1, 2339, 4481, 13633, 619, 1, 1579, 3929, 859, 1, 4091, 563, 1, 46819, 1, 1, 48673, 16433, 1, 857, 5689, 1571, 52489, 17713, 1, 3203, 18371, 18593, 2971, 577, 19267, 1, 1, 739, 3187, 1201, 1, 62659, 2347, 521, 1, 21841, 433, 6089, 22571, 22817, 69193, 409, 23563, 71443, 587, 1, 4337, 1307, 2281, 1289, 8537, 25873, 881, 1553, 2963, 80803, 2473, 1, 83233, 1, 1489, 1993, 28843, 571, 457, 443, 1, 1, 10177, 30817, 499, 31393, 1, 95923, 787, 32561, 1, 1, 3041, 1, 2003, 1, 103969, 34961, 35267, 1, 1, 2129, 1129, 3347, 12377, 112339, 3433, 1, 115201, 12907, 39043, 6947, 2089, 13339, 11003, 1, 1, 11273, 1, 1, 1187, 1, 1, 130051, 43691, 4003, 1987, 4969, 1, 3323, 45763, 809, 139393, 46817, 1097, 8387, 1451, 2539, 691, 4451, 1, 149059, 50051, 50417, 1163, 1, 769, 14153, 52267, 5849, 761, 1, 3163, 162451, 1, 617, 8731, 55681, 1699, 659, 56843, 1, 10169, 1, 58411, 176419, 811, 19867, 180001, 1, 1483, 1321, 1867, 62017, 187273, 3307, 7027, 4441, 64067, 3793, 1609, 1, 65731, 18041, 66571, 1, 11897, 1, 3593, 206083, 23041, 6323, 209953, 1051, 1, 213859, 4219, 72161, 1, 1, 73483, 977, 6761, 1, 1, 6883, 76163, 12097, 8563, 77521, 1, 1913, 26297, 1, 79811, 80273, 1, 27067, 81667, 246403, 1, 27691, 6113, 84017, 7681, 254899, 1499, 1, 259201, 1, 1, 1249, 4649, 88817, 267913, 907, 1, 1, 8297, 1, 4691, 92753, 93251, 281251, 1, 4987, 1, 1, 32089, 26393, 97283, 5147, 7193, 32939, 5843, 299539, 100363, 1019, 2843, 101921, 1, 308899, 3833, 1, 971, 1, 35201, 318403, 9697, 107201, 17011, 1, 1, 328051, 109891, 1, 1, 111521, 112067, 30713, 1, 6689, 2833, 6043, 4273, 347779, 116483, 117041, 20753, 1, 1, 357859, 119851, 1, 362953, 121553, 1259, 368083, 2417, 2099, 5113, 1033, 2203, 1123, 1, 7489, 8923, 1, 6793, 388963, 937, 14537, 1, 1361, 132611, 2137, 14867, 134401, 405001, 135601, 1, 21601, 1, 12547, 415873, 46411, 12713, 1, 141067, 1, 426889, 1, 8443, 1, 1, 1, 438049, 13331, 16363, 26099, 7817, 149153, 449353, 1, 151051, 1009, 8017, 1, 1, 1, 154883, 466579, 52057, 156817, 472393, 158113, 1, 478243, 2713, 1, 484129, 54011, 162691, 1, 164011, 1, 1, 15091, 166667, 2083, 1697, 168673, 508033, 1, 56897, 12539, 4001, 172721, 1, 58027, 174763, 1, 1979, 3467, 28027, 2441, 4363, 538723, 60089, 16451, 32057, 182353, 1, 5683, 2753, 185153, 2393, 1091, 1, 4057, 1, 63131, 1723, 1, 191531, 576739, 1, 11393, 1, 195121, 1, 53609, 2377, 198017, 1, 66491, 1, 1, 201667, 1, 5393, 203873, 1, 616051, 1, 18803, 622729, 5081, 1, 2011, 1, 211313, 636193, 6449, 1, 1, 19553, 71947, 649801, 217361, 3697, 1, 1, 1, 1, 221953, 1, 60953, 224267, 1619, 2699, 1481, 227371, 684451, 12049, 