Inhaltsverzeichnis

Development of
Algorithmic Constructions

03:27:06
Deutsch
29.Mar 2024

Polynom = x^2-276x+1811

0. Sequence

1. Algorithm

2. Mathematical background

3. Correctness of the algorithm

4. Infinity of the sequence

5. Sequence of the polynom with 1

6. Sequence of the polynom (only primes)

7. Distribution of the primes

8. Check for existing Integer Sequences by OEIS

0. Sequence

f(0) = 1811 = 1811
f(1) = 3 = 3
f(2) = 1263 = 3*421
f(3) = 31 = 31
f(4) = 723 = 3*241
f(5) = 57 = 3*19
f(6) = 191 = 191
f(7) = 9 = 3*3
f(8) = 333 = 3*3*37
f(9) = 37 = 37
f(10) = 849 = 3*283
f(11) = 69 = 3*23
f(12) = 1357 = 23*59
f(13) = 201 = 3*67
f(14) = 1857 = 3*619
f(15) = 263 = 263
f(16) = 2349 = 3*3*3*3*29
f(17) = 81 = 3*3*3*3
f(18) = 2833 = 2833
f(19) = 3 = 3
f(20) = 3309 = 3*1103
f(21) = 443 = 443
f(22) = 3777 = 3*1259
f(23) = 501 = 3*167
f(24) = 4237 = 19*223
f(25) = 279 = 3*3*31
f(26) = 4689 = 3*3*521
f(27) = 307 = 307
f(28) = 5133 = 3*29*59
f(29) = 669 = 3*223
f(30) = 5569 = 5569
f(31) = 723 = 3*241
f(32) = 5997 = 3*1999
f(33) = 97 = 97
f(34) = 6417 = 3*3*23*31
f(35) = 207 = 3*3*23
f(36) = 6829 = 6829
f(37) = 879 = 3*293
f(38) = 7233 = 3*2411
f(39) = 929 = 929
f(40) = 7629 = 3*2543
f(41) = 489 = 3*163
f(42) = 8017 = 8017
f(43) = 513 = 3*3*3*19
f(44) = 8397 = 3*3*3*311
f(45) = 1073 = 29*37
f(46) = 8769 = 3*37*79
f(47) = 1119 = 3*373
f(48) = 9133 = 9133
f(49) = 291 = 3*97
f(50) = 9489 = 3*3163
f(51) = 151 = 151
f(52) = 9837 = 3*3*1093
f(53) = 1251 = 3*3*139
f(54) = 10177 = 10177
f(55) = 1293 = 3*431
f(56) = 10509 = 3*31*113
f(57) = 667 = 23*29
f(58) = 10833 = 3*23*157
f(59) = 687 = 3*229
f(60) = 11149 = 11149
f(61) = 1413 = 3*3*157
f(62) = 11457 = 3*3*19*67
f(63) = 1451 = 1451
f(64) = 11757 = 3*3919
f(65) = 93 = 3*31
f(66) = 12049 = 12049
f(67) = 381 = 3*127
f(68) = 12333 = 3*4111
f(69) = 1559 = 1559
f(70) = 12609 = 3*3*3*467
f(71) = 1593 = 3*3*3*59
f(72) = 12877 = 79*163
f(73) = 813 = 3*271
f(74) = 13137 = 3*29*151
f(75) = 829 = 829
f(76) = 13389 = 3*4463
f(77) = 1689 = 3*563
f(78) = 13633 = 13633
f(79) = 1719 = 3*3*191
f(80) = 13869 = 3*3*23*67
f(81) = 437 = 19*23
f(82) = 14097 = 3*37*127
f(83) = 111 = 3*37
f(84) = 14317 = 103*139
f(85) = 1803 = 3*601
f(86) = 14529 = 3*29*167
f(87) = 1829 = 31*59
f(88) = 14733 = 3*3*1637
f(89) = 927 = 3*3*103
f(90) = 14929 = 14929
f(91) = 939 = 3*313
f(92) = 15117 = 3*5039
f(93) = 1901 = 1901
f(94) = 15297 = 3*5099
f(95) = 1923 = 3*641
f(96) = 15469 = 31*499
f(97) = 243 = 3*3*3*3*3
f(98) = 15633 = 3*3*3*3*193
f(99) = 491 = 491
f(100) = 15789 = 3*19*277

1. Algorithm

If you are interested in some better algorithms have a look at quadr_Sieb_x^2+1.php.

