Development of
Algorithmic Constructions

00:33:17

19.Apr 2024

Polynom = x^2-16x+2

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) = 13 = 13
f(2) = 13 = 13
f(3) = 37 = 37
f(4) = 23 = 23
f(5) = 53 = 53
f(6) = 29 = 29
f(7) = 61 = 61
f(8) = 31 = 31
f(9) = 61 = 61
f(10) = 29 = 29
f(11) = 53 = 53
f(12) = 23 = 23
f(13) = 37 = 37
f(14) = 13 = 13
f(15) = 13 = 13
f(16) = 1 = 1
f(17) = 19 = 19
f(18) = 19 = 19
f(19) = 59 = 59
f(20) = 41 = 41
f(21) = 107 = 107
f(22) = 67 = 67
f(23) = 163 = 163
f(24) = 97 = 97
f(25) = 227 = 227
f(26) = 131 = 131
f(27) = 299 = 13*23
f(28) = 169 = 13*13
f(29) = 379 = 379
f(30) = 211 = 211
f(31) = 467 = 467
f(32) = 257 = 257
f(33) = 563 = 563
f(34) = 307 = 307
f(35) = 667 = 23*29
f(36) = 361 = 19*19
f(37) = 779 = 19*41
f(38) = 419 = 419
f(39) = 899 = 29*31
f(40) = 481 = 13*37
f(41) = 1027 = 13*79
f(42) = 547 = 547
f(43) = 1163 = 1163
f(44) = 617 = 617
f(45) = 1307 = 1307
f(46) = 691 = 691
f(47) = 1459 = 1459
f(48) = 769 = 769
f(49) = 1619 = 1619
f(50) = 851 = 23*37
f(51) = 1787 = 1787
f(52) = 937 = 937
f(53) = 1963 = 13*151
f(54) = 1027 = 13*79
f(55) = 2147 = 19*113
f(56) = 1121 = 19*59
f(57) = 2339 = 2339
f(58) = 1219 = 23*53
f(59) = 2539 = 2539
f(60) = 1321 = 1321
f(61) = 2747 = 41*67
f(62) = 1427 = 1427
f(63) = 2963 = 2963
f(64) = 1537 = 29*53
f(65) = 3187 = 3187
f(66) = 1651 = 13*127
f(67) = 3419 = 13*263
f(68) = 1769 = 29*61
f(69) = 3659 = 3659
f(70) = 1891 = 31*61
f(71) = 3907 = 3907
f(72) = 2017 = 2017
f(73) = 4163 = 23*181
f(74) = 2147 = 19*113
f(75) = 4427 = 19*233
f(76) = 2281 = 2281
f(77) = 4699 = 37*127
f(78) = 2419 = 41*59
f(79) = 4979 = 13*383
f(80) = 2561 = 13*197
f(81) = 5267 = 23*229
f(82) = 2707 = 2707
f(83) = 5563 = 5563
f(84) = 2857 = 2857
f(85) = 5867 = 5867
f(86) = 3011 = 3011
f(87) = 6179 = 37*167
f(88) = 3169 = 3169
f(89) = 6499 = 67*97
f(90) = 3331 = 3331
f(91) = 6827 = 6827
f(92) = 3497 = 13*269
f(93) = 7163 = 13*19*29
f(94) = 3667 = 19*193
f(95) = 7507 = 7507
f(96) = 3841 = 23*167
f(97) = 7859 = 29*271
f(98) = 4019 = 4019
f(99) = 8219 = 8219
f(100) = 4201 = 4201