6961, 691489, 5641, 1, 5099, 4099, 3499, 16411, 236017, 1, 712819, 21673, 239201, 1, 7297, 241603, 8171, 2273, 1427, 734473, 14449, 4177, 67433, 1, 13099, 68099, 3739, 83777, 1747, 252971, 14929, 9203, 85147, 23297, 771283, 2659, 7841, 2411, 260417, 261251, 9473, 87641, 263761, 1, 1, 29587, 801379, 1433, 268817, 808993, 1, 271363, 8419, 273067, 1, 1, 1, 1, 75641, 92737, 279073, 839809, 14779, 93889, 1, 283403, 1, 1931, 95339, 2371, 863299, 16979, 10723, 13003, 1787, 15377, 12043, 2969, 2609, 887113, 1, 1, 6833, 2153, 7321, 1409, 1, 302851, 1, 304651, 1, 83579, 307361, 1, 48817, 2521, 1, 2161, 18401, 3169, 49681, 1, 28771, 16139, 1, 1, 960499, 321091, 107339, 22531, 29443, 1, 977203, 1, 3947, 57977, 329473, 1, 994051, 17489, 333233, 91139, 6571, 4049, 1, 8243, 37657, 1811, 1, 341771, 3659, 4243, 1, 7459, 346561, 10531, 1, 4787, 18443, 1, 1753, 2579, 3011, 32297, 118747, 12041, 32563, 359171, 1080451, 1, 1, 57331, 8467, 4507, 99833, 1523, 21649, 5297, 41113, 1, 16657, 373003, 2113, 1125001, 376001, 34273, 1627, 126337, 1, 1143073, 20107, 127681, 1152163, 22651, 1657, 1161289, 1, 9491, 1170451, 1, 43577, 1179649, 394241, 23251, 4201, 1, 398353, 108923, 400417, 7043, 1, 3571, 2953, 2131, 1, 1, 1226179, 4937, 1, 1235593, 1, 37633, 6451, 1, 1, 1254529, 5051, 1, 66529, 3491, 5801, 1273609, 1, 3257, 67537, 1889, 143291, 1, 1, 2657, 118409, 2459, 1, 119291, 25793, 146521, 1321939, 1, 442817, 13729, 148331, 1, 1341523, 23593, 1, 1, 451553, 2777, 31657, 50539, 455953, 80657, 41651, 153089, 1, 41953, 1, 1391113, 3779, 465931, 1401139, 27539, 8233, 128291, 471521, 472643, 129209, 2683, 28001, 34913, 478273, 53267, 1441603, 481667, 43891, 1777, 53897, 44201, 5059, 2729, 163211, 77491, 3539, 493067, 22129, 1, 1, 1492993, 1, 166657, 1, 11681, 503441, 1, 1, 29819, 138569, 1, 56713, 1, 26987, 513923, 5347, 19121, 7723, 21313, 519793, 15787, 1566451, 1, 1, 1907, 10331, 27793, 14051, 530443, 177211, 14939, 1, 1, 1609219, 1, 538801, 1620001, 541201, 20089, 1, 1, 546017, 8779, 60937, 549643, 7907, 6203, 4289, 2251, 1, 556931, 88129, 1, 1, 1685449, 563041, 17099, 4889, 566723, 33409, 6803, 1, 9689, 6073, 1, 63929, 1, 1, 579083, 3049, 1, 1, 1752193, 585313, 4547, 160313, 34651, 590321, 161339, 2707, 594091, 7411, 596611, 1, 1, 600401, 1, 1808803, 201401, 2897, 42331, 8329, 67699, 107747, 14923, 1, 1843201, 1, 10457, 1854739, 3313, 206939, 98227, 623393, 624683, 1, 209089, 628561, 9041, 631153, 12401, 1, 635051, 636353, 1912969, 212987, 640267, 2633, 3331, 1, 3299, 1, 58921, 11953, 72307, 652081, 1960201, 2027, 218681, 1972099, 1, 9851, 9403, 1, 1, 1,

