Development of
Algorithmic Constructions

22:00:50

19.Apr 2024

Polynom = x^2-4x+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) = 3 = 3
f(3) = 1 = 1
f(4) = 1 = 1
f(5) = 3 = 3
f(6) = 13 = 13
f(7) = 11 = 11
f(8) = 33 = 3*11
f(9) = 23 = 23
f(10) = 61 = 61
f(11) = 39 = 3*13
f(12) = 97 = 97
f(13) = 59 = 59
f(14) = 141 = 3*47
f(15) = 83 = 83
f(16) = 193 = 193
f(17) = 111 = 3*37
f(18) = 253 = 11*23
f(19) = 143 = 11*13
f(20) = 321 = 3*107
f(21) = 179 = 179
f(22) = 397 = 397
f(23) = 219 = 3*73
f(24) = 481 = 13*37
f(25) = 263 = 263
f(26) = 573 = 3*191
f(27) = 311 = 311
f(28) = 673 = 673
f(29) = 363 = 3*11*11
f(30) = 781 = 11*71
f(31) = 419 = 419
f(32) = 897 = 3*13*23
f(33) = 479 = 479
f(34) = 1021 = 1021
f(35) = 543 = 3*181
f(36) = 1153 = 1153
f(37) = 611 = 13*47
f(38) = 1293 = 3*431
f(39) = 683 = 683
f(40) = 1441 = 11*131
f(41) = 759 = 3*11*23
f(42) = 1597 = 1597
f(43) = 839 = 839
f(44) = 1761 = 3*587
f(45) = 923 = 13*71
f(46) = 1933 = 1933
f(47) = 1011 = 3*337
f(48) = 2113 = 2113
f(49) = 1103 = 1103
f(50) = 2301 = 3*13*59
f(51) = 1199 = 11*109
f(52) = 2497 = 11*227
f(53) = 1299 = 3*433
f(54) = 2701 = 37*73
f(55) = 1403 = 23*61
f(56) = 2913 = 3*971
f(57) = 1511 = 1511
f(58) = 3133 = 13*241
f(59) = 1623 = 3*541
f(60) = 3361 = 3361
f(61) = 1739 = 37*47
f(62) = 3597 = 3*11*109
f(63) = 1859 = 11*13*13
f(64) = 3841 = 23*167
f(65) = 1983 = 3*661
f(66) = 4093 = 4093
f(67) = 2111 = 2111
f(68) = 4353 = 3*1451
f(69) = 2243 = 2243
f(70) = 4621 = 4621
f(71) = 2379 = 3*13*61
f(72) = 4897 = 59*83
f(73) = 2519 = 11*229
f(74) = 5181 = 3*11*157
f(75) = 2663 = 2663
f(76) = 5473 = 13*421
f(77) = 2811 = 3*937
f(78) = 5773 = 23*251
f(79) = 2963 = 2963
f(80) = 6081 = 3*2027
f(81) = 3119 = 3119
f(82) = 6397 = 6397
f(83) = 3279 = 3*1093
f(84) = 6721 = 11*13*47
f(85) = 3443 = 11*313
f(86) = 7053 = 3*2351
f(87) = 3611 = 23*157
f(88) = 7393 = 7393
f(89) = 3783 = 3*13*97
f(90) = 7741 = 7741
f(91) = 3959 = 37*107
f(92) = 8097 = 3*2699
f(93) = 4139 = 4139
f(94) = 8461 = 8461
f(95) = 4323 = 3*11*131
f(96) = 8833 = 11*11*73
f(97) = 4511 = 13*347
f(98) = 9213 = 3*37*83
f(99) = 4703 = 4703
f(100) = 9601 = 9601

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 following proof does not mean that the primes of the form f(x)=x^2-4x+1 is infinitiv,
but supposes that the primes of the form p | x^2-4x+1 is infinitiv.

i would like to show that the result of the algorithm for f'(x) is either 1 or a prime
and not a result of two primes.

f'(x) is the value which is reduced by the primes of f(0) up to f'(x-1) according the algorithm above.