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^2-16x+2

f(0)=1
f(1)=13
f(2)=1
f(3)=37
f(4)=23
f(5)=53
f(6)=29
f(7)=61
f(8)=31
f(9)=1
f(10)=1
f(11)=1
f(12)=1
f(13)=1
f(14)=1
f(15)=1
f(16)=1
f(17)=19
f(18)=1
f(19)=59
f(20)=41
f(21)=107
f(22)=67
f(23)=163
f(24)=97
f(25)=227
f(26)=131
f(27)=1
f(28)=1
f(29)=379
f(30)=211
f(31)=467
f(32)=257
f(33)=563
f(34)=307
f(35)=1
f(36)=1
f(37)=1
f(38)=419
f(39)=1
f(40)=1
f(41)=79
f(42)=547
f(43)=1163
f(44)=617
f(45)=1307
f(46)=691
f(47)=1459
f(48)=769
f(49)=1619
f(50)=1
f(51)=1787
f(52)=937
f(53)=151
f(54)=1
f(55)=113
f(56)=1
f(57)=2339
f(58)=1
f(59)=2539
f(60)=1321
f(61)=1
f(62)=1427
f(63)=2963
f(64)=1
f(65)=3187
f(66)=127
f(67)=263
f(68)=1
f(69)=3659
f(70)=1
f(71)=3907
f(72)=2017
f(73)=181
f(74)=1
f(75)=233
f(76)=2281
f(77)=1
f(78)=1
f(79)=383
f(80)=197
f(81)=229
f(82)=2707
f(83)=5563
f(84)=2857
f(85)=5867
f(86)=3011
f(87)=167
f(88)=3169
f(89)=1
f(90)=3331
f(91)=6827
f(92)=269
f(93)=1
f(94)=193
f(95)=7507
f(96)=1
f(97)=271
f(98)=4019
f(99)=8219

b) Substitution of the polynom
The polynom f(x)=x^2-16x+2  could be written as f(y)= y^2-62 with x=y+8