6. Sequence of the polynom (only primes)

3, 19, 11, 17, 73, 43, 163, 67, 113, 131, 41, 193, 59, 241, 89, 883, 353, 1153, 139, 1459, 523, 1801, 641, 683, 2179, 257, 2593, 83, 107, 179, 97, 3529, 137, 1291, 4051, 491, 419, 1601, 1667, 601, 1873, 307, 2017, 2243, 211, 379, 827, 233, 467, 2731, 313, 8713, 3083, 3361, 10369, 1217, 11251, 3851, 283, 251, 1193, 4483, 1531, 1283, 4817, 4931, 15139, 1721, 5281, 953, 5521, 17299, 547, 18433, 337, 19603, 2267, 20809, 643, 7211, 22051, 227, 449, 569, 2689, 1297, 8363, 8513, 2939, 8971, 9283, 1049, 347, 9923, 30259, 947, 281, 10753, 331, 401, 593, 673, 34849, 3931, 11971, 36451, 929, 1171, 2339, 4481, 13633, 619, 1579, 3929, 859, 4091, 563, 46819, 48673, 16433, 857, 5689, 1571, 52489, 17713, 3203, 18371, 18593, 2971, 577, 19267, 739, 3187, 1201, 62659, 2347, 521, 21841, 433, 6089, 22571, 22817, 69193, 409, 23563, 71443, 587, 4337, 1307, 2281, 1289, 8537, 25873, 881, 1553, 2963, 80803, 2473, 83233, 1489, 1993, 28843, 571, 457, 443, 10177, 30817, 499, 31393, 95923, 787, 32561, 3041, 2003, 103969, 34961, 35267, 2129, 1129, 3347, 12377, 112339, 3433, 115201, 12907, 39043, 6947, 2089, 13339, 11003, 11273, 1187, 130051, 43691, 4003, 1987, 4969, 3323, 45763, 809, 139393, 46817, 1097, 8387, 1451, 2539, 691, 4451, 149059, 50051, 50417, 1163, 769, 14153, 52267, 5849, 761, 3163, 162451, 617, 8731, 55681, 1699, 659, 56843, 10169, 58411, 176419, 811, 19867, 180001, 1483, 1321, 1867, 62017, 187273, 3307, 7027, 4441, 64067, 3793, 1609, 65731, 18041, 66571, 11897, 3593, 206083, 23041, 6323, 209953, 1051, 213859, 4219, 72161, 73483, 977, 6761, 6883, 76163, 12097, 8563, 77521, 1913, 26297, 79811, 80273, 27067, 81667, 246403, 27691, 6113, 84017, 7681, 254899, 1499, 259201, 1249, 4649, 88817, 267913, 907, 8297, 4691, 92753, 93251, 281251, 4987, 32089, 26393, 97283, 5147, 7193, 32939, 5843, 299539, 100363, 1019, 2843, 101921, 308899, 3833, 971, 35201, 318403, 9697, 107201, 17011, 328051, 109891, 111521, 112067, 30713, 6689, 2833, 6043, 4273, 347779, 116483, 117041, 20753, 357859, 119851, 362953, 121553, 1259, 368083, 2417, 2099, 5113, 1033, 2203, 1123, 7489, 8923, 6793, 388963, 937, 14537, 1361, 132611, 2137, 14867, 134401, 405001, 135601, 21601, 12547, 415873, 46411, 12713, 141067, 426889, 8443, 438049, 13331, 16363, 26099, 7817, 149153, 449353, 151051, 1009, 8017, 154883, 466579, 52057, 156817, 472393, 158113, 478243, 2713, 484129, 54011, 162691, 164011, 15091, 166667, 2083, 1697, 168673, 508033, 56897, 12539, 4001, 172721, 58027, 174763, 1979, 3467, 28027, 2441, 4363, 538723, 60089, 16451, 32057, 182353, 5683, 2753, 185153, 2393, 1091, 4057, 63131, 1723, 191531, 576739, 11393, 195121, 