Development of
Algorithmic Constructions

16:20:23

18.Apr 2024

Polynom = x^2+2x-1

3. Correctness of the algorithm

5. Sequence of the polynom with 1

6. Sequence of the polynom (only primes)

8. Check for existing Integer Sequences by OEIS

0. Sequence

f(0) = 1 = 1
f(1) = 1 = 1
f(2) = 7 = 7
f(3) = 7 = 7
f(4) = 23 = 23
f(5) = 17 = 17
f(6) = 47 = 47
f(7) = 31 = 31
f(8) = 79 = 79
f(9) = 49 = 7*7
f(10) = 119 = 7*17
f(11) = 71 = 71
f(12) = 167 = 167
f(13) = 97 = 97
f(14) = 223 = 223
f(15) = 127 = 127
f(16) = 287 = 7*41
f(17) = 161 = 7*23
f(18) = 359 = 359
f(19) = 199 = 199
f(20) = 439 = 439
f(21) = 241 = 241
f(22) = 527 = 17*31
f(23) = 287 = 7*41
f(24) = 623 = 7*89
f(25) = 337 = 337
f(26) = 727 = 727
f(27) = 391 = 17*23
f(28) = 839 = 839
f(29) = 449 = 449
f(30) = 959 = 7*137
f(31) = 511 = 7*73
f(32) = 1087 = 1087
f(33) = 577 = 577
f(34) = 1223 = 1223
f(35) = 647 = 647
f(36) = 1367 = 1367
f(37) = 721 = 7*103
f(38) = 1519 = 7*7*31
f(39) = 799 = 17*47
f(40) = 1679 = 23*73
f(41) = 881 = 881
f(42) = 1847 = 1847
f(43) = 967 = 967
f(44) = 2023 = 7*17*17
f(45) = 1057 = 7*151
f(46) = 2207 = 2207
f(47) = 1151 = 1151
f(48) = 2399 = 2399
f(49) = 1249 = 1249
f(50) = 2599 = 23*113
f(51) = 1351 = 7*193
f(52) = 2807 = 7*401
f(53) = 1457 = 31*47
f(54) = 3023 = 3023
f(55) = 1567 = 1567
f(56) = 3247 = 17*191
f(57) = 1681 = 41*41
f(58) = 3479 = 7*7*71
f(59) = 1799 = 7*257
f(60) = 3719 = 3719
f(61) = 1921 = 17*113
f(62) = 3967 = 3967
f(63) = 2047 = 23*89
f(64) = 4223 = 41*103
f(65) = 2177 = 7*311
f(66) = 4487 = 7*641
f(67) = 2311 = 2311
f(68) = 4759 = 4759
f(69) = 2449 = 31*79
f(70) = 5039 = 5039
f(71) = 2591 = 2591
f(72) = 5327 = 7*761
f(73) = 2737 = 7*17*23
f(74) = 5623 = 5623
f(75) = 2887 = 2887
f(76) = 5927 = 5927
f(77) = 3041 = 3041
f(78) = 6239 = 17*367
f(79) = 3199 = 7*457
f(80) = 6559 = 7*937
f(81) = 3361 = 3361
f(82) = 6887 = 71*97
f(83) = 3527 = 3527
f(84) = 7223 = 31*233
f(85) = 3697 = 3697
f(86) = 7567 = 7*23*47
f(87) = 3871 = 7*7*79
f(88) = 7919 = 7919
f(89) = 4049 = 4049
f(90) = 8279 = 17*487
f(91) = 4231 = 4231
f(92) = 8647 = 8647
f(93) = 4417 = 7*631
f(94) = 9023 = 7*1289
f(95) = 4607 = 17*271
f(96) = 9407 = 23*409
f(97) = 4801 = 4801
f(98) = 9799 = 41*239
f(99) = 4999 = 4999
f(100) = 10199 = 7*31*47

1. Algorithm

liste_max:=100000;

sieving:=proc (stelle, p)
begin
   while (stelle<=liste_max) do
     erg:=liste[stelle];
     while(erg mod p=0) do
        // Divison of the stored f(x) by the prime
        erg:=erg /p;
     end_while;
     liste[stelle]:=erg;
     stelle:=stelle+p;
   end_while;
end_proc;