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-8
f'(x)>2x-17  with x > 8

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, 13, 1, 37, 23, 53, 29, 61, 31, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 59, 41, 107, 67, 163, 97, 227, 131, 1, 1, 379, 211, 467, 257, 563, 307, 1, 1, 1, 419, 1, 1, 79, 547, 1163, 617, 1307, 691, 1459, 769, 1619, 1, 1787, 937, 151, 1, 113, 1, 2339, 1, 2539, 1321, 1, 1427, 2963, 1, 3187, 127, 263, 1, 3659, 1, 3907, 2017, 181, 1, 233, 2281, 1, 1, 383, 197, 229, 2707, 5563, 2857, 5867, 3011, 167, 3169, 1, 3331, 6827, 269, 1, 193, 7507, 1, 271, 4019, 8219, 4201, 277, 1, 8963, 199, 719, 367, 9739, 4969, 10139, 5171, 1, 283, 577, 1, 1, 5801, 223, 463, 1, 1, 1, 1, 13163, 1, 13627, 239, 613, 1, 1, 7411, 1, 1, 1, 7907, 16067, 8161, 281, 8419, 17099, 8681, 17627, 389, 443, 709, 1439, 9491, 19259, 9769, 19819, 1, 1, 10337, 20963, 10627, 743, 1, 1, 863, 22739, 1, 631, 11827, 773, 1, 1069, 12451, 25219, 1, 1361, 1, 2039, 13417, 1181, 1, 27827, 14081, 28499, 14419, 29179, 509, 29867, 15107, 2351, 1, 31267, 1, 1, 1, 1721, 1, 33427, 1, 1, 557, 521, 1, 1, 487, 461, 18401, 907, 18787, 37963, 1, 38747, 19571, 2081, 1051, 1, 1567, 1789, 1, 1447, 21187, 42787, 21601, 823, 1, 1933, 22441, 45307, 1759, 1, 23297, 1, 1249, 2521, 24169, 48779, 24611, 49667, 25057, 857, 1109, 1, 1997, 52379, 911, 53299, 26881, 1, 1, 55163, 27817, 2953, 1489, 57059, 2213, 4463, 29251, 967, 1, 983, 1, 1033, 991, 2693, 31219, 62939, 31721, 4919, 1, 1, 1723, 1, 811, 2311, 33769, 68059, 647, 2383, 941, 70163, 2719, 5479, 677, 1, 1, 73379, 1607, 659, 37507, 1, 2003, 76667, 38611, 1, 1, 1, 593, 1013, 683, 81163, 40867, 82307, 1429, 83459, 42019, 2287, 1, 6599, 2273, 1, 43777, 1, 44371, 2179, 1, 2447, 1, 3989, 757, 7151, 1, 971, 1, 95419, 1549, 1, 1, 5153, 2593, 1871, 1217, 3463, 1, 7823, 51169, 1, 1, 821, 52457, 105563, 2309, 106867, 1453, 108179, 54419, 8423, 1, 1, 1, 112163, 56417, 1, 1543, 114859, 1409, 1, 58451, 1993, 4549, 9151, 2063, 797, 60521, 1, 2111, 6481, 3259, 1, 62627, 125963, 63337, 1, 1, 128819, 1, 130259, 829, 131707, 2879, 1, 66947, 2207, 1277, 136099, 1, 1, 1, 139067, 1319, 1, 70657, 142067, 71411, 4951, 72169, 3539, 72931, 11279, 5669, 1, 1, 1, 1123, 1, 4001, 152819, 76801, 154387, 77587, 6781, 6029, 12119, 1931, 1, 1, 160739, 1, 5237, 1, 1451, 82387, 165587, 1, 1, 1, 1289, 2293, 1021, 85667, 1609, 1291, 173827, 3797, 175499, 88169, 3343, 1, 13759, 1523, 180563, 1487, 1, 1, 183979, 92419, 185699, 93281, 1, 3037, 14551, 7309, 1, 95891, 1, 96769, 1, 97651, 196187, 1, 197963, 5233, 10513, 7717, 1, 1, 203339, 102121, 1, 1, 3089, 4519, 208787, 104851, 210619, 105769, 1, 1, 5227, 4679, 1, 1, 1, 109481, 2267, 110419, 977, 1, 223667, 1, 17351, 3061, 227467, 1, 9973, 1, 231299, 116131, 1, 6163, 12377, 3191, 1, 9157, 3919, 1, 8311, 121001, 1259, 1, 8447, 3967, 246947, 1, 2203, 9613, 1, 5477, 13313, 1, 254963, 1, 3253, 129001, 259019, 