2. Mathematical background

Lemma: If p | f(x) then also p | f(x+p) and p | f(-x-b/a) a) p | f(x) <=> ax^2 + bx + c = 0 mod p p | f(x+p) <=> a(x+p)^2 + b(x+p) + c = 0 mod p <=> ax^2 + 2axp + ap^2 + bx + bp + c = 0 mod p <=> ax^2 + bx + c = 0 mod p Thus if p | f(x) then p | f(x+p) b) if b = 0 mod a p | f(x) <=> ax^2 + bx + c = 0 mod p p | f(-x-b/a) <=> a(-x-b/a)^2 + b(-x-b/a) + c = 0 mod p <=> ax^2 + 2bx + b^2/a - bx - b^2/a + c = 0 mod p <=> ax^2 + bx + c = 0 mod p Thus if p | f(x) then p | f(-x-b/a)

3. Correctness of the algorithm

The proof for this polynom is similar to the proof for the polynom f(x)=x^2-4x+1. a) First terms for the polynom f(x) = x^2-276x+1811

f(0)=1811
f(1)=3
f(2)=421
f(3)=31
f(4)=241
f(5)=19
f(6)=191
f(7)=1
f(8)=37
f(9)=1
f(10)=283
f(11)=23
f(12)=59
f(13)=67
f(14)=619
f(15)=263
f(16)=29
f(17)=1
f(18)=2833
f(19)=1
f(20)=1103
f(21)=443
f(22)=1259
f(23)=167
f(24)=223
f(25)=1
f(26)=521
f(27)=307
f(28)=1
f(29)=1
f(30)=5569
f(31)=1
f(32)=1999
f(33)=97
f(34)=1
f(35)=1
f(36)=6829
f(37)=293
f(38)=2411
f(39)=929
f(40)=2543
f(41)=163
f(42)=8017
f(43)=1
f(44)=311
f(45)=1
f(46)=79
f(47)=373
f(48)=9133
f(49)=1
f(50)=3163
f(51)=151
f(52)=1093
f(53)=139
f(54)=10177
f(55)=431
f(56)=113
f(57)=1
f(58)=157
f(59)=229
f(60)=11149
f(61)=1
f(62)=1
f(63)=1451
f(64)=3919
f(65)=1
f(66)=12049
f(67)=127
f(68)=4111
f(69)=1559
f(70)=467
f(71)=1
f(72)=1
f(73)=271
f(74)=1
f(75)=829
f(76)=4463
f(77)=563
f(78)=13633
f(79)=1
f(80)=1
f(81)=1
f(82)=1
f(83)=1
f(84)=103
f(85)=601
f(86)=1
f(87)=1
f(88)=1637
f(89)=1
f(90)=14929
f(91)=313
f(92)=5039
f(93)=1901
f(94)=5099
f(95)=641
f(96)=499
f(97)=1
f(98)=193
f(99)=491

b) Substitution of the polynom
The polynom f(x)=x^2-276x+1811 could be written as f(y)= y^2-17233 with x=y+138

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-138
f'(x)>2x-277 with x > 131

