素数生成多项式

yangchuanju · 发表于 2025-4-30 15:00

本帖最后由 yangchuanju 于 2025-5-1 03:52 编辑

Prime-Generating Polynomial
https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
网页《Prime-Generating Polynomial》给出23个素数生成多项式，
n 素数个数多项式
56 57 1/4*(n^5-133n^4+6729n^3-158379n^2+1720294n-6823316)
54 55 1/36*(n^6-126n^5+6217n^4-153066n^3+1987786n^2-13055316n+34747236)
49 49 n^4-97n^3+3294n^2-45458n+213589
46 47 n^5-99n^4+3588n^3-56822n^2+348272n-286397
45 46 66n^3+3845n^2-60897n+251831
44 45 36n^2-810n+2753
46 43 3n^3-183n^2+3318n-18757
42 43 47n^2-1701n+10181
42 43 103n^2-4707n+50383
40 40 n^2-n+41
39 40 42n^3+270n^2-26436n+250703
34 35 43n^2-537n+2971
61 31 8n^2-488n+7243
57 29 6n^2-342n+4903
28 29 2n^2+29
23 24 7n^2-371n+4871
21 22 3n^2+3n+23
19 20 n^4+29n^3+101
17 18 3n^2+39n+37
15 16 n^2+n+17
13 14 4n^2+4n+59
10 11 2n^2+11
10 11 n^3+n^2+17

yangchuanju · 发表于 2025-4-30 15:02

本帖最后由 yangchuanju 于 2025-4-30 18:02 编辑

各种各样的多项式大都能生成无穷多个素数，其中必然还要夹带大量的（或无穷多个）合数；
在多项式计算值的前部一般会有几个或几十个连续素数，如2x^2+29开始的29个数字(x=0-28)都是素数，x^2+x+41开始的40个数字(x=0-39)都是素数；
现在已经找到一个能够产生57个连续素数的多项式，但要找到一个能够产生无穷多个连续素数的多项式是不可能的！

太阳先生试图找到不含合数的多项式，有可能吗？
别人找到了产生57个连续素数的多项式，太阳先生若能找到一个产生58个连续素数的多项式，也算得上破世界纪录啦！
若能找到第二个产生57个连续素数的多项式，刷新一下世界纪录也是世界名人啦！

yangchuanju · 发表于 2025-4-30 15:11

本帖最后由 yangchuanju 于 2025-4-30 18:03 编辑

仿太阳“求证：c=p”之命题——

已知：2*a^2+p=c，自然数a≥0，素数p≥3，奇数c≥3，
求：c中的连续素数
例：当p=3，a=0,1,2时，c=3,5,11有3个连续素数；
当p=5，a=0,1,2,3,4时，c=5,7,13,23,37有5个连续素数。

yangchuanju · 发表于 2025-4-30 15:15

A165234给出能够产生1-15个连续素数的最小素数
Least prime p such that 2x^2 + p produces primes for x=0..n-1 and composite for x=n.
1 2
2 17
3 3
4 1481
5 5
6 149
7 569
8 2081
9 2339
10 5939831
11 11
12 33164857769
13 3217755097229
14 272259344081
15 17762917045631

16-28不知
29 29

yangchuanju · 发表于 2025-4-30 15:19

A271348给出22个至少产生11个连续素数的二次多项式2x^2+p之素数p——
1 11 11
2 29 29
3 438926021 11
4 1210400879 11
5 7446335849 11
6 31757068151 11
7 33090566651 11
8 33164857769 12
9 40137398219 11
10 45133754591 11
11 46642404071 11
12 100444384301 11
13 114546675671 11
14 144553207691 11
15 159587584529 12
16 161557039991 11
17 166054101539 11
18 210447830009 11
19 227625400031 11
20 236241327599 12
21 254850262949 11
22 272259344081 14

yangchuanju · 发表于 2025-5-1 04:42

由1#多项式生成的57个连续素数（负数取绝对值）是——
(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316)/4.
0 -1705829 prime
1 -1313701 prime
2 -991127 prime
3 -729173 prime
4 -519643 prime
5 -355049 prime
6 -228581 prime
7 -134077 prime
8 -65993 prime
9 -19373 prime
10 10181 prime
11 26539 prime
12 33073 prime
13 32687 prime
14 27847 prime
15 20611 prime
16 12659 prime
17 5323 prime
18 -383 prime
19 -3733 prime
20 -4259 prime
21 -1721 prime
22 3923 prime
23 12547 prime
24 23887 prime
25 37571 prime
26 53149 prime
27 70123 prime
28 87977 prime
29 106207 prime
30 124351 prime
31 142019 prime
32 158923 prime
33 174907 prime
34 189977 prime
35 204331 prime
36 218389 prime
37 232823 prime
38 248587 prime
39 266947 prime
40 289511 prime
41 318259 prime
42 355573 prime
43 404267 prime
44 467617 prime
45 549391 prime
46 653879 prime
47 785923 prime
48 950947 prime
49 1154987 prime
50 1404721 prime
51 1707499 prime
52 2071373 prime
53 2505127 prime
54 3018307 prime
55 3621251 prime
56 4325119 prime