5653, 261059, 131041, 1, 10159, 265163, 133097, 267227, 1187, 1367, 1, 6619, 1, 1, 4733, 11981, 10639, 1, 139361, 279779, 140419, 1231, 141481, 1, 142547, 286163, 143617, 288307, 1, 22343, 11213, 15401, 1, 294787, 147937, 7243, 149027, 299147, 1, 10391, 1, 303539, 11717, 1, 153427, 1381, 6719, 310187, 5021, 2069, 1, 16561, 3851, 316907, 159017, 24551, 1, 321427, 161281, 323699, 162419, 14173, 163561, 1, 164707, 330563, 165857, 2621, 1, 1, 1, 1, 5839, 339827, 170497, 1301, 1, 9311, 1, 1, 174019, 26863, 13477, 351587, 7669, 353963, 1, 9631, 1, 1, 179969, 6121, 7877, 8867, 14029, 28151, 183587, 12703, 1, 6079, 186019, 1, 1657, 375707, 1, 378163, 189697, 1, 1, 383099, 6199, 385579, 1, 388067, 5261, 16981, 195907, 9587, 197161, 3697, 15263, 1, 1, 7559, 5431, 403163, 1, 1, 3449, 7703, 1, 410819, 206051, 31799, 1, 415963, 208627, 1511, 9127, 2789, 211219, 13669, 1, 1, 213827, 1, 1, 33199, 216451, 2399, 3691, 11807, 219091, 1, 1, 1, 221747, 2729, 5441, 1, 1, 1, 3701, 3457, 11953, 23977, 228457, 1, 2909, 460979, 231169, 1, 1, 35879, 3491, 11443, 1, 471907, 236641, 474659, 238019, 1699, 4517, 1, 1, 1, 1433, 8233, 1, 488539, 1, 491339, 246371, 494147, 1951, 496963, 249187, 499787, 1, 1, 1669, 505459, 13339, 1, 1, 511163, 1, 22349, 257731, 8761, 259169, 39983, 20047, 1, 1, 4651, 6427, 18223, 264961, 1, 1, 1, 1, 8807, 20719, 1, 2531, 1, 1, 10303, 11903, 549019, 275251, 1, 1, 1, 278227, 1, 21517, 560939, 1, 1, 9749, 566947, 284227, 24781, 1, 7253, 287251, 576019, 1, 44543, 2713, 25309, 291817, 585163, 1, 14347, 2251, 31121, 15601, 594379, 8053, 45959, 23039, 19373, 301057, 603667, 1, 1601, 304169, 21031, 8263, 2749, 1, 21247, 1, 47639, 16339, 1, 4657, 15259, 1, 628787, 5167, 2459, 316777, 635147, 6007, 49103, 1, 27893, 321571, 644747, 2467, 647963, 10477, 34273, 1, 1, 11311, 657659, 25357, 50839, 11423, 11257, 1, 667427, 4993, 18127, 2647, 673979, 1, 8573, 14759, 4027, 1, 35993, 342761, 687179, 2113, 5437, 1, 693827, 347747, 2917, 4423, 1, 1, 1867, 1949, 707219, 354451, 1, 356137, 1, 1, 31189, 1, 17579, 361219, 1, 27917, 11927, 364627, 1, 9901, 1, 1907, 737819, 1877, 4909, 371491, 744707, 1, 1, 1777, 751627, 1, 7057, 16453, 758579, 13109, 18587, 6473, 1, 383657, 59159, 1, 4003, 387169, 776099, 388931, 1, 20563, 11689, 392467, 786707, 394241, 8147, 1, 1, 1999, 1, 399587, 2609, 6803, 27743, 7607, 1, 404969, 1, 1, 3301, 1, 818963, 410387, 822587, 412201, 1, 414019, 1, 1, 4231, 1, 10597, 1, 64679, 421331, 844499, 11437, 44641, 22369, 5101, 1, 3769, 14783, 859267, 1, 66383, 1, 21139, 1, 14753, 436147, 6673, 438017, 877907, 439891, 38333, 23251, 46601, 34127, 68399, 445537, 1, 447427, 1, 449321, 900539, 4217, 1, 6763, 3209, 4691, 1, 35149, 1, 3613, 1, 24251, 29789, 20117, 5689, 464617, 931163, 466547, 1, 36037, 72223, 1, 942779, 1, 946667, 474307, 15583, 4451, 15647, 25169, 50441, 480169, 1, 37087, 966227, 16693, 42181, 5011, 974107, 16829, 4271, 490019, 14657,