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

1811, 3, 421, 31, 241, 19, 191, 1, 37, 1, 283, 23, 59, 67, 619, 263, 29, 1, 2833, 1, 1103, 443, 1259, 167, 223, 1, 521, 307, 1, 1, 5569, 1, 1999, 97, 1, 1, 6829, 293, 2411, 929, 2543, 163, 8017, 1, 311, 1, 79, 373, 9133, 1, 3163, 151, 1093, 139, 10177, 431, 113, 1, 157, 229, 11149, 1, 1, 1451, 3919, 1, 12049, 127, 4111, 1559, 467, 1, 1, 271, 1, 829, 4463, 563, 13633, 1, 1, 1, 1, 1, 103, 601, 1, 1, 1637, 1, 14929, 313, 5039, 1901, 5099, 641, 499, 1, 193, 491, 277, 661, 15937, 1, 233, 1009, 1801, 1, 16333, 683, 5483, 2063, 5519, 173, 16657, 1, 1861, 2099, 181, 1, 457, 353, 5659, 1063, 631, 1, 743, 1, 5711, 1, 1, 179, 593, 239, 1913, 2153, 5743, 359, 907, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 331, 977, 1, 1, 1, 1361, 547, 1, 1, 1, 1, 1, 1, 2161, 1, 7103, 1, 859, 503, 2789, 1, 9011, 389, 3221, 1249, 1, 1, 379, 1, 3889, 1, 1, 1, 13043, 1, 509, 881, 4817, 617, 523, 647, 5297, 1, 1847, 1, 1, 739, 6037, 2311, 1, 401, 1, 1, 2267, 1, 1, 1, 1, 1, 7589, 1, 1, 1, 659, 1033, 1, 1601, 8677, 1, 1, 1, 3079, 3517, 9521, 1, 29423, 1, 439, 1, 3463, 1, 32051, 677, 1, 2087, 1, 1429, 1, 1, 1321, 1129, 12197, 1, 1, 1583, 12821, 1, 1, 1, 1091, 1, 599, 5227, 1, 1783, 43283, 1, 4919, 1399, 1, 1907, 46271, 1949, 15761, 1, 1789, 1, 49331, 1, 1, 6361, 17137, 541, 2281, 1, 1, 1, 587, 2297, 55667, 1171, 18917, 3581, 6427, 811, 58943, 1, 1, 947, 20389, 643, 1, 1, 1, 1, 21521, 1, 2857, 1381, 719, 8431, 7559, 953, 69203, 727, 809, 1109, 23857, 1, 72767, 1019, 8219, 1, 1319, 1579, 76403, 3209, 1, 9781, 1, 1, 80111, 1, 733, 1, 27541, 3469, 83891, 1, 9463, 1, 28817, 1, 87743, 1, 1, 2803, 10039, 1, 2957, 3847, 1, 1, 1367, 1979, 1, 1, 3593, 12211, 32789, 4127, 1, 1, 1, 3181, 1, 1433, 1, 4357, 35089, 1, 1, 2237, 3727, 1511, 1, 1, 1, 1163, 1, 1, 1, 1, 1423, 1, 773, 2447, 39397, 7433, 1, 1, 121151, 1693, 13627, 1, 41381, 1301, 125651, 1, 1, 15991, 1, 1, 130223, 2729, 1, 16567, 1201, 1, 134867, 1, 1, 1, 1, 5783, 1439, 5849, 47057, 1, 1, 997, 6277, 1, 48661, 1, 49201, 1, 1889, 1, 16763, 1, 1753, 6389, 4973, 3229, 51941, 9791, 1, 1, 159167, 1, 53617, 1, 54181, 1, 1, 2293, 18439, 1, 55889, 3511, 1, 3547, 57041, 21499, 19207, 1, 1, 1, 58789, 1, 59377, 7459, 179903, 1, 2243, 