// Calculation of the values of the polynom for x from 0 to liste_max
for x from 0 to liste_max do
   p:=abs (a*x^2+b*x+c);
   while (p mod 2=0) p:=p/2;
   liste [x]:=p;
end_for;

for x from 0 to liste_max do
    p:=liste[x];

    if (p>1) then

      // Printing the Primes
      print (x, p);

      // 1. Sieving
      sieving (x+p,  p);

      t:=(-x-b/a) mod p;
      if t=0 then t:=p; end_if;

      // 2. Sieving
      sieving (t, p);

    end_if;

end_for;

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+2x-1

f(0)=1
f(1)=1
f(2)=7
f(3)=1
f(4)=23
f(5)=17
f(6)=47
f(7)=31
f(8)=79
f(9)=1
f(10)=1
f(11)=71
f(12)=167
f(13)=97
f(14)=223
f(15)=127
f(16)=41
f(17)=1
f(18)=359
f(19)=199
f(20)=439
f(21)=241
f(22)=1
f(23)=1
f(24)=89
f(25)=337
f(26)=727
f(27)=1
f(28)=839
f(29)=449
f(30)=137
f(31)=73
f(32)=1087
f(33)=577
f(34)=1223
f(35)=647
f(36)=1367
f(37)=103
f(38)=1
f(39)=1
f(40)=1
f(41)=881
f(42)=1847
f(43)=967
f(44)=1
f(45)=151
f(46)=2207
f(47)=1151
f(48)=2399
f(49)=1249
f(50)=113
f(51)=193
f(52)=401
f(53)=1
f(54)=3023
f(55)=1567
f(56)=191
f(57)=1
f(58)=1
f(59)=257
f(60)=3719
f(61)=1
f(62)=3967
f(63)=1
f(64)=1
f(65)=311
f(66)=641
f(67)=2311
f(68)=4759
f(69)=1
f(70)=5039
f(71)=2591
f(72)=761
f(73)=1
f(74)=5623
f(75)=2887
f(76)=5927
f(77)=3041
f(78)=367
f(79)=457
f(80)=937
f(81)=3361
f(82)=1
f(83)=3527
f(84)=233
f(85)=3697
f(86)=1
f(87)=1
f(88)=7919
f(89)=4049
f(90)=487
f(91)=4231
f(92)=8647
f(93)=631
f(94)=1289
f(95)=271
f(96)=409
f(97)=4801
f(98)=239
f(99)=4999