The proof bases on the algorithm above especially that a prime p occurs periodically with periode p on the polynom.
If p | f(x) then also p | f(x+p)

a) As i regard f(x)=abs (x^2-4x+1) i check the sequence of primes up to the point
that f(x)=x^2-4x+1

f(0)=1
f(1)=2
f(2)=3
f(3)=2
f(4)=1
f(x)>0 for x>=5

b) Substitution of the polynom

The polynom f(x)=x^2-4x+1 could be written as f(y)=y^2-3 with x=y+2

Supposing that the values f(0), f'(1), f'(2), ..., f'(y-1) are either primes or 1.
Supposing f'(y)=a*b, a, b element of N>1
f'(y) is the value which is reduced by the primes of f(0) up to f'(y-1)

a>y-1 and b>y-1  because the values of f(0) up to f'(y-1) were already calculated by the algorithm
if a<=y-1 or b<=y-1 then they would be eleminated by the algorithm before.
( if a<=y-1 then y-a>=1 that means f'(y-a) is a prime and was already sieved out by the algorithm before. )

The values of the polynom f(y) are symmetrical to the y-axis, that means f(-y)=f(y)

Therefore
a>2(y-1) and b>2(y-1)
if a<=2(y-1) or b<=2(y-1) then they would be eleminated by the algorithm before.

f(y)=abs(y^2-3) therefore f(0)=3
That means that you could add the nullpoint of the polynom to the estimation
a>2(y-1)+1 and b>2(y-1)+1
<=> a>2y-1 and b>2y-1

Supposing f'(y)=a*b, a, b element of N>1 you could lead this equilation to a contradiction
f'(y)>(2y-1)^2
<=> f'(y)>4y^2-4y+1
But f(y) is at least f(y)=y^2-4y+1 with y>=3
f'(y) would be greater than f(y)
But this is a contradiction.

Therefore the supposition is wrong and f'(y) is either 1 or a prime
and not a result of two primes.

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)  y=x-2  with x>=5 respectively y>=3
f'(x)>(2(x-2)-1)
f'(x)> 2x-5      with x>=5

