Development of
Algorithmic Constructions

11:28:25

18.Apr 2024

Polynom = x^2-10x+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) = 7 = 7
f(2) = 7 = 7
f(3) = 19 = 19
f(4) = 11 = 11
f(5) = 23 = 23
f(6) = 11 = 11
f(7) = 19 = 19
f(8) = 7 = 7
f(9) = 7 = 7
f(10) = 1 = 1
f(11) = 13 = 13
f(12) = 13 = 13
f(13) = 41 = 41
f(14) = 29 = 29
f(15) = 77 = 7*11
f(16) = 49 = 7*7
f(17) = 121 = 11*11
f(18) = 73 = 73
f(19) = 173 = 173
f(20) = 101 = 101
f(21) = 233 = 233
f(22) = 133 = 7*19
f(23) = 301 = 7*43
f(24) = 169 = 13*13
f(25) = 377 = 13*29
f(26) = 209 = 11*19
f(27) = 461 = 461
f(28) = 253 = 11*23
f(29) = 553 = 7*79
f(30) = 301 = 7*43
f(31) = 653 = 653
f(32) = 353 = 353
f(33) = 761 = 761
f(34) = 409 = 409
f(35) = 877 = 877
f(36) = 469 = 7*67
f(37) = 1001 = 7*11*13
f(38) = 533 = 13*41
f(39) = 1133 = 11*103
f(40) = 601 = 601
f(41) = 1273 = 19*67
f(42) = 673 = 673
f(43) = 1421 = 7*7*29
f(44) = 749 = 7*107
f(45) = 1577 = 19*83
f(46) = 829 = 829
f(47) = 1741 = 1741
f(48) = 913 = 11*83
f(49) = 1913 = 1913
f(50) = 1001 = 7*11*13
f(51) = 2093 = 7*13*23
f(52) = 1093 = 1093
f(53) = 2281 = 2281
f(54) = 1189 = 29*41
f(55) = 2477 = 2477
f(56) = 1289 = 1289
f(57) = 2681 = 7*383
f(58) = 1393 = 7*199
f(59) = 2893 = 11*263
f(60) = 1501 = 19*79
f(61) = 3113 = 11*283
f(62) = 1613 = 1613
f(63) = 3341 = 13*257
f(64) = 1729 = 7*13*19
f(65) = 3577 = 7*7*73
f(66) = 1849 = 43*43
f(67) = 3821 = 3821
f(68) = 1973 = 1973
f(69) = 4073 = 4073
f(70) = 2101 = 11*191
f(71) = 4333 = 7*619
f(72) = 2233 = 7*11*29
f(73) = 4601 = 43*107
f(74) = 2369 = 23*103
f(75) = 4877 = 4877
f(76) = 2509 = 13*193
f(77) = 5161 = 13*397
f(78) = 2653 = 7*379
f(79) = 5453 = 7*19*41
f(80) = 2801 = 2801
f(81) = 5753 = 11*523
f(82) = 2953 = 2953
f(83) = 6061 = 11*19*29
f(84) = 3109 = 3109
f(85) = 6377 = 7*911
f(86) = 3269 = 7*467
f(87) = 6701 = 6701
f(88) = 3433 = 3433
f(89) = 7033 = 13*541
f(90) = 3601 = 13*277
f(91) = 7373 = 73*101
f(92) = 3773 = 7*7*7*11
f(93) = 7721 = 7*1103
f(94) = 3949 = 11*359
f(95) = 8077 = 41*197
f(96) = 4129 = 4129
f(97) = 8441 = 23*367
f(98) = 4313 = 19*227
f(99) = 8813 = 7*1259
f(100) = 4501 = 7*643