53609, 2377, 198017, 66491, 201667, 5393, 203873, 616051, 18803, 622729, 5081, 2011, 211313, 636193, 6449, 19553, 71947, 649801, 217361, 3697, 221953, 60953, 224267, 1619, 2699, 1481, 227371, 684451, 12049, 6961, 691489, 5641, 5099, 4099, 3499, 16411, 236017, 712819, 21673, 239201, 7297, 241603, 8171, 2273, 1427, 734473, 14449, 4177, 67433, 13099, 68099, 3739, 83777, 1747, 252971, 14929, 9203, 85147, 23297, 771283, 2659, 7841, 2411, 260417, 261251, 9473, 87641, 263761, 29587, 801379, 1433, 268817, 808993, 271363, 8419, 273067, 75641, 92737, 279073, 839809, 14779, 93889, 283403, 1931, 95339, 2371, 863299, 16979, 10723, 13003, 1787, 15377, 12043, 2969, 2609, 887113, 6833, 2153, 7321, 1409, 302851, 304651, 83579, 307361, 48817, 2521, 2161, 18401, 3169, 49681, 28771, 16139, 960499, 321091, 107339, 22531, 29443, 977203, 3947, 57977, 329473, 994051, 17489, 333233, 91139, 6571, 4049, 8243, 37657, 1811, 341771, 3659, 4243, 7459, 346561, 10531, 4787, 18443, 1753, 2579, 3011, 32297, 118747, 12041, 32563, 359171, 1080451, 57331, 8467, 4507, 99833, 1523, 21649, 5297, 41113, 16657, 373003, 2113, 1125001, 376001, 34273, 1627, 126337, 1143073, 20107, 127681, 1152163, 22651, 1657, 1161289, 9491, 1170451, 43577, 1179649, 394241, 23251, 4201, 398353, 108923, 400417, 7043, 3571, 2953, 2131, 1226179, 4937, 1235593, 37633, 6451, 1254529, 5051, 66529, 3491, 5801, 1273609, 3257, 67537, 1889, 143291, 2657, 118409, 2459, 119291, 25793, 146521, 1321939, 442817, 13729, 148331, 1341523, 23593, 451553, 2777, 31657, 50539, 455953, 80657, 41651, 153089, 41953, 1391113, 3779, 465931, 1401139, 27539, 8233, 128291, 471521, 472643, 129209, 2683, 28001, 34913, 478273, 53267, 1441603, 481667, 43891, 1777, 53897, 44201, 5059, 2729, 163211, 77491, 3539, 493067, 22129, 1492993, 166657, 11681, 503441, 29819, 138569, 56713, 26987, 513923, 5347, 19121, 7723, 21313, 519793, 15787, 1566451, 1907, 10331, 27793, 14051, 530443, 177211, 14939, 1609219, 538801, 1620001, 541201, 20089, 546017, 8779, 60937, 549643, 7907, 6203, 4289, 2251, 556931, 88129, 1685449, 563041, 17099, 4889, 566723, 33409, 6803, 9689, 6073, 63929, 579083, 3049, 1752193, 585313, 4547, 160313, 34651, 590321, 161339, 2707, 594091, 7411, 596611, 600401, 1808803, 201401, 2897, 42331, 8329, 67699, 107747, 14923, 1843201, 10457, 1854739, 3313, 206939, 98227, 623393, 624683, 209089, 628561, 9041, 631153, 12401, 635051, 636353, 1912969, 212987, 640267, 2633, 3331, 3299, 58921, 11953, 72307, 652081, 1960201, 2027, 218681, 1972099, 9851, 9403,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= 2x^2x+1 and
the reducible primes which appear as divisor for the first time
p | 2x^2x+1 and p < 2x^2x+1