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, 3, 1, 1, 1, 13, 11, 1, 23, 61, 1, 97, 59, 47, 83, 193, 37, 1, 1, 107, 179, 397, 73, 1, 263, 191, 311, 673, 1, 71, 419, 1, 479, 1021, 181, 1153, 1, 431, 683, 131, 1, 1597, 839, 587, 1, 1933, 337, 2113, 1103, 1, 109, 227, 433, 1, 1, 971, 1511, 241, 541, 3361, 1, 1, 1, 167, 661, 4093, 2111, 1451, 2243, 4621, 1, 1, 229, 157, 2663, 421, 937, 251, 2963, 2027, 3119, 6397, 1093, 1, 313, 2351, 1, 7393, 1, 7741, 1, 2699, 4139, 8461, 1, 1, 347, 1, 4703, 9601, 1, 769, 5099, 3467, 5303, 983, 1, 239, 1, 1, 5939, 12097, 2053, 12541, 491, 1, 601, 1223, 2281, 13921, 7079, 4799, 563, 647, 2521, 15373, 1, 1, 733, 16381, 1, 277, 373, 5807, 1, 1381, 3037, 18493, 853, 577, 743, 19597, 3313, 20161, 10223, 6911, 457, 21313, 1, 1, 1009, 7499, 11399, 1777, 1, 1031, 12011, 8111, 12323, 1, 383, 1, 12959, 8747, 359, 26893, 349, 467, 1, 409, 1, 1, 443, 29581, 1151, 10091, 1, 659, 1, 2437, 1, 10799, 1489, 3011, 5581, 1, 1, 887, 17483, 599, 5953, 36097, 1, 1117, 1693, 37633, 6337, 1, 19403, 1, 1523, 1, 6733, 40801, 1873, 1, 21011, 42433, 1, 43261, 21839, 14699, 22259, 3457, 7561, 1, 1, 15551, 1811, 47521, 7993, 48397, 24419, 16427, 1, 1, 1, 4643, 25763, 17327, 1, 1, 8893, 503, 27143, 18251, 27611, 1, 1, 4357, 28559, 1, 1, 1, 757, 1609, 30011, 877, 1, 5591, 10333, 62497, 2423, 1, 32003, 64513, 10837, 1, 33023, 2017, 3049, 2939, 11353, 827, 34583, 1787, 35111, 70753, 1, 71821, 1, 1, 1, 1, 12421, 2029, 37811, 25391, 1, 709, 12973, 7127, 1, 2039, 1741, 1367, 13537, 1, 41183, 27647, 41759, 6469, 1283, 1, 42923, 613, 3347, 87613, 1, 88801, 44699, 1, 1, 8291, 1, 92413, 46511, 1, 47123, 7297, 15913, 96097, 1307, 1, 1, 8963, 719, 7681, 1069, 911, 1, 102397, 1321, 103681, 52163, 3181, 4801, 106273, 1, 107581, 1, 36299, 54779, 110221, 18481, 8581, 5101, 1, 56783, 4967, 1, 115597, 1237, 2999, 997, 1621, 1, 10883, 1, 1091, 60899, 122497, 20533, 5387, 62303, 41771, 1, 126733, 1931, 1, 64439, 3323, 2833, 131041, 21961, 2819, 1129, 44651, 67343, 947, 2063, 136897, 68819, 1, 5351, 2293, 1019, 1297, 1, 1, 6529, 13127, 1861, 1, 1033, 2137, 2003, 1, 1, 3203, 75659, 1, 6949, 4153, 25741, 11941, 1, 1, 78803, 1, 1, 829, 7309, 1, 81203, 1, 27337, 863, 1, 55487, 83639, 1571, 1, 1187, 7753, 57131, 86111, 173053, 1, 174721, 1439, 4523, 3853, 4813, 2711, 1, 6947, 1, 91163, 2207, 1, 8039, 1, 62207, 7211, 17123, 1, 190093, 95483, 4919, 1, 193597, 32413, 1, 98123, 2857, 9001, 1, 33301, 1, 100799, 67499, 7823, 204301, 1487, 2903, 103511, 6301, 1, 4463, 1, 211597, 1801, 71147, 107183, 16561, 36037, 5869, 1, 6637, 1549, 220897, 36973, 17137, 111863, 74891, 1163, 9851, 2917, 20771, 10429, 1259, 115679, 4943, 1, 1, 1, 1, 118583, 238141, 3623, 1, 120539, 80687, 121523, 244033, 1, 1, 123503, 6359, 5413, 22727, 3803, 252001, 1, 1, 2713, 256033, 42841, 258061, 129539, 1, 1, 23831, 1907, 1, 132611, 6827, 1249, 1499, 