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^2-10x+2

f(0)=1
f(1)=7
f(2)=1
f(3)=19
f(4)=11
f(5)=23
f(6)=1
f(7)=1
f(8)=1
f(9)=1
f(10)=1
f(11)=13
f(12)=1
f(13)=41
f(14)=29
f(15)=1
f(16)=1
f(17)=1
f(18)=73
f(19)=173
f(20)=101
f(21)=233
f(22)=1
f(23)=43
f(24)=1
f(25)=1
f(26)=1
f(27)=461
f(28)=1
f(29)=79
f(30)=1
f(31)=653
f(32)=353
f(33)=761
f(34)=409
f(35)=877
f(36)=67
f(37)=1
f(38)=1
f(39)=103
f(40)=601
f(41)=1
f(42)=673
f(43)=1
f(44)=107
f(45)=83
f(46)=829
f(47)=1741
f(48)=1
f(49)=1913
f(50)=1
f(51)=1
f(52)=1093
f(53)=2281
f(54)=1
f(55)=2477
f(56)=1289
f(57)=383
f(58)=199
f(59)=263
f(60)=1
f(61)=283
f(62)=1613
f(63)=257
f(64)=1
f(65)=1
f(66)=1
f(67)=3821
f(68)=1973
f(69)=4073
f(70)=191
f(71)=619
f(72)=1
f(73)=1
f(74)=1
f(75)=4877
f(76)=193
f(77)=397
f(78)=379
f(79)=1
f(80)=2801
f(81)=523
f(82)=2953
f(83)=1
f(84)=3109
f(85)=911
f(86)=467
f(87)=6701
f(88)=3433
f(89)=541
f(90)=277
f(91)=1
f(92)=1
f(93)=1103
f(94)=359
f(95)=197
f(96)=4129
f(97)=367
f(98)=227
f(99)=1259

b) Substitution of the polynom
The polynom f(x)=x^2-10x+2  could be written as f(y)= y^2-23 with x=y+5