yangchuanju · 发表于 2025-5-1 04:47

本帖最后由 yangchuanju 于 2025-5-1 04:48 编辑

由多项式生成57个连续素数的记录已经被打破，生成58个连续素数的记录已产生——
A330363提供（n等于-45至12）
-45 39300979
-44 32074681
-43 25874993
-42 20591567
-41 16122559
-40 12374209
-39 9260431
-38 6702413
-37 4628227
-36 2972449
-35 1675789
-34 684731
-33 -48817
-32 -567863
-31 -910661
-30 -1111031
-29 -1198669
-28 -1199447
-27 -1135703
-26 -1026521
-25 -888001
-24 -733519
-23 -573977
-22 -418043
-21 -272381
-20 -141871
-19 -29819
-18 61843
-17 132367
-16 182009
-15 211859
-14 223681
-13 219763
-12 202777
-11 175649
-10 141439
-9 103231
-8 64033
-7 26687
-6 -6211
-5 -32381
-4 -49919
-3 -57347
-2 -53653
-1 -38321
0 -11351
1 26731
2 74873
3 131507
4 194569
5 261529
6 329431
7 394943
8 454417
9 503959
10 539509
11 556931
12 552113

yangchuanju · 发表于 2025-5-1 04:56

由二次三项式a*x^2-b*x+p生成的多个连续素数之最小p
A268101提供
1 2 a(1) = 2 (a prime), x^2 + 2 gives a prime for x = 1.
2 3 a(2) = 3 (a prime), 2*x^2 + 3 gives distinct primes for x = 1 to 2.
3 5
4 5 a(4) = 5 (a prime), 2*x^2 + 5 gives distinct primes for x = 1 to 4.
5 7
6 7 a(6) = 7 (a prime), 4*x^2 + 7 gives distinct primes for x = 1 to 6.
7 11
8 11
9 11
10 11 a(10) = 11 (a prime), 2*x^2 + 11 gives distinct primes for x = 1 to 10.
11 13
12 13 a(12) = 13 (a prime), 6*x^2 + 13 gives distinct primes for x = 1 to 12.
13 17
14 17
15 17
16 17 a(16) = 17 (a prime), 6*x^2 + 17 gives distinct primes for x = 1 to 16.
17 19
18 19 a(18) = 19 (a prime), 10*x^2 + 19 gives distinct primes for x = 1 to 18.
19 23
20 23
21 23
22 23 a(22) = 23 (a prime), 3*x^2 - 3*x + 23 gives distinct primes for x = 1 to 22.
23 29
24 29
25 29
26 29
27 29
28 29 a(28) = 29 (a prime), 2*x^2 + 29 gives distinct primes for x = 1 to 28.
29 31 a(29) = 31 (a prime), 2*x^2 - 4*x + 31 gives distinct primes for x = 1 to 29.
30 41
31 41
32 41
33 41
34 41
35 41
36 41
37 41
38 41
39 41
40 41 a(40) = 41 (a prime), x^2 - x + 41 gives distinct primes for x = 1 to 40.
41 647 a(41) = 647 (a prime), abs(36*x^2 - 594*x + 647) gives distinct primes for x = 1 to 41.
42 1277 a(42) = 1277 (a prime), abs(36*x^2 - 666*x + 1277) gives distinct primes for x = 1 to 42.
43 1979 a(43) = 1979 (a prime), abs(36*x^2 - 738*x + 1979) gives distinct primes for x = 1 to 43.
44 2753 a(44) = 2753 (a prime), abs(36*x^2 - 810*x + 2753) gives distinct primes for x = 1 to 44.

yangchuanju · 发表于 2025-5-1 05:06

虽然由多项式n^2-79n+1601可生成80个连续素数，
然它不被认为是世界纪录，
它是由生成40个连续素数的欧拉多项式n^2±n+41演变而来的，80个素数两两相等！

yangchuanju · 发表于 2025-5-1 05:19

由2#多项式1/36*(n^6 - 126n^5 + 6217n^4 - 153066n^3 + 1987786n^2 - 13055316n + 34747236)生成的55个连续素数（负数取绝对值）是——
A272555提供
1 965201
2 653687
3 429409
4 272563
5 166693
6 98321
7 56597
8 32969
9 20873
10 15443
11 13241
12 12007
13 10429
14 7933
15 4493
16 461
17 3583
18 6961
19 9007
20 9157
21 7019
22 2423
23 4549
24 13553
25 23993
26 35051
27 45737
28 54959
29 61613
30 64693
31 63421
32 57397
33 46769
34 32423
35 16193
36 1091
37 8443
38 6271
39 15733
40 67993
41 163561
42 318467
43 552089
44 887543
45 1352093
46 1977581
47 2800877
48 3864349
49 5216353
50 6911743
51 9012401
52 11587787
53 14715509
54 18481913
55 22982693

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