44893, 4583, 135719, 1, 12433, 21121, 1, 276673, 138863, 92927, 1, 280897, 46993, 3877, 1, 8641, 143111, 12491, 3697, 1201, 145259, 1, 2399, 1, 1, 26903, 1, 99371, 149603, 300301, 1, 23269, 151799, 2161, 1429, 306913, 1, 1, 2543, 103787, 6793, 313597, 52453, 2411, 1, 106031, 1, 29123, 4871, 1, 161879, 2927, 163019, 3373, 54721, 329473, 165311, 1, 1, 1, 1, 336397, 12983, 4909, 169943, 1427, 57037, 343393, 172283, 10477, 1213, 348097, 2531, 3613, 1, 1, 177011, 5003, 1, 1, 1, 10909, 3061, 1, 60601, 1, 1, 1, 1, 369661, 61813, 33827, 1, 124847, 187883, 1, 1, 1657, 1453, 2087, 191579, 29569, 5843, 35171, 4129, 1, 195311, 391873, 65521, 1319, 1, 132299, 1, 3301, 1, 401953, 201611, 1847, 8821, 6899, 68053, 409597, 15803, 137387, 18793, 1, 69337, 1, 1, 139967, 210599, 1, 1, 425101, 213203, 1, 19501, 430333, 71941, 432961, 16703, 1, 218459, 438241, 1, 440893, 20101, 13441, 1, 446221, 1, 9551, 225119, 11579, 226463, 19751, 75937, 41543, 1, 2099, 3779, 35569, 2089, 6551, 10141, 1, 18047, 470593, 7151, 1, 2447, 158699, 1, 478861, 1, 481633, 241511, 161471, 1, 3407, 1, 3121, 1, 164267, 2267, 495613, 82837, 1667, 249923, 167087, 1, 45827, 6481, 506941, 1619, 1559, 1, 1, 1823, 13933, 1, 1, 23629, 521281, 87121, 1, 7103, 175691, 264263, 529981, 3851, 3191, 1, 1, 268643, 6491, 1, 541693, 1607, 2557, 273059, 1, 91513, 50051, 1, 184511, 21347, 556513, 93001, 4271, 280499, 14423, 3863, 1, 1, 51683, 7703, 3229, 1583, 1, 96013, 6959, 289559, 5231, 22391, 1, 8867, 1, 2749, 1, 1, 592897, 7621, 595981, 1789, 1, 1, 4211, 100621, 10259, 1, 202799, 304979, 8377, 102181, 3637, 5051, 1, 1, 3251, 1, 624097, 312839, 5651, 314423, 630433, 105337, 633613, 2221, 1, 8627, 639997, 1753, 49477, 322403, 215471, 324011, 649633, 108541, 59351, 1, 16823, 6997, 1, 110161, 1669, 1, 221951, 4021, 669121, 10163, 5557, 14653, 225227, 1, 1, 113437, 1, 341963, 17579, 1, 62627, 10463, 692221, 346943, 4933, 1, 1, 1979, 6563, 351959, 235199, 2473, 64451, 1, 30971, 1, 3911, 358703, 719101, 9241, 3023, 362099, 21997, 33073, 56101, 1, 2341, 2339, 1, 368939, 1, 123553, 1, 1, 22621, 374111, 20269, 1, 753421, 1, 1, 379319, 4201, 3433, 1, 2677, 1, 1, 770881, 1, 1, 1, 1, 4019, 781453, 11867, 71363, 393383, 20219, 395159, 34439, 132313, 795661, 30671, 266411, 5641, 1, 1, 1871, 404099, 3253, 31223, 813601, 3673, 35531, 8713, 1, 1, 1, 137713, 22381, 414959, 277247, 18121, 1, 139537, 1993, 1, 25537, 2953, 846397, 1, 850081, 6983, 284591, 1, 10331, 1, 1, 39229, 26209, 2269, 1, 2459, 872353, 437111, 12697, 438983, 4861, 146953, 1, 3659, 295787, 444623, 891133, 1, 2579, 12119, 299567, 450299, 39239, 1, 1, 2687, 303371, 1, 913933, 152641, 1, 19993, 3167, 1, 84131, 14051, 929293, 465611, 1, 467543, 15361, 156493, 12889, 36263, 314927, 1, 86243, 3371, 952573, 12899, 1, 1, 9901, 1, 41927, 5821, 1, 44101, 2377, 1, 2617, 1, 1, 491039, 1, 164341, 988033, 4091, 1, 38231,