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

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+1
f'(x)>2x+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, 1, 7, 1, 23, 17, 47, 31, 79, 1, 1, 71, 167, 97, 223, 127, 41, 1, 359, 199, 439, 241, 1, 1, 89, 337, 727, 1, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 1, 1, 1, 881, 1847, 967, 1, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 1, 3023, 1567, 191, 1, 1, 257, 3719, 1, 3967, 1, 1, 311, 641, 2311, 4759, 1, 5039, 2591, 761, 1, 5623, 2887, 5927, 3041, 367, 457, 937, 3361, 1, 3527, 233, 3697, 1, 1, 7919, 4049, 487, 4231, 8647, 631, 1289, 271, 409, 4801, 239, 4999, 1, 743, 10607, 5407, 1, 1, 11447, 1, 1697, 263, 1, 6271, 751, 1, 1889, 1, 13687, 6961, 14159, 313, 14639, 1063, 2161, 7687, 919, 7937, 16127, 8191, 2377, 1, 17159, 281, 769, 1, 18223, 1321, 383, 9521, 19319, 1, 1, 593, 1, 1481, 21023, 10657, 1, 1, 1, 1607, 3257, 11551, 1, 1, 24023, 1, 503, 1783, 1487, 12799, 25919, 13121, 857, 1, 3889, 599, 353, 1, 28559, 14449, 4177, 2113, 29927, 15137, 1, 911, 31327, 1, 1, 1, 1, 16561, 33487, 16927, 4889, 1, 479, 431, 1553, 18049, 36479, 2633, 1, 607, 809, 19207, 1, 1153, 5657, 2857, 569, 887, 463, 20807, 42023, 433, 6121, 1, 1409, 1297, 44519, 977, 6481, 3271, 2719, 23327, 47087, 23761, 1, 3457, 6977, 601, 49727, 25087, 1, 25537, 1, 1, 1279, 26449, 53359, 1583, 54287, 3911, 1, 27847, 56167, 1, 57119, 929, 8297, 1, 1, 1, 1, 1, 61007, 1, 521, 31249, 863, 31751, 64007, 32257, 1327, 1, 66047, 1447, 67079, 1, 4007, 4903, 1, 34847, 70223, 2081, 71287, 35911, 10337, 1, 1, 1, 74527, 37537, 1609, 5441, 1, 2273, 77839, 39199, 3433, 39761, 673, 823, 81223, 40897, 2657, 1, 1777, 6007, 12097, 1, 85847, 1879, 5119, 617, 12601, 6343, 89399, 2647, 90599, 1471, 91807, 1, 1, 46817, 1193, 47431, 95479, 48049, 1, 1, 97967, 49297, 99223, 49927, 1, 1, 14537, 51199, 1, 1103, 104327, 719, 1, 7591, 4649, 53791, 6367, 54449, 1231, 7873, 1, 1, 112223, 1201, 113567, 1, 16417, 1, 116279, 58481, 1657, 59167, 2903, 1, 1, 1, 3929, 2663, 7247, 1511, 2543, 8951, 126023, 63367, 127447, 64081, 128879, 9257, 18617, 65521, 1, 2137, 1049, 66977, 1, 1, 2897, 68449, 137639, 69191, 1, 1, 20089, 70687, 1, 1, 1, 1, 1, 1489, 3121, 73727, 8719, 1, 149767, 10753, 21617, 1, 152879, 76831, 3767, 77617, 1, 1, 1, 79201, 159199, 79999, 1423, 1, 23201, 1033, 164023, 1129, 1097, 83231, 1039, 12007, 5449, 84871, 1, 1, 172223, 1, 24841, 1, 1, 991, 177239, 89041, 25561, 12841, 180623, 2927, 4447, 91591, 184039, 1, 1, 4057, 1, 2297, 189223, 5591, 27281, 1, 1009, 96799, 194479, 1, 196247, 14081, 28289, 1, 199807, 5903, 201599, 983, 1, 14593, 12071, 1, 9001, 103967, 6737, 14983, 30097, 1, 953, 106721, 4561, 107647, 1, 15511, 218087, 2671, 219959, 1, 2287, 15913, 1031, 112337, 5503, 113287, 1, 4967, 1, 2351, 231359, 6833, 2953, 117127, 3313, 16871, 1993, 1, 239119, 120049, 241079, 1, 34721, 17431, 245023, 123007, 247007, 124001, 1, 2551, 1559, 126001, 2239, 1, 255023, 1, 36721, 18433, 1, 2767, 11353, 