6. Sequence of the polynom (only primes)

13, 37, 23, 53, 29, 61, 31, 19, 59, 41, 107, 67, 163, 97, 227, 131, 379, 211, 467, 257, 563, 307, 419, 79, 547, 1163, 617, 1307, 691, 1459, 769, 1619, 1787, 937, 151, 113, 2339, 2539, 1321, 1427, 2963, 3187, 127, 263, 3659, 3907, 2017, 181, 233, 2281, 383, 197, 229, 2707, 5563, 2857, 5867, 3011, 167, 3169, 3331, 6827, 269, 193, 7507, 271, 4019, 8219, 4201, 277, 8963, 199, 719, 367, 9739, 4969, 10139, 5171, 283, 577, 5801, 223, 463, 13163, 13627, 239, 613, 7411, 7907, 16067, 8161, 281, 8419, 17099, 8681, 17627, 389, 443, 709, 1439, 9491, 19259, 9769, 19819, 10337, 20963, 10627, 743, 863, 22739, 631, 11827, 773, 1069, 12451, 25219, 1361, 2039, 13417, 1181, 27827, 14081, 28499, 14419, 29179, 509, 29867, 15107, 2351, 31267, 1721, 33427, 557, 521, 487, 461, 18401, 907, 18787, 37963, 38747, 19571, 2081, 1051, 1567, 1789, 1447, 21187, 42787, 21601, 823, 1933, 22441, 45307, 1759, 23297, 1249, 2521, 24169, 48779, 24611, 49667, 25057, 857, 1109, 1997, 52379, 911, 53299, 26881, 55163, 27817, 2953, 1489, 57059, 2213, 4463, 29251, 967, 983, 1033, 991, 2693, 31219, 62939, 31721, 4919, 1723, 811, 2311, 33769, 68059, 647, 2383, 941, 70163, 2719, 5479, 677, 73379, 1607, 659, 37507, 2003, 76667, 38611, 593, 1013, 683, 81163, 40867, 82307, 1429, 83459, 42019, 2287, 6599, 2273, 43777, 44371, 2179, 2447, 3989, 757, 7151, 971, 95419, 1549, 5153, 2593, 1871, 1217, 3463, 7823, 51169, 821, 52457, 105563, 2309, 106867, 1453, 108179, 54419, 8423, 112163, 56417, 1543, 114859, 1409, 58451, 1993, 4549, 9151, 2063, 797, 60521, 2111, 6481, 3259, 62627, 125963, 63337, 128819, 130259, 829, 131707, 2879, 66947, 2207, 1277, 136099, 139067, 1319, 70657, 142067, 71411, 4951, 72169, 3539, 72931, 11279, 5669, 1123, 4001, 152819, 76801, 154387, 77587, 6781, 6029, 12119, 1931, 160739, 5237, 1451, 82387, 165587, 1289, 2293, 1021, 85667, 1609, 1291, 173827, 3797, 175499, 88169, 3343, 13759, 1523, 180563, 1487, 183979, 92419, 185699, 93281, 3037, 14551, 7309, 95891, 96769, 97651, 196187, 197963, 5233, 10513, 7717, 203339, 102121, 3089, 4519, 208787, 104851, 210619, 105769, 5227, 4679, 109481, 2267, 110419, 977, 223667, 17351, 3061, 227467, 9973, 231299, 116131, 6163, 12377, 3191, 9157, 3919, 8311, 121001, 1259, 8447, 3967, 246947, 2203, 9613, 5477, 13313, 254963, 3253, 129001, 259019, 5653, 261059, 131041, 10159, 265163, 133097, 267227, 1187, 1367, 6619, 4733, 11981, 10639, 139361, 279779, 140419, 1231, 141481, 142547, 286163, 143617, 288307, 22343, 11213, 15401, 294787, 147937, 7243, 149027, 299147, 10391, 303539, 11717, 153427, 1381, 6719, 310187, 5021, 2069, 16561, 3851, 316907, 159017, 24551, 321427, 161281, 323699, 162419, 14173, 163561, 164707, 330563, 165857, 2621, 5839, 339827, 170497, 1301, 9311, 174019, 26863, 13477, 351587, 7669, 353963, 9631, 179969, 6121, 7877, 8867, 14029, 28151, 183587, 12703, 6079, 186019, 1657, 375707, 378163, 189697, 383099, 6199, 385579, 388067, 5261, 16981, 195907, 9587, 197161, 3697, 15263, 7559, 5431, 403163, 3449, 7703, 410819, 206051, 31799, 415963, 208627, 1511, 9127, 2789, 211219, 13669, 213827, 33199, 216451, 2399, 3691, 11807, 219091, 221747, 2729, 5441, 3701, 3457, 11953, 23977, 228457, 2909, 460979, 231169, 35879, 3491, 11443, 471907, 236641, 474659, 238019, 1699, 4517, 1433, 8233, 488539, 491339, 246371, 494147, 1951, 496963, 249187, 499787, 1669, 505459, 13339, 511163, 22349, 257731, 8761, 259169, 39983, 20047, 4651, 6427, 18223, 264961, 8807, 20719, 2531, 10303, 11903, 549019, 275251, 278227, 21517, 560939, 9749, 566947, 284227, 24781, 7253, 287251, 576019, 44543, 2713, 25309, 291817, 585163, 14347, 2251, 31121, 15601, 594379, 8053, 45959, 23039, 19373, 301057, 603667, 1601, 304169, 21031, 8263, 2749, 21247, 47639, 16339, 4657, 15259, 628787, 5167, 2459, 316777, 635147, 6007, 49103, 27893, 321571, 644747, 2467, 647963, 10477, 34273, 11311, 657659, 25357, 50839, 11423, 11257, 667427, 4993, 18127, 2647, 673979, 8573, 14759, 4027, 35993, 342761, 687179, 2113, 5437, 693827, 347747, 2917, 4423, 1867, 1949, 707219, 354451, 356137, 31189, 17579, 361219, 27917, 11927, 364627, 9901, 1907, 737819, 1877, 4909, 371491, 744707, 1777, 751627, 7057, 16453, 758579, 13109, 18587, 6473, 383657, 59159, 4003, 387169, 776099, 388931, 20563, 11689, 392467, 786707, 394241, 8147, 1999, 399587, 2609, 6803, 27743, 7607, 404969, 3301, 818963, 410387, 822587, 412201, 414019, 4231, 10597, 64679, 421331, 844499, 11437, 44641, 22369, 5101, 3769, 14783, 859267, 66383, 21139, 14753, 436147, 6673, 438017, 877907, 439891, 38333, 23251, 46601, 34127, 68399, 445537, 447427, 449321, 900539, 4217, 6763, 3209, 4691, 35149, 3613, 24251, 29789, 20117, 5689, 464617, 931163, 466547, 36037, 72223, 942779, 946667, 474307, 15583, 4451, 15647, 25169, 50441, 480169, 37087, 966227, 16693, 42181, 5011, 974107, 16829, 4271, 490019, 14657,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-16x+2 and
the reducible primes which appear as divisor for the first time
p | x^2-16x+2 and p < x^2-16x+2