6. Sequence of the polynom (only primes)

3, 13, 11, 23, 61, 97, 59, 47, 83, 193, 37, 107, 179, 397, 73, 263, 191, 311, 673, 71, 419, 479, 1021, 181, 1153, 431, 683, 131, 1597, 839, 587, 1933, 337, 2113, 1103, 109, 227, 433, 971, 1511, 241, 541, 3361, 167, 661, 4093, 2111, 1451, 2243, 4621, 229, 157, 2663, 421, 937, 251, 2963, 2027, 3119, 6397, 1093, 313, 2351, 7393, 7741, 2699, 4139, 8461, 347, 4703, 9601, 769, 5099, 3467, 5303, 983, 239, 5939, 12097, 2053, 12541, 491, 601, 1223, 2281, 13921, 7079, 4799, 563, 647, 2521, 15373, 733, 16381, 277, 373, 5807, 1381, 3037, 18493, 853, 577, 743, 19597, 3313, 20161, 10223, 6911, 457, 21313, 1009, 7499, 11399, 1777, 1031, 12011, 8111, 12323, 383, 12959, 8747, 359, 26893, 349, 467, 409, 443, 29581, 1151, 10091, 659, 2437, 10799, 1489, 3011, 5581, 887, 17483, 599, 5953, 36097, 1117, 1693, 37633, 6337, 19403, 1523, 6733, 40801, 1873, 21011, 42433, 43261, 21839, 14699, 22259, 3457, 7561, 15551, 1811, 47521, 7993, 48397, 24419, 16427, 4643, 25763, 17327, 8893, 503, 27143, 18251, 27611, 4357, 28559, 757, 1609, 30011, 877, 5591, 10333, 62497, 2423, 32003, 64513, 10837, 33023, 2017, 3049, 2939, 11353, 827, 34583, 1787, 35111, 70753, 71821, 12421, 2029, 37811, 25391, 709, 12973, 7127, 2039, 1741, 1367, 13537, 41183, 27647, 41759, 6469, 1283, 42923, 613, 3347, 87613, 88801, 44699, 8291, 92413, 46511, 47123, 7297, 15913, 96097, 1307, 8963, 719, 7681, 1069, 911, 102397, 1321, 103681, 52163, 3181, 4801, 106273, 107581, 36299, 54779, 110221, 18481, 8581, 5101, 56783, 4967, 115597, 1237, 2999, 997, 1621, 10883, 1091, 60899, 122497, 20533, 5387, 62303, 41771, 126733, 1931, 64439, 3323, 2833, 131041, 21961, 2819, 1129, 44651, 67343, 947, 2063, 136897, 68819, 5351, 2293, 1019, 1297, 6529, 13127, 1861, 1033, 2137, 2003, 3203, 75659, 6949, 4153, 25741, 11941, 78803, 829, 7309, 81203, 27337, 863, 55487, 83639, 1571, 1187, 7753, 57131, 86111, 173053, 174721, 1439, 4523, 3853, 4813, 2711, 6947, 91163, 2207, 8039, 62207, 7211, 17123, 190093, 95483, 4919, 193597, 32413, 98123, 2857, 9001, 33301, 100799, 67499, 7823, 204301, 1487, 2903, 103511, 6301, 4463, 211597, 1801, 71147, 107183, 16561, 36037, 5869, 6637, 1549, 220897, 36973, 17137, 111863, 74891, 1163, 9851, 2917, 20771, 10429, 1259, 115679, 4943, 118583, 238141, 3623, 120539, 80687, 121523, 244033, 123503, 6359, 5413, 22727, 3803, 252001, 2713, 256033, 42841, 258061, 129539, 23831, 1907, 132611, 6827, 1249, 1499, 44893, 4583, 135719, 12433, 21121, 276673, 138863, 92927, 280897, 46993, 3877, 8641, 143111, 12491, 3697, 1201, 145259, 2399, 26903, 99371, 149603, 300301, 23269, 151799, 2161, 1429, 306913, 2543, 103787, 6793, 313597, 52453, 2411, 106031, 29123, 4871, 161879, 2927, 163019, 3373, 54721, 329473, 165311, 336397, 12983, 4909, 169943, 1427, 57037, 343393, 172283, 10477, 1213, 348097, 2531, 3613, 177011, 5003, 10909, 3061, 60601, 369661, 61813, 33827, 124847, 187883, 1657, 1453, 2087, 191579, 29569, 5843, 35171, 4129, 195311, 391873, 65521, 1319, 132299, 3301, 401953, 201611, 1847, 8821, 6899, 68053, 409597, 15803, 137387, 18793, 69337, 139967, 210599, 425101, 213203, 19501, 430333, 71941, 432961, 