131071, 263167, 1, 37889, 1, 1951, 134161, 8689, 1, 2281, 19463, 273527, 1, 275623, 138337, 1439, 19913, 5711, 140449, 281959, 141511, 1, 6199, 1319, 20521, 1, 8513, 1, 145799, 292679, 20983, 1, 1873, 297023, 149057, 13009, 150151, 6151, 1, 1303, 2087, 2969, 153457, 18119, 1, 1, 1, 3511, 156799, 314719, 1999, 45281, 22721, 319223, 5167, 18911, 1663, 323759, 1, 1, 9623, 328327, 1, 330623, 165887, 1, 1, 2447, 1, 10889, 169361, 339887, 1433, 48889, 171697, 344567, 172871, 20407, 174049, 1217, 25033, 15289, 176417, 354023, 1831, 11497, 1, 51257, 179999, 21247, 181201, 15809, 3881, 52289, 1543, 368447, 184831, 370879, 186049, 1, 1, 53681, 1, 378223, 1, 1913, 1, 1, 27457, 385639, 193441, 1, 194687, 5351, 1, 1, 6361, 1, 1, 2063, 4871, 57241, 28711, 23719, 1, 405767, 1, 1, 1721, 1, 206081, 13337, 207367, 416023, 208657, 8543, 1, 18313, 1399, 1, 12503, 426407, 1, 1, 2089, 25391, 216481, 14009, 1, 62417, 1361, 439567, 220447, 1, 1, 444887, 31873, 3761, 224449, 450239, 9817, 11047, 1, 65089, 4663, 458327, 229841, 1753, 5639, 2777, 33223, 1, 1, 1, 13841, 5303, 1, 1, 1, 28087, 239431, 4951, 1, 21001, 4943, 69401, 1, 1, 1, 2039, 3119, 4153, 35401, 16033, 249217, 2617, 1, 3329, 36007, 1, 5393, 508367, 6217, 511223, 1871, 1, 1601, 516959, 1, 1, 3671, 1, 37441, 1, 263537, 528527, 264991, 4703, 266449, 3319, 38273, 537287, 269377, 1, 8737, 1, 38903, 78017, 2423, 23873, 16193, 552047, 276767, 1, 1, 558007, 279751, 1, 281249, 563999, 1, 81001, 1, 13903, 285767, 573047, 287281, 1, 41257, 579119, 1, 582167, 12689, 1, 41911, 2711, 294911, 8329, 1, 1, 297991, 85361, 6113, 600623, 1, 26249, 2383, 606839, 43457, 1, 305761, 19777, 1, 616223, 2999, 3847, 2609, 622519, 3217, 4567, 6673, 1, 1, 90289, 7727, 635207, 10271, 8081, 13913, 1, 45943, 15727, 323207, 38119, 6911, 651247, 46633, 93497, 1, 657719, 329671, 1801, 10687, 94889, 1, 667487, 334561, 2143, 336199, 3529, 1, 1, 1759, 680623, 3833, 40231, 342791, 3167, 49207, 2687, 346111, 30169, 1, 9551, 49921, 2441, 351121, 1, 4969, 707279, 3137, 14503, 1, 1, 15559, 3079, 2153, 1, 51607, 103457, 362951, 1, 364657, 1823, 1, 1, 52583, 7607, 1, 43607, 3607, 744767, 1, 1, 374977, 2417, 1, 32833, 378449, 108377, 7759, 1, 381937, 765623, 12377, 3449, 1, 4799, 387199, 776159, 388961, 1, 390727, 2729, 1, 8111, 394271, 7673, 23297, 25609, 1, 113921, 399617, 47119, 401407, 804607, 2111, 1, 1, 811799, 1, 2129, 408607, 6449, 58631, 1, 5647, 3049, 414049, 11689, 1, 1, 59671, 1, 419527, 11519, 421361, 20599, 60457, 2473, 13711, 1, 25111, 37201, 10457, 122761, 1, 50767, 432449, 866759, 434311, 18521, 62311, 1, 438047, 1, 1, 11161, 441799, 1, 1, 889247, 445567, 893023, 26321, 19081, 2791, 128657, 451249, 904399, 3001, 29297, 5113, 1, 1, 11593, 26993, 919679, 11239, 40153, 66103, 7793, 464647, 9041, 4129, 22807, 1, 4327, 9601, 40993, 472391, 946727, 474337, 1, 68041, 136361, 478241, 958439, 1, 13183, 6791, 138041, 1, 2879, 2017, 974167, 1, 5857, 1, 140297, 1, 986047, 1, 990023, 496007, 8353, 71143, 3209, 1,