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^2-16x+2	P(x) \| x^2-16x+2	C/B	D/B	E/B	C(n) / C(n-1)	D(n) / D(n-1)	E(n) / E(n-1)
1	10	8	5	3	0,800000	0,500000	0,300000
2	100	70	29	41	0,700000	0,290000	0,410000	8,750000	5,800000	13,666667
3	1.000	719	168	551	0,719000	0,168000	0,551000	10,271429	5,793103	13,439024
4	10.000	7.200	1.202	5.998	0,720000	0,120200	0,599800	10,013908	7,154762	10,885662
5	100.000	71.724	9.337	62.387	0,717240	0,093370	0,623870	9,961667	7,767887	10,401300
6	1.000.000	712.508	76.320	636.188	0,712508	0,076320	0,636188	9,934025	8,173932	10,197445
7	10.000.000	7.093.022	645.063	6.447.959	0,709302	0,064506	0,644796	9,955007	8,452083	10,135304
8	100.000.000	70.710.106	5.596.204	65.113.902	0,707101	0,055962	0,651139	9,968968	8,675438	10,098374
9	1.000.000.000	705.431.613	49.375.937	656.055.676	0,705432	0,049376	0,656056	9,976390	8,823112	10,075509
10	10.000.000.000	7.040.983.987	441.890.595	6.599.093.392	0,704098	0,044189	0,659909	9,981101	8,949513	10,058740
11	100.000.000.000	70.302.773.396	3.998.928.092	66.303.845.304	0,703028	0,039989	0,663038	9,984794	9,049589	10,047417
12	1.000.000.000.000	702.147.035.673	36.519.999.574	665.627.036.099	0,702147	0,036520	0,665627	9,987473	9,132447	10,039041