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

I

J

K

 

10^n

x

all Primes

P(x)=2x^2+1

P(x) | 2x^2+1

C/B

D/B

E/B

C(n) / C(n-1)

D(n) / D(n-1)

E(n) / E(n-1)

 

1

10

8

4

4

0,800000

0,400000

0,400000

   

 

2

100

76

19

57

0,760000

0,190000

0,570000

9,500000

4,750000

14,250000

 

3

1.000

760

117

643

0,760000

0,117000

0,643000

10,000000

6,157895

11,280702

 

4

10.000

7.445

841

6.604

0,744500

0,084100

0,660400

9,796053

7,188034

10,270607

 

5

100.000

73.477

6.606

66.871

0,734770

0,066060

0,668710

9,869308

7,854935

10,125833

 

6

1.000.000

726.948

54.629

672.319

0,726948

0,054629

0,672319

9,893545

8,269603

10,053970

 

7

10.000.000

7.218.256

462.547

6.755.709

0,721826

0,046255

0,675571

9,929536

8,467060

10,048368

 

8

100.000.000

71.801.859

4.027.074

67.774.785

0,718019

0,040271

0,677748

9,947259

8,706302

10,032224

 

9

1.000.000.000

715.087.632

35.630.373

679.457.259

0,715088

0,035630

0,679457

9,959180

8,847708

10,025222

 

10

10.000.000.000

7.127.665.635

319.422.804

6.808.242.831

0,712767

0,031942

0,680824

9,967541

8,964902

10,020119

 

11

100.000.000.000

71.089.166.879

2.895.004.512

68.194.162.367

0,710892

0,028950

0,681942

9,973696

9,063237

10,016412

 

12

1.000.000.000.000

709.344.259.821

26.471.105.994

682.873.153.827

0,709344

0,026471

0,682873

9,978233

9,143718

10,013660

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A

B

C

D

E

F

G

H

I

J

K

 

2^n

x

all Primes

P(x)=2x^2+1

P(x) | 2x^2+1

C/B

D/B

E/B

C(n) / C(n-1)

D(n) / D(n-1)

E(n) / E(n-1)

 

1

2

1

1

0

0,500000

0,500000

0,000000

   

 

2

4

3

2

1

0,750000

0,500000

0,250000

3,000000

2,000000

 

 

3

8

6

3

3

0,750000

0,375000

0,375000

2,000000

1,500000

3,000000

 

4

16

11

4

7

0,687500

0,250000

0,437500

1,833333

1,333333

2,333333

 

5

32

24

8

16

0,750000

0,250000

0,500000

2,181818

2,000000

2,285714

 

6

64

50

12

38

0,781250

0,187500

0,593750

2,083333

1,500000

2,375000

 

7

128

99

22

77

0,773438

0,171875

0,601563

1,980000

1,833333

2,026316

 

8

256

194

37

157

0,757813

0,144531

0,613281

1,959596

1,681818

2,038961

 

9

512

388

70

318

0,757813

0,136719

0,621094

2,000000

1,891892

2,025478

 

10

1.024

776

120

656

0,757813

0,117188

0,640625

2,000000

1,714286

2,062893

 

11

2.048

1.540

223

1.317

0,751953

0,108887

0,643066

1,984536

1,858333

2,007622

 

12

4.096

3.073

392

2.681

0,750244

0,095703

0,654541

1,995455

1,757848

2,035687

 

13

8.192

6.099

714

5.385

0,744507

0,087158

0,657349

1,984705

1,821429

2,008579

 

14

16.384

12.160

1.293

10.867

0,742188

0,078918

0,663269

1,993769

1,810924

2,018013

 

15

32.768

24.225

2.372

21.853

0,739288

0,072388

0,666901

1,992188

1,834493

2,010951

 

16

65.536

48.262

4.484

43.778

0,736420

0,068420

0,667999

1,992239

1,890388

2,003295

 

17

131.072

96.117

8.450

87.667

0,733315

0,064468

0,668846

1,991567

1,884478

2,002536

 

18

262.144

191.661

15.902

175.759

0,731129

0,060661

0,670467

1,994039

1,881893

2,004848

 

19

524.288

382.127

29.956

352.171

0,728849

0,057137

0,671713

1,993765

1,883788

2,003715

 

20

1.048.576

762.133

57.002

705.131

0,726827

0,054361

0,672465

1,994449

1,902858

2,002240

 