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-5
f'(x)>2x-11  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, 7, 1, 19, 11, 23, 1, 1, 1, 1, 1, 13, 1, 41, 29, 1, 1, 1, 73, 173, 101, 233, 1, 43, 1, 1, 1, 461, 1, 79, 1, 653, 353, 761, 409, 877, 67, 1, 1, 103, 601, 1, 673, 1, 107, 83, 829, 1741, 1, 1913, 1, 1, 1093, 2281, 1, 2477, 1289, 383, 199, 263, 1, 283, 1613, 257, 1, 1, 1, 3821, 1973, 4073, 191, 619, 1, 1, 1, 4877, 193, 397, 379, 1, 2801, 523, 2953, 1, 3109, 911, 467, 6701, 3433, 541, 277, 1, 1, 1103, 359, 197, 4129, 367, 227, 1259, 643, 317, 1, 1, 4889, 907, 727, 1483, 1, 251, 5501, 11213, 1, 1663, 1, 929, 1, 659, 6373, 12973, 1, 1, 6833, 13901, 7069, 1307, 7309, 1, 1, 1181, 269, 1, 8053, 16361, 1187, 2411, 1, 17401, 1, 1, 479, 1, 1, 827, 9649, 19577, 9929, 1831, 1459, 1, 10501, 1, 1, 21881, 853, 1, 1627, 23081, 1063, 1, 1091, 593, 1759, 509, 1, 25577, 563, 2017, 1021, 349, 1, 2503, 13933, 28201, 751, 431, 2087, 1, 787, 30253, 1, 2381, 1423, 4523, 2287, 32377, 16369, 419, 577, 1471, 1, 449, 1, 1, 1373, 839, 18229, 1, 2659, 1297, 19001, 38393, 1, 39181, 1, 5711, 1553, 3137, 20593, 41593, 21001, 1, 1, 3931, 1, 4007, 1171, 1549, 1, 503, 1777, 46633, 1, 47501, 2179, 6911, 1, 49261, 857, 50153, 25301, 2687, 1, 571, 26209, 1, 26669, 1, 631, 1117, 3943, 55673, 1, 1381, 28549, 1, 1, 8363, 2683, 59513, 1579, 60493, 30493, 8783, 1, 62477, 31489, 1, 1, 1, 4643, 1, 33013, 66541, 33529, 67577, 1, 9803, 1, 1, 3191, 5441, 2741, 3779, 5167, 1, 36709, 73961, 37253, 6823, 1, 1, 5479, 77261, 1, 6029, 1361, 743, 1, 11519, 3691, 81773, 1, 821, 1, 12011, 6047, 1, 3301, 86413, 1061, 7963, 6299, 1153, 1, 89977, 45289, 4799, 45893, 1, 1, 1, 4283, 94841, 4339, 3313, 6907, 13903, 48973, 98573, 1, 99833, 1, 1, 1, 1, 2711, 4507, 52153, 104953, 1, 1, 53453, 3709, 4919, 108877, 1, 1, 7919, 111533, 56101, 2753, 56773, 114221, 1, 1, 58129, 10631, 1, 1, 1, 17099, 8599, 121081, 60889, 122477, 1, 1697, 809, 2557, 1, 126713, 1, 9857, 3391, 1, 1, 1, 3467, 12043, 66601, 1999, 9619, 1, 1583, 10529, 1, 1667, 6323, 19979, 1, 141353, 1733, 1, 71809, 1, 1481, 1097, 5641, 1031, 74101, 1, 74873, 21503, 1, 152077, 3323, 153641, 77213, 1, 1013, 1723, 1, 1, 79589, 159977, 4231, 1, 1657, 163193, 1907, 14983, 82813, 15131, 919, 1847, 84449, 3947, 1, 7451, 2969, 1301, 1129, 4261, 1, 9283, 88609, 13697, 983, 25679, 1237, 6257, 91153, 16651, 92009, 1, 13267, 186601, 1399, 188333, 1, 14621, 1, 1, 1, 193577, 8839, 195341, 2393, 971, 14143, 1931, 1, 1, 7753, 1, 1, 1, 102593, 10847, 1, 1, 1, 1, 1, 9199, 1, 16417, 9743, 215273, 15443, 31019, 1019, 219001, 1, 220877, 110909, 1, 1, 1571, 8677, 226553, 5987, 228461, 2341, 32911, 1123, 997, 1, 1, 10691, 33739, 1303, 1409, 119549, 2377, 2803, 12739, 17359, 3169, 122501, 1, 123493, 1, 124489, 1, 1, 8689, 126493, 253993, 1, 11131, 1669, 1, 129529, 1, 6871, 262121, 