A	B	C	D	E	F	G	H	I	J	K
2^n	x	all Primes	P(x)=x^2-16x+2	P(x) \| x^2-16x+2	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	4	3	1	1,000000	0,750000	0,250000	2,000000	1,500000
3	8	8	5	3	1,000000	0,625000	0,375000	2,000000	1,666667	3,000000
4	16	8	5	3	0,500000	0,312500	0,187500	1,000000	1,000000	1,000000
5	32	21	12	9	0,656250	0,375000	0,281250	2,625000	2,400000	3,000000
6	64	42	21	21	0,656250	0,328125	0,328125	2,000000	1,750000	2,333333
7	128	88	34	54	0,687500	0,265625	0,421875	2,095238	1,619048	2,571429
8	256	183	58	125	0,714844	0,226563	0,488281	2,079545	1,705882	2,314815
9	512	364	97	267	0,710938	0,189453	0,521484	1,989071	1,672414	2,136000
10	1.024	736	171	565	0,718750	0,166992	0,551758	2,021978	1,762887	2,116105
11	2.048	1.469	313	1.156	0,717285	0,152832	0,564453	1,995924	1,830409	2,046018
12	4.096	2.951	562	2.389	0,720459	0,137207	0,583252	2,008850	1,795527	2,066609
13	8.192	5.909	1.003	4.906	0,721313	0,122437	0,598877	2,002372	1,784698	2,053579
14	16.384	11.782	1.850	9.932	0,719116	0,112915	0,606201	1,993908	1,844467	2,024460
15	32.768	23.546	3.426	20.120	0,718567	0,104553	0,614014	1,998472	1,851892	2,025775
16	65.536	47.059	6.355	40.704	0,718063	0,096970	0,621094	1,998598	1,854933	2,023062
17	131.072	93.937	11.909	82.028	0,716682	0,090858	0,625824	1,996154	1,873958	2,015232
18	262.144	187.390	22.221	165.169	0,714836	0,084766	0,630070	1,994848	1,865900	2,013569
19	524.288	374.082	42.075	332.007	0,713505	0,080252	0,633253	1,996275	1,893479	2,010105
20	1.048.576	746.958	79.781	667.177	0,712355	0,076085	0,636270	1,996776	1,896162	2,009527
21	2.097.152	1.491.510	151.120	1.340.390	0,711207	0,072060	0,639148	1,996779	1,894185	2,009047
22	4.194.304	2.979.058	287.216	2.691.842	0,710263	0,068478	0,641785	1,997344	1,900582	2,008253
23	8.388.608	5.951.774	547.406	5.404.368	0,709507	0,065256	0,644251	1,997871	1,905904	2,007684
24	16.777.216	11.890.419	1.046.472	10.843.947	0,708724	0,062375	0,646350	1,997794	1,911693	2,006515
25	33.554.432	23.758.408	2.004.260	21.754.148	0,708056	0,059732	0,648324	1,998114	1,915254	2,006110
26	67.108.864	47.475.497	3.844.550	43.630.947	0,707440	0,057288	0,650152	1,998261	1,918189	2,005638
27	134.217.728	94.875.809	7.384.191	87.491.618	0,706880	0,055017	0,651863	1,998416	1,920691	2,005265
28	268.435.456	189.607.422	14.208.341	175.399.081	0,706343	0,052930	0,653412	1,998480	1,924157	2,004753
29	536.870.912	378.949.878	27.378.369	351.571.509	0,705849	0,050996	0,654853	1,998603	1,926922	2,004409
30	1.073.741.824	757.402.963	52.825.776	704.577.187	0,705386	0,049198	0,656189	1,998689	1,929471	2,004079
31	2.147.483.648	1.513.881.145	102.057.253	1.411.823.892	0,704956	0,047524	0,657432	1,998779	1,931959	2,003789
32	4.294.967.296	3.026.021.209	197.399.844	2.828.621.365	0,704550	0,045961	0,658590	1,998850	1,934207	2,003523
33	8.589.934.592	6.048.852.483	382.231.908	5.666.620.575	0,704179	0,044498	0,659681	1,998946	1,936333	2,003315
34	17.179.869.184	12.091.659.008	740.857.699	11.350.801.309	0,703827	0,043124	0,660704	1,999000	1,938241	2,003099
35	34.359.738.368	24.171.998.907	1.437.402.424	22.734.596.483	0,703498	0,041834	0,661664	1,999064	1,940187	2,002907
36	68.719.476.736	48.322.679.667	2.791.252.099	45.531.427.568	0,703188	0,040618	0,662569	1,999118	1,941872	2,002737
37	137.438.953.472	96.605.348.500	5.424.891.506	91.180.456.994	0,702896	0,039471	0,663425	1,999172	1,943533	2,002583
38	274.877.906.944	193.134.895.286	10.551.959.945	182.582.935.341	0,702621	0,038388	0,664233	1,999215	1,945101	2,002435
39	549.755.813.888	386.126.797.856	20.539.917.412	365.586.880.444	0,702361	0,037362	0,664999	1,999260	1,946550	2,002306
40	1.099.511.627.776	771.982.707.683	40.011.095.600	731.971.612.083	0,702114	0,036390	0,665724	1,999298	1,947968	2,002182

8. Check for existing Integer Sequences by OEIS
Found in Database : 1, 13, 1, 37, 23, 53, 29, 61, 31, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 59,
Found in Database : 13, 37, 23, 53, 29, 61, 31, 19, 59, 41, 107, 67, 163, 97, 227, 131, 379, 211, 467, 257, 563, 307, 419,
Found in Database : 13, 19, 23, 29, 31, 37, 41, 53, 59, 61, 67, 79, 97, 107, 113, 127, 131,

Development ofAlgorithmic Constructions

Polynom = x^2-16x+2

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