11411, 2659, 1, 185267, 7757, 1, 23497, 1, 1, 10037, 1, 2213, 24181, 967, 1, 5303, 1, 1, 12437, 1, 8369, 201791, 8447, 2341, 1, 1, 1, 207443, 1, 2251, 1, 1051, 4421, 3613, 1487, 23899, 1, 72341, 1, 218963, 1, 73637, 3467, 24763, 3109, 224831, 1, 1, 1, 76261, 4787, 230771, 1, 1, 29221, 1, 1, 236783, 2477, 79601, 1, 26759, 3359, 242867, 5081, 81637, 1, 1, 10333, 1, 1, 1213, 7879, 84389, 1, 255251, 1, 85781, 32299, 3203, 1, 1, 5471, 1, 33091, 4663, 11119, 2371, 1, 1, 1, 2927, 1, 1, 1, 92177, 17351, 1, 1, 280883, 1, 1, 35521, 95089, 1, 1, 1, 10729, 1, 97301, 1, 294131, 6151, 1, 18593, 33179, 1, 1, 12583, 1279, 2377, 101797, 1, 16193, 4289, 34439, 1, 1, 6529, 314543, 6577, 3407, 1, 11821, 1483, 5449, 3361, 107941, 2539, 1, 1, 328511, 1, 1, 20753, 1, 6967, 5009, 14033, 1, 42397, 1, 1, 11057, 3583, 115057, 1493, 115861, 14533, 350003, 1, 1, 1, 1, 1, 5333, 14939, 119921, 1, 1, 1, 364691, 1, 122389, 46051, 123217, 7727, 1, 2593, 41627, 2473, 125717, 15767, 379667, 1, 1, 1, 14249, 1787, 387263, 16189, 129937, 24443, 4219, 1, 394931, 5503, 1523, 1, 7019, 1, 10883, 1, 135089, 50821, 45319, 5683, 1, 1, 5987, 1, 138577, 17377, 1, 1, 1, 6599, 141221, 1, 1621, 1, 1, 53791, 47963, 1, 4217, 1, 145681, 54799, 146581, 18379, 19237, 1, 49463, 6977, 149297, 18719, 1, 1, 151121, 1, 1877, 1, 15823, 1, 153877, 1867, 1, 1, 467183, 1627, 1, 58921, 2671, 1, 1, 1, 1, 1, 1, 6703, 3083, 1, 162289, 15259, 1, 1, 492563, 2287, 1, 1, 1, 1, 3001, 1, 1, 1, 1, 1, 509843, 2663, 2551, 16069, 171889, 1, 2687, 1, 57947, 32687, 174821, 10957, 527411, 22037, 176789, 3499, 19753, 1, 2963, 2801, 2683, 2939, 1, 1, 545267, 3797, 60919, 34361, 183761, 1, 554303, 23159, 5021, 1, 1, 1951, 563411, 23539, 6091, 70999, 1, 11897, 5903, 1, 1, 3137, 192917, 24179, 581843, 1, 194981, 1, 1, 1, 4253, 24697, 198097, 1, 2053, 12479, 19373, 8363, 67079, 2609, 202289, 3169, 610031, 1, 8887, 1, 1, 2861, 32609, 12941, 207589, 39023, 208657, 1, 1, 8761, 2423, 19813, 211877, 3319, 2081, 26687, 1, 1, 71707, 4493, 28201, 1, 1, 1, 218389, 27367, 5827, 1, 24509, 20731, 7151, 27779, 1, 27917, 1, 42083, 2027, 1, 11497, 1, 11959, 1, 228337, 1, 1, 1, 1, 86677, 231701, 29033, 698483, 1, 7547, 43973, 8707, 1, 2141, 29599, 237361, 1, 3019, 1, 718931, 10009, 80263, 90511,