6. Sequence of the polynom (only primes)

7, 23, 17, 47, 31, 79, 71, 167, 97, 223, 127, 41, 359, 199, 439, 241, 89, 337, 727, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 881, 1847, 967, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 3023, 1567, 191, 257, 3719, 3967, 311, 641, 2311, 4759, 5039, 2591, 761, 5623, 2887, 5927, 3041, 367, 457, 937, 3361, 3527, 233, 3697, 7919, 4049, 487, 4231, 8647, 631, 1289, 271, 409, 4801, 239, 4999, 743, 10607, 5407, 11447, 1697, 263, 6271, 751, 1889, 13687, 6961, 14159, 313, 14639, 1063, 2161, 7687, 919, 7937, 16127, 8191, 2377, 17159, 281, 769, 18223, 1321, 383, 9521, 19319, 593, 1481, 21023, 10657, 1607, 3257, 11551, 24023, 503, 1783, 1487, 12799, 25919, 13121, 857, 3889, 599, 353, 28559, 14449, 4177, 2113, 29927, 15137, 911, 31327, 16561, 33487, 16927, 4889, 479, 431, 1553, 18049, 36479, 2633, 607, 809, 19207, 1153, 5657, 2857, 569, 887, 463, 20807, 42023, 433, 6121, 1409, 1297, 44519, 977, 6481, 3271, 2719, 23327, 47087, 23761, 3457, 6977, 601, 49727, 25087, 25537, 1279, 26449, 53359, 1583, 54287, 3911, 27847, 56167, 57119, 929, 8297, 61007, 521, 31249, 863, 31751, 64007, 32257, 1327, 66047, 1447, 67079, 4007, 4903, 34847, 70223, 2081, 71287, 35911, 10337, 74527, 37537, 1609, 5441, 2273, 77839, 39199, 3433, 39761, 673, 823, 81223, 40897, 2657, 1777, 6007, 12097, 85847, 1879, 5119, 617, 12601, 6343, 89399, 2647, 90599, 1471, 91807, 46817, 1193, 47431, 95479, 48049, 97967, 49297, 99223, 49927, 14537, 51199, 1103, 104327, 719, 7591, 4649, 53791, 6367, 54449, 1231, 7873, 112223, 1201, 113567, 16417, 116279, 58481, 1657, 59167, 2903, 3929, 2663, 7247, 1511, 2543, 8951, 126023, 63367, 127447, 64081, 128879, 9257, 18617, 65521, 2137, 1049, 66977, 2897, 68449, 137639, 69191, 20089, 70687, 1489, 3121, 73727, 8719, 149767, 10753, 21617, 152879, 76831, 3767, 77617, 79201, 159199, 79999, 1423, 23201, 1033, 164023, 1129, 1097, 83231, 1039, 12007, 5449, 84871, 172223, 24841, 991, 177239, 89041, 25561, 12841, 180623, 2927, 4447, 91591, 184039, 4057, 2297, 189223, 5591, 27281, 1009, 96799, 194479, 196247, 14081, 28289, 199807, 5903, 201599, 983, 14593, 12071, 9001, 103967, 6737, 14983, 30097, 953, 106721, 4561, 107647, 15511, 218087, 2671, 219959, 2287, 15913, 1031, 112337, 5503, 113287, 4967, 2351, 231359, 6833, 2953, 117127, 3313, 16871, 1993, 239119, 120049, 241079, 34721, 17431, 245023, 123007, 247007, 124001, 2551, 1559, 126001, 2239, 255023, 36721, 18433, 2767, 11353, 131071, 263167, 37889, 1951, 134161, 8689, 2281, 19463, 273527, 275623, 138337, 1439, 19913, 5711, 140449, 281959, 141511, 6199, 1319, 20521, 8513, 145799, 292679, 20983, 1873, 297023, 149057, 13009, 150151, 6151, 1303, 2087, 2969, 153457, 18119, 3511, 156799, 314719, 1999, 45281, 22721, 319223, 5167, 18911, 1663, 323759, 9623, 328327, 330623, 165887, 2447, 10889, 169361, 339887, 1433, 48889, 171697, 344567, 172871, 20407, 174049, 1217, 25033, 15289, 176417, 354023, 1831, 11497, 51257, 179999, 21247, 181201, 15809, 3881, 52289, 1543, 368447, 184831, 370879, 186049, 53681, 378223, 1913, 27457, 385639, 193441, 194687, 5351, 6361, 2063, 4871, 57241, 28711, 23719, 405767, 1721, 206081, 13337, 207367, 416023, 208657, 8543, 18313, 1399, 12503, 426407, 2089, 