16703, 218459, 438241, 440893, 20101, 13441, 446221, 9551, 225119, 11579, 226463, 19751, 75937, 41543, 2099, 3779, 35569, 2089, 6551, 10141, 18047, 470593, 7151, 2447, 158699, 478861, 481633, 241511, 161471, 3407, 3121, 164267, 2267, 495613, 82837, 1667, 249923, 167087, 45827, 6481, 506941, 1619, 1559, 1823, 13933, 23629, 521281, 87121, 7103, 175691, 264263, 529981, 3851, 3191, 268643, 6491, 541693, 1607, 2557, 273059, 91513, 50051, 184511, 21347, 556513, 93001, 4271, 280499, 14423, 3863, 51683, 7703, 3229, 1583, 96013, 6959, 289559, 5231, 22391, 8867, 2749, 592897, 7621, 595981, 1789, 4211, 100621, 10259, 202799, 304979, 8377, 102181, 3637, 5051, 3251, 624097, 312839, 5651, 314423, 630433, 105337, 633613, 2221, 8627, 639997, 1753, 49477, 322403, 215471, 324011, 649633, 108541, 59351, 16823, 6997, 110161, 1669, 221951, 4021, 669121, 10163, 5557, 14653, 225227, 113437, 341963, 17579, 62627, 10463, 692221, 346943, 4933, 1979, 6563, 351959, 235199, 2473, 64451, 30971, 3911, 358703, 719101, 9241, 3023, 362099, 21997, 33073, 56101, 2341, 2339, 368939, 123553, 22621, 374111, 20269, 753421, 379319, 4201, 3433, 2677, 770881, 4019, 781453, 11867, 71363, 393383, 20219, 395159, 34439, 132313, 795661, 30671, 266411, 5641, 1871, 404099, 3253, 31223, 813601, 3673, 35531, 8713, 137713, 22381, 414959, 277247, 18121, 139537, 1993, 25537, 2953, 846397, 850081, 6983, 284591, 10331, 39229, 26209, 2269, 2459, 872353, 437111, 12697, 438983, 4861, 146953, 3659, 295787, 444623, 891133, 2579, 12119, 299567, 450299, 39239, 2687, 303371, 913933, 152641, 19993, 3167, 84131, 14051, 929293, 465611, 467543, 15361, 156493, 12889, 36263, 314927, 86243, 3371, 952573, 12899, 9901, 41927, 5821, 44101, 2377, 2617, 491039, 164341, 988033, 4091, 38231,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-4x+1 and
the reducible primes which appear as divisor for the first time
p | x^2-4x+1 and p < x^2-4x+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^2-4x+1	P(x) \| x^2-4x+1	C/B	D/B	E/B	C(n) / C(n-1)	D(n) / D(n-1)	E(n) / E(n-1)
1	10	6	4	2	0,600000	0,400000	0,200000
2	100	72	21	51	0,720000	0,210000	0,510000	12,000000	5,250000	25,500000
3	1.000	730	121	609	0,730000	0,121000	0,609000	10,138889	5,761905	11,941176
4	10.000	7.250	851	6.399	0,725000	0,085100	0,639900	9,931507	7,033058	10,507389
5	100.000	71.897	6.665	65.232	0,718970	0,066650	0,652320	9,916828	7,831962	10,194093
6	1.000.000	714.326	54.516	659.810	0,714326	0,054516	0,659810	9,935408	8,179445	10,114821
7	10.000.000	7.108.758	460.021	6.648.737	0,710876	0,046002	0,664874	9,951700	8,438275	10,076745
8	100.000.000	70.834.766	3.982.975	66.851.791	0,708348	0,039830	0,668518	9,964436	8,658246	10,054811
9	1.000.000.000	706.466.981	35.174.009	671.292.972	0,706467	0,035174	0,671293	9,973450	8,831090	10,041511
10	10.000.000.000	7.049.881.447	314.770.815	6.735.110.632	0,704988	0,031477	0,673511	9,979067	8,948960	10,033042
11	100.000.000.000	70.380.549.172	2.848.504.520	67.532.044.652	0,703805	0,028485	0,675320	9,983224	9,049456	10,026865
12	1.000.000.000.000	702.837.765.233	26.013.764.611	676.824.000.622	0,702838	0,026014	0,676824	9,986250	9,132429	10,022264