21

2.097.152

1.520.931

108.191

1.412.740

0,725236

0,051589

0,673647

1,995624

1,898021

2,003514

 

22

4.194.304

3.035.027

205.398

2.829.629

0,723607

0,048971

0,674636

1,995506

1,898476

2,002937

 

23

8.388.608

6.058.303

392.498

5.665.805

0,722206

0,046789

0,675417

1,996128

1,910914

2,002314

 

24

16.777.216

12.094.615

751.210

11.343.405

0,720895

0,044776

0,676120

1,996370

1,913921

2,002082

 

25

33.554.432

24.149.710

1.439.421

22.710.289

0,719717

0,042898

0,676819

1,996732

1,916137

2,002070

 

26

67.108.864

48.226.395

2.764.920

45.461.475

0,718629

0,041201

0,677429

1,996976

1,920856

2,001801

 

27

134.217.728

96.316.171

5.316.867

90.999.304

0,717611

0,039614

0,677998

1,997167

1,922973

2,001680

 

28

268.435.456

192.381.714

10.237.140

182.144.574

0,716678

0,038136

0,678541

1,997398

1,925408

2,001604

 

29

536.870.912

384.295.122

19.746.417

364.548.705

0,715805

0,036781

0,679025

1,997566

1,928900

2,001425

 

30

1.073.741.824

767.729.885

38.119.892

729.609.993

0,715004

0,035502

0,679502

1,997761

1,930471

2,001406

 

31

2.147.483.648

1.533.851.929

73.682.997

1.460.168.932

0,714255

0,034311

0,679944

1,997906

1,932928

2,001301

 

32

4.294.967.296

3.064.705.816

142.607.233

2.922.098.583

0,713557

0,033203

0,680354

1,998045

1,935416

2,001206

 

33

8.589.934.592

6.123.796.529

276.272.479

5.847.524.050

0,712904

0,032162

0,680741

1,998168

1,937296

2,001139

 

34

17.179.869.184

12.237.071.081

535.760.398

11.701.310.683

0,712291

0,031185

0,681106

1,998282

1,939246

2,001071

 

35

34.359.738.368

24.454.319.674

1.039.909.544

23.414.410.130

0,711714

0,030265

0,681449

1,998380

1,940997

2,001007

 

36

68.719.476.736

48.871.362.246

2.020.283.544

46.851.078.702

0,711172

0,029399

0,681773

1,998476

1,942749

2,000951

 

37

137.438.953.472

97.672.370.547

3.928.018.222

93.744.352.325

0,710660

0,028580

0,682080

1,998560

1,944291

2,000901

 

38

274.877.906.944

195.211.920.067

7.643.397.852

187.568.522.215

0,710177

0,027807

0,682370

1,998640

1,945866

2,000851

 

39

549.755.813.888

390.172.411.497

14.883.784.616

375.288.626.881

0,709719

0,027073

0,682646

1,998712

1,947273

2,000808

 

40

1.099.511.627.776

779.868.595.785

29.003.055.988

750.865.539.797

0,709286

0,026378

0,682908

1,998779

1,948634

2,000768

 

           

 

           

 

            
            