1, 2903, 1, 24203, 133633, 24391, 134669, 2677, 19387, 38923, 136753, 3761, 3361, 1637, 1, 5689, 1, 14783, 4861, 283001, 1, 1, 20443, 41039, 144173, 1, 11173, 2039, 1, 1447, 21059, 1487, 148501, 298093, 1, 1, 1, 6173, 13799, 1, 1, 306893, 154001, 44159, 22159, 311341, 156229, 1, 157349, 28711, 22639, 45439, 12277, 1, 2399, 16979, 161869, 46411, 1, 1, 14923, 3079, 7187, 331753, 1, 3671, 12893, 1279, 168769, 1, 169933, 1, 24443, 343373, 9067, 1523, 5981, 348077, 1, 3851, 1453, 352813, 16091, 355193, 2441, 2221, 1, 12413, 1, 362381, 1699, 2551, 2011, 1, 4493, 4679, 185429, 19583, 186649, 53503, 26839, 1, 17191, 379433, 1, 1, 27367, 1277, 1, 1367, 4733, 4691, 1, 1, 1, 35851, 1, 30529, 15313, 399401, 28619, 1, 1, 1, 18443, 2131, 1, 58511, 29347, 1, 15901, 1, 208001, 417293, 1, 1, 1, 1, 211889, 4127, 213193, 61099, 30643, 1, 1, 14929, 19739, 435577, 2837, 62603, 1, 10753, 221101, 443533, 1, 63743, 2459, 1, 225109, 41051, 226453, 454253, 4649, 2251, 229153, 2657, 1, 462377, 1, 1, 1, 467833, 1, 24767, 1, 473321, 1, 2957, 238729, 1, 240113, 43783, 1429, 5323, 4957, 2473, 1, 489977, 1, 492781, 3209, 1, 1, 1, 3011, 38557, 19333, 1, 36107, 1, 1, 1, 1, 1607, 1, 73643, 258469, 27283, 19993, 1, 261353, 1, 3413, 527053, 24023, 529961, 265709, 1, 38167, 76543, 1, 1, 1, 2141, 271573, 1, 2053, 547577, 274529, 550541, 1, 553513, 1, 1, 1951, 43037, 1, 562477, 281989, 80783, 40499, 568493, 2767, 13291, 286513, 2749, 1789, 1, 22273, 30559, 10037, 583673, 292601, 1, 42019, 8803, 26879, 592877, 1, 595961, 1, 1, 300301, 602153, 15887, 605261, 303409, 7901, 2293, 2417, 306533, 614633, 308101, 47521, 1, 1, 1, 1, 28439, 1, 1, 90059, 6449, 33347, 317593, 636781, 1, 2591, 45827, 8353, 11117, 58763, 14087, 649613, 1, 93263, 46747, 656077, 328849, 1, 2311, 8387, 1, 1, 3119, 1747, 1, 672377, 11621, 13789, 1, 61723, 26177, 1, 341953, 1, 1, 98411, 1, 23869, 346933, 36607, 1, 99839, 4549, 2843, 27073, 10531, 353629, 16487, 1, 14537, 4519, 65063, 1, 65371, 1811, 3559, 1, 55837, 1, 729293, 1, 732713, 1, 1, 3049, 1, 12781, 2129, 5101, 1, 4111, 1889, 375833, 68491, 9209, 1, 7741, 5717, 4591, 33211, 382801, 40387, 29581, 1, 1, 774377, 35279, 777901, 1, 781433, 1, 112139, 393373, 788521, 5413, 1, 1, 10333, 1, 1, 1, 802793, 17491, 806381, 57727, 1, 36899, 12143, 2851, 62861, 1, 117259, 1, 28429, 413129, 1, 414949, 75611, 59539, 1, 418601, 1, 32341, 842701, 14561, 1, 1, 850061, 38723, 1, 38891, 857453, 61379, 2861, 1, 66529, 433369, 3229, 22907, 11329, 1, 1, 15137, 879821, 440849, 883577, 63247, 1, 1, 1933, 40591, 894893, 40763, 1, 64327, 902477, 1, 47699, 5471, 910093, 5011, 1, 457913, 83431, 459829, 1, 4483, 1, 66239, 21611, 1, 1, 2237, 72077, 1, 1, 1, 11959, 473353, 14159, 475301, 3319, 2351, 86951, 479209, 7937, 37013, 74177, 69019, 138319, 485101, 2693, 487073, 2719, 1, 7369, 1, 24001, 493013, 76001,