A	B	C	D	E	F	G	H	I	J	K
2^n	x	all Primes	P(x)=x^2-4x+1	P(x) \| x^2-4x+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	2	2	0	0,500000	0,500000	0,000000	1,000000	1,000000
3	8	4	3	1	0,500000	0,375000	0,125000	2,000000	1,500000
4	16	11	6	5	0,687500	0,375000	0,312500	2,750000	2,000000	5,000000
5	32	22	8	14	0,687500	0,250000	0,437500	2,000000	1,333333	2,800000
6	64	45	14	31	0,703125	0,218750	0,484375	2,045455	1,750000	2,214286
7	128	93	25	68	0,726563	0,195313	0,531250	2,066667	1,785714	2,193548
8	256	186	41	145	0,726563	0,160156	0,566406	2,000000	1,640000	2,132353
9	512	378	69	309	0,738281	0,134766	0,603516	2,032258	1,682927	2,131034
10	1.024	746	122	624	0,728516	0,119141	0,609375	1,973545	1,768116	2,019417
11	2.048	1.496	217	1.279	0,730469	0,105957	0,624512	2,005362	1,778689	2,049679
12	4.096	2.996	388	2.608	0,731445	0,094727	0,636719	2,002674	1,788018	2,039093
13	8.192	5.943	713	5.230	0,725464	0,087036	0,638428	1,983645	1,837629	2,005368
14	16.384	11.868	1.304	10.564	0,724365	0,079590	0,644775	1,996971	1,828892	2,019885
15	32.768	23.690	2.421	21.269	0,722961	0,073883	0,649078	1,996124	1,856595	2,013347
16	65.536	47.199	4.515	42.684	0,720200	0,068893	0,651306	1,992360	1,864932	2,006864
17	131.072	94.145	8.562	85.583	0,718269	0,065323	0,652946	1,994640	1,896346	2,005037
18	262.144	187.935	15.995	171.940	0,716915	0,061016	0,655899	1,996229	1,868138	2,009044
19	524.288	375.124	30.069	345.055	0,715492	0,057352	0,658140	1,996031	1,879900	2,006834
20	1.048.576	748.938	57.009	691.929	0,714243	0,054368	0,659875	1,996508	1,895939	2,005272
21	2.097.152	1.495.456	107.840	1.387.616	0,713089	0,051422	0,661667	1,996769	1,891631	2,005431
22	4.194.304	2.986.502	204.655	2.781.847	0,712038	0,048794	0,663244	1,997051	1,897765	2,004767
23	8.388.608	5.965.103	390.252	5.574.851	0,711096	0,046522	0,664574	1,997354	1,906877	2,004011
24	16.777.216	11.915.756	745.708	11.170.048	0,710234	0,044448	0,665787	1,997578	1,910837	2,003650
25	33.554.432	23.805.274	1.427.090	22.378.184	0,709452	0,042531	0,666922	1,997798	1,913738	2,003410
26	67.108.864	47.562.506	2.736.167	44.826.339	0,708737	0,040772	0,667965	1,997982	1,917305	2,003127
27	134.217.728	95.037.108	5.257.030	89.780.078	0,708082	0,039168	0,668914	1,998152	1,921312	2,002842
28	268.435.456	189.910.577	10.119.280	179.791.297	0,707472	0,037697	0,669775	1,998278	1,924904	2,002575
29	536.870.912	379.531.096	19.500.520	360.030.576	0,706932	0,036323	0,670609	1,998473	1,927066	2,002492
30	1.073.741.824	758.505.599	37.631.611	720.873.988	0,706413	0,035047	0,671366	1,998533	1,929775	2,002258
31	2.147.483.648	1.515.990.840	72.702.752	1.443.288.088	0,705938	0,033855	0,672083	1,998655	1,931960	2,002136
32	4.294.967.296	3.030.069.117	140.612.644	2.889.456.473	0,705493	0,032739	0,672754	1,998738	1,934076	2,001996
33	8.589.934.592	6.056.548.773	272.278.713	5.784.270.060	0,705075	0,031697	0,673378	1,998815	1,936374	2,001854
34	17.179.869.184	12.106.430.840	527.737.486	11.578.693.354	0,704687	0,030718	0,673969	1,998899	1,938225	2,001755
35	34.359.738.368	24.200.370.632	1.023.919.349	23.176.451.283	0,704323	0,029800	0,674524	1,998968	1,940206	2,001647
36	68.719.476.736	48.377.290.173	1.988.288.904	46.389.001.269	0,703982	0,028933	0,675049	1,999031	1,941841	2,001558
37	137.438.953.472	96.710.404.839	3.864.234.742	92.846.170.097	0,703661	0,028116	0,675545	1,999087	1,943498	2,001469
38	274.877.906.944	193.337.551.506	7.516.299.681	185.821.251.825	0,703358	0,027344	0,676014	1,999139	1,945094	2,001388
39	549.755.813.888	386.517.873.054	14.631.035.754	371.886.837.300	0,703072	0,026614	0,676458	1,999187	1,946574	2,001315
40	1.099.511.627.776	772.738.562.104	28.500.611.740	744.237.950.364	0,702802	0,025921	0,676880	1,999231	1,947956	2,001248