25391, 216481, 14009, 62417, 1361, 439567, 220447, 444887, 31873, 3761, 224449, 450239, 9817, 11047, 65089, 4663, 458327, 229841, 1753, 5639, 2777, 33223, 13841, 5303, 28087, 239431, 4951, 21001, 4943, 69401, 2039, 3119, 4153, 35401, 16033, 249217, 2617, 3329, 36007, 5393, 508367, 6217, 511223, 1871, 1601, 516959, 3671, 37441, 263537, 528527, 264991, 4703, 266449, 3319, 38273, 537287, 269377, 8737, 38903, 78017, 2423, 23873, 16193, 552047, 276767, 558007, 279751, 281249, 563999, 81001, 13903, 285767, 573047, 287281, 41257, 579119, 582167, 12689, 41911, 2711, 294911, 8329, 297991, 85361, 6113, 600623, 26249, 2383, 606839, 43457, 305761, 19777, 616223, 2999, 3847, 2609, 622519, 3217, 4567, 6673, 90289, 7727, 635207, 10271, 8081, 13913, 45943, 15727, 323207, 38119, 6911, 651247, 46633, 93497, 657719, 329671, 1801, 10687, 94889, 667487, 334561, 2143, 336199, 3529, 1759, 680623, 3833, 40231, 342791, 3167, 49207, 2687, 346111, 30169, 9551, 49921, 2441, 351121, 4969, 707279, 3137, 14503, 15559, 3079, 2153, 51607, 103457, 362951, 364657, 1823, 52583, 7607, 43607, 3607, 744767, 374977, 2417, 32833, 378449, 108377, 7759, 381937, 765623, 12377, 3449, 4799, 387199, 776159, 388961, 390727, 2729, 8111, 394271, 7673, 23297, 25609, 113921, 399617, 47119, 401407, 804607, 2111, 811799, 2129, 408607, 6449, 58631, 5647, 3049, 414049, 11689, 59671, 419527, 11519, 421361, 20599, 60457, 2473, 13711, 25111, 37201, 10457, 122761, 50767, 432449, 866759, 434311, 18521, 62311, 438047, 11161, 441799, 889247, 445567, 893023, 26321, 19081, 2791, 128657, 451249, 904399, 3001, 29297, 5113, 11593, 26993, 919679, 11239, 40153, 66103, 7793, 464647, 9041, 4129, 22807, 4327, 9601, 40993, 472391, 946727, 474337, 68041, 136361, 478241, 958439, 13183, 6791, 138041, 2879, 2017, 974167, 5857, 140297, 986047, 990023, 496007, 8353, 71143, 3209,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+2x-1 and
the reducible primes which appear as divisor for the first time
p | x^2+2x-1 and p < x^2+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)=x^2+2x-1	P(x) \| x^2+2x-1	C/B	D/B	E/B	C(n) / C(n-1)	D(n) / D(n-1)	E(n) / E(n-1)
1	10	7	5	2	0,700000	0,500000	0,200000
2	100	77	26	51	0,770000	0,260000	0,510000	11,000000	5,200000	25,500000
3	1.000	750	157	593	0,750000	0,157000	0,593000	9,740260	6,038462	11,627451
4	10.000	7.379	1.153	6.226	0,737900	0,115300	0,622600	9,838667	7,343949	10,499157
5	100.000	72.842	8.888	63.954	0,728420	0,088880	0,639540	9,871527	7,708586	10,272085
6	1.000.000	721.999	72.928	649.071	0,721999	0,072928	0,649071	9,911850	8,205221	10,149029
7	10.000.000	7.175.153	615.643	6.559.510	0,717515	0,061564	0,655951	9,937899	8,441792	10,105998
8	100.000.000	71.419.128	5.328.644	66.090.484	0,714191	0,053286	0,660905	9,953673	8,655412	10,075521
9	1.000.000.000	711.654.088	47.034.083	664.620.005	0,711654	0,047034	0,664620	9,964475	8,826651	10,056213
10	10.000.000.000	7.096.508.558	420.950.239	6.675.558.319	0,709651	0,042095	0,667556	9,971851	8,949898	10,044173
11	100.000.000.000	70.804.162.734	3.809.286.970	66.994.875.764	0,708042	0,038093	0,669949	9,977324	9,049257	10,035846
12	1.000.000.000.000	706.718.272.259	34.788.375.186	671.929.897.073	0,706718	0,034788	0,671930	9,981310	9,132516	10,029572