6. Sequence of the polynom (only primes)

7, 19, 11, 23, 13, 41, 29, 73, 173, 101, 233, 43, 461, 79, 653, 353, 761, 409, 877, 67, 103, 601, 673, 107, 83, 829, 1741, 1913, 1093, 2281, 2477, 1289, 383, 199, 263, 283, 1613, 257, 3821, 1973, 4073, 191, 619, 4877, 193, 397, 379, 2801, 523, 2953, 3109, 911, 467, 6701, 3433, 541, 277, 1103, 359, 197, 4129, 367, 227, 1259, 643, 317, 4889, 907, 727, 1483, 251, 5501, 11213, 1663, 929, 659, 6373, 12973, 6833, 13901, 7069, 1307, 7309, 1181, 269, 8053, 16361, 1187, 2411, 17401, 479, 827, 9649, 19577, 9929, 1831, 1459, 10501, 21881, 853, 1627, 23081, 1063, 1091, 593, 1759, 509, 25577, 563, 2017, 1021, 349, 2503, 13933, 28201, 751, 431, 2087, 787, 30253, 2381, 1423, 4523, 2287, 32377, 16369, 419, 577, 1471, 449, 1373, 839, 18229, 2659, 1297, 19001, 38393, 39181, 5711, 1553, 3137, 20593, 41593, 21001, 3931, 4007, 1171, 1549, 503, 1777, 46633, 47501, 2179, 6911, 49261, 857, 50153, 25301, 2687, 571, 26209, 26669, 631, 1117, 3943, 55673, 1381, 28549, 8363, 2683, 59513, 1579, 60493, 30493, 8783, 62477, 31489, 4643, 33013, 66541, 33529, 67577, 9803, 3191, 5441, 2741, 3779, 5167, 36709, 73961, 37253, 6823, 5479, 77261, 6029, 1361, 743, 11519, 3691, 81773, 821, 12011, 6047, 3301, 86413, 1061, 7963, 6299, 1153, 89977, 45289, 4799, 45893, 4283, 94841, 4339, 3313, 6907, 13903, 48973, 98573, 99833, 2711, 4507, 52153, 104953, 53453, 3709, 4919, 108877, 7919, 111533, 56101, 2753, 56773, 114221, 58129, 10631, 17099, 8599, 121081, 60889, 122477, 1697, 809, 2557, 126713, 9857, 3391, 3467, 12043, 66601, 1999, 9619, 1583, 10529, 1667, 6323, 19979, 141353, 1733, 71809, 1481, 1097, 5641, 1031, 74101, 74873, 21503, 152077, 3323, 153641, 77213, 1013, 1723, 79589, 159977, 4231, 1657, 163193, 1907, 14983, 82813, 15131, 919, 1847, 84449, 3947, 7451, 2969, 1301, 1129, 4261, 9283, 88609, 13697, 983, 25679, 1237, 6257, 91153, 16651, 92009, 13267, 186601, 1399, 188333, 14621, 193577, 8839, 195341, 2393, 971, 14143, 1931, 7753, 102593, 10847, 9199, 16417, 9743, 215273, 15443, 31019, 1019, 219001, 220877, 110909, 1571, 8677, 226553, 5987, 228461, 2341, 32911, 1123, 997, 10691, 33739, 1303, 1409, 119549, 2377, 2803, 12739, 17359, 3169, 122501, 123493, 124489, 8689, 126493, 253993, 11131, 1669, 129529, 6871, 262121, 2903, 24203, 133633, 24391, 134669, 2677, 19387, 38923, 136753, 3761, 3361, 1637, 5689, 14783, 4861, 283001, 20443, 41039, 144173, 11173, 2039, 1447, 21059, 1487, 148501, 298093, 6173, 13799, 306893, 154001, 44159, 22159, 311341, 156229, 157349, 28711, 22639, 45439, 12277, 2399, 16979, 161869, 46411, 14923, 3079, 7187, 331753, 3671, 12893, 1279, 168769, 169933, 24443, 343373, 9067, 1523, 5981, 348077, 3851, 1453, 352813, 16091, 355193, 2441, 2221, 12413, 362381, 1699, 2551, 2011, 4493, 4679, 185429, 19583, 186649, 53503, 26839, 17191, 379433, 27367, 1277, 1367, 4733, 4691, 35851, 30529, 15313, 399401, 28619, 18443, 2131, 58511, 29347, 15901, 208001, 417293, 211889, 4127, 213193, 61099, 30643, 14929, 19739, 435577, 2837, 62603, 10753, 221101, 