ABCDEFGHI
exponent =log2 (x) <=xnumber of primes with p=f(x) number of primes with p=f(x) and p%6=1 number of primes with p=f(x) and p%6=5 number of primes with p=f(x) and p%8=1 number of primes with p=f(x) and p%8=3 number of primes with p=f(x) and p%8=5 number of primes with p=f(x) and p%8=7
121000100
242100200
383201200
4164301300
5328703500
664121105700
71282221091300
825637360162100
951270690313900
101.0241201190556500
112.048223222011011300
124.096392391019519700
138.192714713035635800
1416.3841.2931.292065963400
1532.7682.3722.37101.1901.18200
1665.5364.4844.48302.2422.24200
17131.0728.4508.44904.2344.21600
18262.14415.90215.90107.9557.94700
19524.28829.95629.955015.02514.93100
201.048.57657.00257.001028.57828.42400
212.097.152108.191108.190054.15054.04100
224.194.304205.398205.3970102.700102.69800
238.388.608392.498392.4970196.330196.16800
2416.777.216751.210751.2090375.676375.53400
2533.554.4321.439.4211.439.4200719.941719.48000
2667.108.8642.764.9202.764.91901.382.1231.382.79700
27134.217.7285.316.8675.316.86602.657.4022.659.46500
28268.435.45610.237.14010.237.13905.117.2985.119.84200
29536.870.91219.746.41719.746.41609.870.4969.875.92100
301.073.741.82438.119.89238.119.891019.054.82219.065.07000
312.147.483.64873.682.99773.682.996036.837.59736.845.40000
324.294.967.296142.607.232142.607.231071.296.02971.311.20300
338.589.934.592276.272.478276.272.4770138.126.078138.146.40000
3417.179.869.184535.760.396535.760.3950267.872.126267.888.27000
3534.359.738.3681.039.909.5401.039.909.5390519.954.072519.955.46800
3668.719.476.7362.020.283.5342.020.283.53301.010.145.4891.010.138.04500


ABCDEFGHI
exponent =log2 (x) <=xnumber of primes with p|f(x) number of primes with p=f(x) and p%6=1 number of primes with p=f(x) and p%6=5 number of primes with p=f(x) and p%8=1 number of primes with p=f(x) and p%8=3 number of primes with p=f(x) and p%8=5 number of primes with p=f(x) and p%8=7
120000000
241010100
383121200
4167253400
532166108800
664381523162200
7128773146344300
82561577186778000
951231815216615616200
101.02465631134532932700
112.0481.31764567265466300
124.0962.6811.2781.4031.3301.35100
138.1925.3852.5922.7932.6892.69600
1416.38410.8675.2425.6255.4535.41400
1532.76821.85310.54811.30510.91910.93400
1665.53643.77821.22522.55321.84221.93600
17131.07287.66742.64845.01943.88843.77900
18262.144175.75985.56990.19088.01587.74400
19524.288352.171171.705180.466176.145176.02600
201.048.576705.131344.399360.732352.910352.22100
212.097.1521.412.740692.049720.691706.459706.28100
224.194.3042.829.6291.387.2031.442.4261.415.1181.414.51100
238.388.6085.665.8052.779.7952.886.0102.832.4812.833.32400
2416.777.21611.343.4055.569.3505.774.0555.669.3015.674.10400
2533.554.43222.710.28911.160.78511.549.50411.351.85011.358.43900
2667.108.86445.461.47522.359.08523.102.39022.731.11622.730.35900
27134.217.72890.999.30444.787.72446.211.58045.501.81145.497.49300
28268.435.456182.144.57489.716.86292.427.71291.072.24391.072.33100
29536.870.912364.548.705179.667.932184.880.773182.269.851182.278.85400
301.073.741.824729.609.993359.793.771369.816.222364.797.540364.812.45300
312.147.483.6481.460.168.932720.413.324739.755.608730.080.596730.088.33600
324.294.967.2962.922.098.5831.442.349.8851.479.748.6981.461.023.1931.461.075.39000
338.589.934.5925.847.524.0502.887.639.7822.959.884.2682.923.714.6572.923.809.39300
3417.179.869.18411.701.310.6815.780.791.6235.920.519.0585.850.587.5815.850.723.10000
3534.359.738.36823.414.410.12211.571.855.02311.842.555.09911.707.141.80111.707.268.32100
3668.719.476.73646.851.078.66823.163.013.69423.688.064.97423.425.547.55923.425.531.10900


8. Check for existing Integer Sequences by OEIS

Found in Database : 1, 3, 1, 19, 11, 17, 73, 1, 43, 163, 67, 1, 1, 113, 131, 41, 1, 193, 59, 241,
Found in Database : 3, 19, 11, 17, 73, 43, 163, 67, 113, 131, 41, 193, 59, 241, 89, 883, 353, 1153, 139, 1459, 523, 1801, 641, 683, 2179, 257, 2593, 83, 107, 179,
Found in Database : 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139,