8. Check for existing Integer Sequences by OEIS
Found in Database : 1, 1, 3, 1, 1, 1, 13, 11, 1, 23, 61, 1, 97, 59, 47, 83, 193, 37, 1, 1,
Found in Database : 3, 13, 11, 23, 61, 97, 59, 47, 83, 193, 37, 107, 179, 397, 73, 263, 191, 311, 673, 71, 419, 479, 1021, 181, 1153, 431, 683,
Found in Database : 3, 11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131,

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
3	8	3	1	0	1	0	1	0	0	0	0	0	0	0	0
4	16	8	4	0	3	0	2	0	2	0	2	0	1	0	1
5	32	13	7	0	4	0	4	0	9	1	7	1	3	0	4
6	64	19	13	0	7	0	7	0	26	7	18	4	8	3	10
7	128	30	24	0	11	0	14	0	63	19	43	10	21	9	22
8	256	46	40	0	20	0	21	0	140	50	89	24	45	26	44
9	512	74	68	0	33	0	36	0	304	113	190	51	98	62	92
10	1024	127	121	0	58	0	64	0	619	232	386	107	194	125	192
11	2048	222	216	0	109	0	108	0	1274	529	744	273	359	256	385
12	4096	393	387	0	190	0	198	0	2603	1099	1503	543	741	556	762
13	8192	718	712	0	357	0	356	0	5225	2243	2981	1115	1484	1128	1497
14	16384	1309	1303	0	649	0	655	0	10559	4581	5977	2285	2972	2296	3005
15	32768	2426	2420	0	1202	0	1219	0	21264	9301	11962	4644	5994	4657	5968
16	65536	4520	4514	0	2225	0	2290	0	42679	19005	23673	9485	11819	9520	11854
17	131072	8567	8561	0	4261	0	4301	0	85578	38528	47049	19252	23510	19276	23539
18	262144	16000	15994	0	7956	0	8039	0	171935	77845	94089	38969	47147	38876	46942
19	524288	30074	30068	0	14998	0	15071	0	345050	157119	187930	78588	93995	78531	93935
20	1048576	57014	57008	0	28535	0	28474	0	691924	317219	374704	158636	187242	158583	187462
21	2097152	107845	107839	0	53893	0	53947	0	1387611	639532	748078	319565	373633	319967	374445
22	4194304	204660	204654	0	102108	0	102547	0	2781842	1287780	1494061	643704	747121	644076	746940
23	8388608	390257	390251	0	194859	0	195393	0	5574846	2590859	2983986	1295745	1492740	1295114	1491246
24	16777216	745713	745707	0	372597	0	373111	0	11170043	5209425	5960617	2605390	2980620	2604035	2979997
25	33554432	1427095	1427089	0	713573	0	713517	0	22378179	10471844	11906334	5234629	5952400	5237215	5953934
26	67108864	2736172	2736166	0	1367721	0	1368446	0	44826334	21036236	23790097	10517552	11897025	10518684	11893072
27	134217728	5257035	5257029	0	2628563	0	2628467	0	89780073	42242960	47537112	21123441	23768896	21119519	23768216
28	268435456	10119285	10119279	0	5061308	0	5057972	0	179791292	84802770	94988521	42403913	47492942	42398857	47495579
29	536870912	19500525	19500519	0	9751880	0	9748640	0	360030571	170199307	189831263	85106259	94918761	85093048	94912502
30	1073741824	37631616	37631610	0	18818345	0	18813266	0	720873983	341482315	379391667	170750735	189691130	170731580	189700537
31	2147483648	72702757	72702751	0	36351900	0	36350852	0	1443288083	685049477	758238605	342528926	379109843	342520551	379128762
32	4294967296	140612649	140612643	0	70307603	0	70305041	0	2889456468	1373984183	1515472284	686984397	757721432	686999786	757750852
33	8589934592	272278718	272278712	0	136134932	0	136143781	0	5784270055	2755187795	3029082259	1377585503	1514505081	1377602292	1514577178
34	17179869184	527737491	527737485	0	263867956	0	263869530	0	11578693349	5523878707	6054814641	2761939621	3027400359	2761939086	3027414282
35	34359738368	1023919354	1023919348	0	511952959	0	511966390	0	23176451274	11073256757	12103194516	5536618434	6051613245	5536638323	6051581271

Development ofAlgorithmic Constructions

Polynom = x^2-4x+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