443533, 63743, 2459, 225109, 41051, 226453, 454253, 4649, 2251, 229153, 2657, 462377, 467833, 24767, 473321, 2957, 238729, 240113, 43783, 1429, 5323, 4957, 2473, 489977, 492781, 3209, 3011, 38557, 19333, 36107, 1607, 73643, 258469, 27283, 19993, 261353, 3413, 527053, 24023, 529961, 265709, 38167, 76543, 2141, 271573, 2053, 547577, 274529, 550541, 553513, 1951, 43037, 562477, 281989, 80783, 40499, 568493, 2767, 13291, 286513, 2749, 1789, 22273, 30559, 10037, 583673, 292601, 42019, 8803, 26879, 592877, 595961, 300301, 602153, 15887, 605261, 303409, 7901, 2293, 2417, 306533, 614633, 308101, 47521, 28439, 90059, 6449, 33347, 317593, 636781, 2591, 45827, 8353, 11117, 58763, 14087, 649613, 93263, 46747, 656077, 328849, 2311, 8387, 3119, 1747, 672377, 11621, 13789, 61723, 26177, 341953, 98411, 23869, 346933, 36607, 99839, 4549, 2843, 27073, 10531, 353629, 16487, 14537, 4519, 65063, 65371, 1811, 3559, 55837, 729293, 732713, 3049, 12781, 2129, 5101, 4111, 1889, 375833, 68491, 9209, 7741, 5717, 4591, 33211, 382801, 40387, 29581, 774377, 35279, 777901, 781433, 112139, 393373, 788521, 5413, 10333, 802793, 17491, 806381, 57727, 36899, 12143, 2851, 62861, 117259, 28429, 413129, 414949, 75611, 59539, 418601, 32341, 842701, 14561, 850061, 38723, 38891, 857453, 61379, 2861, 66529, 433369, 3229, 22907, 11329, 15137, 879821, 440849, 883577, 63247, 1933, 40591, 894893, 40763, 64327, 902477, 47699, 5471, 910093, 5011, 457913, 83431, 459829, 4483, 66239, 21611, 2237, 72077, 11959, 473353, 14159, 475301, 3319, 2351, 86951, 479209, 7937, 37013, 74177, 69019, 138319, 485101, 2693, 487073, 2719, 7369, 24001, 493013, 76001,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-10x+2 and
the reducible primes which appear as divisor for the first time
p | x^2-10x+2 and p < x^2-10x+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^2-10x+2	P(x) \| x^2-10x+2	C/B	D/B	E/B	C(n) / C(n-1)	D(n) / D(n-1)	E(n) / E(n-1)
1	10	5	4	1	0,500000	0,400000	0,100000
2	100	66	20	46	0,660000	0,200000	0,460000	13,200000	5,000000	46,000000
3	1.000	696	130	566	0,696000	0,130000	0,566000	10,545455	6,500000	12,304348
4	10.000	7.013	849	6.164	0,701300	0,084900	0,616400	10,076149	6,530769	10,890459
5	100.000	70.084	6.642	63.442	0,700840	0,066420	0,634420	9,993441	7,823322	10,292343
6	1.000.000	698.560	54.459	644.101	0,698560	0,054459	0,644101	9,967468	8,199187	10,152596
7	10.000.000	6.976.745	461.539	6.515.206	0,697675	0,046154	0,651521	9,987324	8,474981	10,115193
8	100.000.000	69.689.495	3.999.141	65.690.354	0,696895	0,039991	0,656904	9,988826	8,664795	10,082621
9	1.000.000.000	696.315.808	35.316.159	660.999.649	0,696316	0,035316	0,661000	9,991690	8,830936	10,062355
10	10.000.000.000	6.958.903.909	316.068.230	6.642.835.679	0,695890	0,031607	0,664284	9,993890	8,949677	10,049681
11	100.000.000.000	69.556.070.550	2.860.327.345	66.695.743.205	0,695561	0,028603	0,666957	9,995262	9,049715	10,040252
12	1.000.000.000.000	695.300.472.186	26.121.920.194	669.178.551.992	0,695300	0,026122	0,669179	9,996259	9,132493	10,033302