A	B	C	D	E	F	G	H	I	J	K
2^n	x	all Primes	P(x)=x^2+2x-1	P(x) \| x^2+2x-1	C/B	D/B	E/B	C(n) / C(n-1)	D(n) / D(n-1)	E(n) / E(n-1)
1	2	2	2	0	1,000000	1,000000	0,000000
2	4	3	3	0	0,750000	1,000000	0,000000	1,500000	1,500000
3	8	7	5	2	0,875000	0,714286	0,250000	2,333333	1,666667
4	16	13	7	6	0,812500	0,538462	0,375000	1,857143	1,400000	3,000000
5	32	25	12	13	0,781250	0,480000	0,406250	1,923077	1,714286	2,166667
6	64	47	20	27	0,734375	0,425532	0,421875	1,880000	1,666667	2,076923
7	128	99	32	67	0,773438	0,323232	0,523438	2,106383	1,600000	2,481481
8	256	192	54	138	0,750000	0,281250	0,539063	1,939394	1,687500	2,059701
9	512	385	97	288	0,751953	0,251948	0,562500	2,005208	1,796296	2,086957
10	1.024	766	161	605	0,748047	0,210183	0,590820	1,989610	1,659794	2,100694
11	2.048	1.529	289	1.240	0,746582	0,189012	0,605469	1,996084	1,795031	2,049587
12	4.096	3.031	545	2.486	0,739990	0,179809	0,606934	1,982341	1,885813	2,004839
13	8.192	6.055	961	5.094	0,739136	0,158712	0,621826	1,997691	1,763303	2,049075
14	16.384	12.028	1.792	10.236	0,734131	0,148986	0,624756	1,986457	1,864724	2,009423
15	32.768	23.979	3.332	20.647	0,731781	0,138955	0,630096	1,993598	1,859375	2,017097
16	65.536	47.802	6.081	41.721	0,729401	0,127212	0,636612	1,993494	1,825030	2,020681
17	131.072	95.373	11.388	83.985	0,727638	0,119405	0,640755	1,995168	1,872718	2,013015
18	262.144	190.165	21.318	168.847	0,725422	0,112103	0,644100	1,993908	1,871970	2,010442
19	524.288	379.424	40.220	339.204	0,723694	0,106003	0,646980	1,995236	1,886669	2,008943
20	1.048.576	756.830	76.224	680.606	0,721769	0,100715	0,649076	1,994681	1,895177	2,006480
21	2.097.152	1.510.983	144.182	1.366.801	0,720493	0,095423	0,651742	1,996463	1,891556	2,008212
22	4.194.304	3.016.015	274.311	2.741.704	0,719074	0,090951	0,653673	1,996062	1,902533	2,005928
23	8.388.608	6.021.461	522.318	5.499.143	0,717814	0,086743	0,655549	1,996496	1,904109	2,005739
24	16.777.216	12.024.149	998.286	11.025.863	0,716695	0,083023	0,657193	1,996882	1,911261	2,005015
25	33.554.432	24.014.126	1.910.299	22.103.827	0,715677	0,079549	0,658745	1,997158	1,913579	2,004725
26	67.108.864	47.964.118	3.660.832	44.303.286	0,714721	0,076324	0,660170	1,997329	1,916366	2,004326
27	134.217.728	95.809.662	7.032.025	88.777.637	0,713838	0,073396	0,661445	1,997528	1,920882	2,003861
28	268.435.456	191.402.094	13.530.333	177.871.761	0,713028	0,070691	0,662624	1,997733	1,924102	2,003565
29	536.870.912	382.404.759	26.076.182	356.328.577	0,712284	0,068190	0,663714	1,997913	1,927239	2,003289
30	1.073.741.824	764.060.580	50.322.316	713.738.264	0,711587	0,065862	0,664721	1,998042	1,929819	2,003034
31	2.147.483.648	1.526.721.093	97.225.942	1.429.495.151	0,710935	0,063683	0,665661	1,998168	1,932064	2,002828
32	4.294.967.296	3.050.855.611	188.053.713	2.862.801.898	0,710333	0,061640	0,666548	1,998306	1,934193	2,002666
33	8.589.934.592	6.096.874.075	364.114.625	5.732.759.450	0,709770	0,059722	0,667381	1,998414	1,936227	2,002500
34	17.179.869.184	12.184.701.718	705.755.394	11.478.946.324	0,709243	0,057921	0,668163	1,998516	1,938278	2,002342
35	34.359.738.368	24.352.361.189	1.369.222.761	22.983.138.428	0,708747	0,056225	0,668897	1,998601	1,940081	2,002199
36	68.719.476.736	48.672.746.201	2.658.892.862	46.013.853.339	0,708282	0,054628	0,669590	1,998687	1,941899	2,002070
37	137.438.953.472	97.285.241.791	5.167.654.611	92.117.587.180	0,707843	0,053119	0,670244	1,998762	1,943536	2,001953
38	274.877.906.944	194.456.825.258	10.051.629.448	184.405.195.810	0,707430	0,051691	0,670862	1,998832	1,945105	2,001846
39	549.755.813.888	388.698.691.722	19.566.137.344	369.132.554.378	0,707039	0,050338	0,671448	1,998895	1,946564	2,001747
40	1.099.511.627.776	776.990.581.755	38.114.025.483	738.876.556.272	0,706669	0,049053	0,672004	1,998953	1,947959	2,001656