6. Sequence of the polynom (only primes)

1811, 3, 421, 31, 241, 19, 191, 37, 283, 23, 59, 67, 619, 263, 29, 2833, 1103, 443, 1259, 167, 223, 521, 307, 5569, 1999, 97, 6829, 293, 2411, 929, 2543, 163, 8017, 311, 79, 373, 9133, 3163, 151, 1093, 139, 10177, 431, 113, 157, 229, 11149, 1451, 3919, 12049, 127, 4111, 1559, 467, 271, 829, 4463, 563, 13633, 103, 601, 1637, 14929, 313, 5039, 1901, 5099, 641, 499, 193, 491, 277, 661, 15937, 233, 1009, 1801, 16333, 683, 5483, 2063, 5519, 173, 16657, 1861, 2099, 181, 457, 353, 5659, 1063, 631, 743, 5711, 179, 593, 239, 1913, 2153, 5743, 359, 907, 331, 977, 1361, 547, 2161, 7103, 859, 503, 2789, 9011, 389, 3221, 1249, 379, 3889, 13043, 509, 881, 4817, 617, 523, 647, 5297, 1847, 739, 6037, 2311, 401, 2267, 7589, 659, 1033, 1601, 8677, 3079, 3517, 9521, 29423, 439, 3463, 32051, 677, 2087, 1429, 1321, 1129, 12197, 1583, 12821, 1091, 599, 5227, 1783, 43283, 4919, 1399, 1907, 46271, 1949, 15761, 1789, 49331, 6361, 17137, 541, 2281, 587, 2297, 55667, 1171, 18917, 3581, 6427, 811, 58943, 947, 20389, 643, 21521, 2857, 1381, 719, 8431, 7559, 953, 69203, 727, 809, 1109, 23857, 72767, 1019, 8219, 1319, 1579, 76403, 3209, 9781, 80111, 733, 27541, 3469, 83891, 9463, 28817, 87743, 2803, 10039, 2957, 3847, 1367, 1979, 3593, 12211, 32789, 4127, 3181, 1433, 4357, 35089, 2237, 3727, 1511, 1163, 1423, 773, 2447, 39397, 7433, 121151, 1693, 13627, 41381, 1301, 125651, 15991, 130223, 2729, 16567, 1201, 134867, 5783, 1439, 5849, 47057, 997, 6277, 48661, 49201, 1889, 16763, 1753, 6389, 4973, 3229, 51941, 9791, 159167, 53617, 54181, 2293, 18439, 55889, 3511, 3547, 57041, 21499, 19207, 58789, 59377, 7459, 179903, 2243, 11411, 2659, 185267, 7757, 23497, 10037, 2213, 24181, 967, 5303, 12437, 8369, 201791, 8447, 2341, 207443, 2251, 1051, 4421, 3613, 1487, 23899, 72341, 218963, 73637, 3467, 24763, 3109, 224831, 76261, 4787, 230771, 29221, 236783, 2477, 79601, 26759, 3359, 242867, 5081, 81637, 10333, 1213, 7879, 84389, 255251, 85781, 32299, 3203, 5471, 33091, 4663, 11119, 2371, 2927, 92177, 17351, 280883, 35521, 95089, 10729, 97301, 294131, 6151, 18593, 33179, 12583, 1279, 2377, 101797, 16193, 4289, 34439, 6529, 314543, 6577, 3407, 11821, 1483, 5449, 3361, 107941, 2539, 328511, 20753, 6967, 5009, 14033, 42397, 11057, 3583, 115057, 1493, 115861, 14533, 350003, 5333, 14939, 119921, 364691, 122389, 46051, 123217, 7727, 2593, 41627, 2473, 125717, 15767, 379667, 14249, 1787, 387263, 16189, 129937, 24443, 4219, 394931, 5503, 1523, 7019, 10883, 135089, 50821, 45319, 5683, 5987, 138577, 17377, 6599, 141221, 1621, 53791, 47963, 4217, 145681, 54799, 146581, 18379, 19237, 49463, 6977, 149297, 18719, 151121, 1877, 15823, 153877, 1867, 467183, 1627, 58921, 2671, 6703, 3083, 162289, 15259, 492563, 2287, 3001, 509843, 2663, 2551, 16069, 171889, 2687, 57947, 32687, 174821, 10957, 527411, 22037, 176789, 3499, 19753, 2963, 2801, 2683, 2939, 545267, 3797, 60919, 34361, 183761, 554303, 23159, 5021, 1951, 563411, 23539, 6091, 70999, 11897, 5903, 3137, 192917, 24179, 581843, 194981, 4253, 24697, 198097, 2053, 12479, 19373, 8363, 67079, 2609, 202289, 3169, 610031, 8887, 2861, 32609, 12941, 207589, 39023, 208657, 8761, 2423, 19813, 211877, 3319, 2081, 26687, 71707, 4493, 28201, 218389, 27367, 5827, 24509, 20731, 7151, 27779, 27917, 42083, 2027, 11497, 11959, 228337, 86677, 231701, 29033, 698483, 7547, 43973, 8707, 2141, 29599, 237361, 3019, 718931, 10009, 80263, 90511,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-276x+1811 and
the reducible primes which appear as divisor for the first time
p | x^2-276x+1811 and p < x^2-276x+1811

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

8. Check for existing Integer Sequences by OEIS

Found in Database : 1811, 3, 421, 31, 241, 19, 191, 1, 37, 1, 283, 23, 59, 67, 619, 263, 29, 1, 2833, 1,
Found in Database : 1811, 3, 421, 31, 241, 19, 191, 37, 283, 23, 59, 67, 619, 263, 29, 2833, 1103, 443, 1259, 167, 223, 521, 307, 5569, 1999, 97, 6829, 293, 2411, 929,
Found in Database : 3, 19, 23, 29, 31, 37, 59, 67, 79, 97, 103, 113, 127, 139,