A	B	C	D	E	F	G	H	I	J	K
2^n	x	all Primes	P(x)=x^2-10x+2	P(x) \| x^2-10x+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	5	4	1	0,625000	0,500000	0,125000	1,250000	1,333333	1,000000
4	16	8	6	2	0,500000	0,375000	0,125000	1,600000	1,500000	2,000000
5	32	17	10	7	0,531250	0,312500	0,218750	2,125000	1,666667	3,500000
6	64	39	16	23	0,609375	0,250000	0,359375	2,294118	1,600000	3,285714
7	128	84	23	61	0,656250	0,179688	0,476563	2,153846	1,437500	2,652174
8	256	178	43	135	0,695313	0,167969	0,527344	2,119048	1,869565	2,213115
9	512	356	75	281	0,695313	0,146484	0,548828	2,000000	1,744186	2,081481
10	1.024	715	132	583	0,698242	0,128906	0,569336	2,008427	1,760000	2,074733
11	2.048	1.427	234	1.193	0,696777	0,114258	0,582520	1,995804	1,772727	2,046312
12	4.096	2.868	405	2.463	0,700195	0,098877	0,601318	2,009811	1,730769	2,064543
13	8.192	5.756	719	5.037	0,702637	0,087769	0,614868	2,006974	1,775309	2,045067
14	16.384	11.496	1.325	10.171	0,701660	0,080872	0,620789	1,997220	1,842837	2,019257
15	32.768	22.996	2.434	20.562	0,701782	0,074280	0,627502	2,000348	1,836981	2,021630
16	65.536	45.946	4.553	41.393	0,701080	0,069473	0,631607	1,998000	1,870583	2,013082
17	131.072	91.874	8.521	83.353	0,700943	0,065010	0,635933	1,999608	1,871513	2,013698
18	262.144	183.433	16.039	167.394	0,699741	0,061184	0,638557	1,996571	1,882291	2,008254
19	524.288	366.473	30.096	336.377	0,698992	0,057404	0,641588	1,997858	1,876426	2,009493
20	1.048.576	732.422	56.980	675.442	0,698492	0,054340	0,644152	1,998570	1,893275	2,007991
21	2.097.152	1.464.563	107.863	1.356.700	0,698358	0,051433	0,646925	1,999616	1,892998	2,008611
22	4.194.304	2.927.645	205.247	2.722.398	0,698005	0,048935	0,649070	1,998989	1,902849	2,006632
23	8.388.608	5.852.683	391.675	5.461.008	0,697694	0,046691	0,651003	1,999110	1,908310	2,005955
24	16.777.216	11.701.189	748.810	10.952.379	0,697445	0,044633	0,652813	1,999286	1,911815	2,005560
25	33.554.432	23.394.392	1.432.772	21.961.620	0,697207	0,042700	0,654507	1,999318	1,913399	2,005192
26	67.108.864	46.774.459	2.747.186	44.027.273	0,696994	0,040936	0,656057	1,999388	1,917392	2,004737
27	134.217.728	93.523.222	5.278.615	88.244.607	0,696802	0,039329	0,657474	1,999451	1,921463	2,004317
28	268.435.456	186.996.749	10.160.115	176.836.634	0,696617	0,037849	0,658768	1,999469	1,924769	2,003937
29	536.870.912	373.904.646	19.580.675	354.323.971	0,696452	0,036472	0,659980	1,999525	1,927210	2,003680
30	1.073.741.824	747.646.516	37.784.271	709.862.245	0,696300	0,035189	0,661111	1,999565	1,929672	2,003427
31	2.147.483.648	1.494.986.785	72.992.230	1.421.994.555	0,696157	0,033990	0,662168	1,999590	1,931815	2,003198
32	4.294.967.296	2.989.433.068	141.186.109	2.848.246.959	0,696032	0,032872	0,663159	1,999638	1,934262	2,002994
33	8.589.934.592	5.977.865.647	273.395.448	5.704.470.199	0,695915	0,031827	0,664088	1,999665	1,936419	2,002800
34	17.179.869.184	11.953.831.597	529.936.294	11.423.895.303	0,695805	0,030846	0,664958	1,999682	1,938351	2,002622
35	34.359.738.368	23.904.195.871	1.028.133.233	22.876.062.638	0,695704	0,029923	0,665781	1,999710	1,940107	2,002475
36	68.719.476.736	47.801.903.792	1.996.517.596	45.805.386.196	0,695609	0,029053	0,666556	1,999729	1,941886	2,002328
37	137.438.953.472	95.591.810.760	3.880.275.280	91.711.535.480	0,695522	0,028233	0,667289	1,999749	1,943522	2,002200
38	274.877.906.944	191.161.013.353	7.547.544.461	183.613.468.892	0,695440	0,027458	0,667982	1,999763	1,945105	2,002076
39	549.755.813.888	382.279.766.577	14.691.832.804	367.587.933.773	0,695363	0,026724	0,668639	1,999779	1,946571	2,001966
40	1.099.511.627.776	764.480.587.987	28.619.064.640	735.861.523.347	0,695291	0,026029	0,669262	1,999793	1,947957	2,001865

8. Check for existing Integer Sequences by OEIS
Found in Database : 1, 7, 1, 19, 11, 23, 1, 1, 1, 1, 1, 13, 1, 41, 29, 1, 1, 1, 73, 173,
Found in Database : 7, 19, 11, 23, 13, 41, 29, 73, 173, 101, 233, 43, 461, 79, 653, 353, 761, 409, 877, 67, 103,
Found in Database : 7, 11, 13, 19, 23, 29, 41, 43, 67, 73, 79, 83, 101, 103, 107,

Development ofAlgorithmic Constructions

Polynom = x^2-10x+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