8. Check for existing Integer Sequences by OEIS
Found in Database : 1, 1, 7, 1, 23, 17, 47, 31, 79, 1, 1, 71, 167, 97, 223, 127, 41, 1, 359, 199,
Found in Database : 7, 23, 17, 47, 31, 79, 71, 167, 97, 223, 127, 41, 359, 199, 439, 241, 89, 337, 727, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103,
Found in Database : 7, 17, 23, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 127, 137,

A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P
exponent =log2 (x)	<=x	number 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	number 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
1	2	1	1	0	0	0	0	1	0	0	0	0	0	0	0
2	4	3	1	1	0	0	0	2	0	0	0	0	0	0	0
3	8	5	2	2	0	0	0	4	2	1	1	1	0	0	1
4	16	7	3	3	0	0	0	6	6	3	3	3	0	0	3
5	32	12	6	5	0	0	0	11	13	7	6	9	0	0	4
6	64	20	7	12	0	0	0	19	27	14	13	16	0	0	11
7	128	32	11	20	0	0	0	31	67	39	28	35	0	0	32
8	256	54	16	37	0	0	0	53	138	74	64	78	0	0	60
9	512	97	34	62	0	0	0	96	288	166	122	157	0	0	131
10	1024	161	53	107	0	0	0	160	605	339	266	329	0	0	276
11	2048	289	99	189	0	0	0	288	1240	667	573	673	0	0	567
12	4096	545	174	370	0	0	0	544	2486	1319	1167	1328	0	0	1158
13	8192	961	325	635	0	0	0	960	5094	2693	2401	2690	0	0	2404
14	16384	1792	596	1195	0	0	0	1791	10236	5407	4829	5427	0	0	4809
15	32768	3332	1130	2201	0	0	0	3331	20647	10880	9767	10961	0	0	9686
16	65536	6081	2052	4028	0	0	0	6080	41721	21861	19860	22067	0	0	19654
17	131072	11388	3821	7566	0	0	0	11387	83985	43935	40050	44261	0	0	39724
18	262144	21318	7104	14213	0	0	0	21317	168847	88042	80805	88539	0	0	80308
19	524288	40220	13407	26812	0	0	0	40219	339204	176357	162847	177343	0	0	161861
20	1048576	76224	25369	50854	0	0	0	76223	680606	353404	327202	354537	0	0	326069
21	2097152	144182	48016	96165	0	0	0	144181	1366801	707916	658885	710090	0	0	656711
22	4194304	274311	91482	182828	0	0	0	274310	2741704	1417831	1323873	1420999	0	0	1320705
23	8388608	522318	174005	348312	0	0	0	522317	5499143	2839123	2660020	2845057	0	0	2654086
24	16777216	998286	332691	665594	0	0	0	998285	11025863	5681297	5344566	5694090	0	0	5331773
25	33554432	1910299	636407	1273891	0	0	0	1910298	22103827	11376190	10727637	11396466	0	0	10707361
26	67108864	3660832	1220311	2440520	0	0	0	3660831	44303286	22773903	21529383	22809928	0	0	21493358
27	134217728	7032025	2344656	4687368	0	0	0	7032024	88777637	45584795	43192842	45652041	0	0	43125596
28	268435456	13530333	4512555	9017777	0	0	0	13530332	177871761	91229860	86641901	91365653	0	0	86506108
29	536870912	26076182	8693736	17382445	0	0	0	26076181	356328577	182599803	173728774	182839571	0	0	173489006
30	1073741824	50322316	16774903	33547412	0	0	0	50322315	713738264	365413480	348324784	365877881	0	0	347860383
31	2147483648	97225942	32402950	64822991	0	0	0	97225941	1429495151	731257688	698237463	732137077	0	0	697358074
32	4294967296	188053712	62679024	125374687	0	0	0	188053711	2862801898	1463319239	1399482659	1464982662	0	0	1397819236
33	8589934592	364114624	121362059	242752564	0	0	0	364114623	5732759450	2928163309	2804596141	2931270011	0	0	2801489439
34	17179869184	705755392	235234917	470520474	0	0	0	705755391	11478946323	5859095078	5619851245	5865089708	0	0	5613856615
35	34359738368	1369222759	456381230	912841528	0	0	0	1369222758	22983138420	11723566945	11259571475	11734943136	0	0	11248195284

Development ofAlgorithmic Constructions

Polynom = x^2+2x-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

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

Development of
Algorithmic Constructions