Problems & Puzzles: Puzzles

 Puzzle 355. Puzzle 232 in 4th order Seiji Tomita sends the following puzzle: I made a puzzle about prime-generating polynomials. When I say easily, it is the fourth degree version of the puzzle 232. At fourth polynomials f(x) =ax^4+bx^3+cx^2+dx+e, looking for those producing only primes in the x range [0, k]. T: Total primes(The quantity that absolute of f(x) is prime) N: Negative primes in the Total primes D: Distinct primes in the Total primes Search contents: 1. Find the maximum number of T. 2. Find the maximum number of D. 3. Find the maximum number of T(Only positive primes). The search results are as the next. No a b c d e T N D 1. 1 -134 6272 -119461 793669 68 4 34 2. 1 -63 1155 -7449 11593 43 31 42 3. 1 -63 1165 -7649 15073 42 25 42 4. 1 -70 1429 -7140 11287 38 0 20 5. 1 -74 1525 -5772 10159 38 0 19 As for the details, see my homepage.   Questions: 1.Find better polynomials than No.1(T>68) 2.Find better polynomials than No.2,3(D>42) 3.Find better polynomials than No.4,5(T>38,Only positive primes)

Parviz Afereidoon wrote (22/4/6:

I found  better polynomials for No.4,5 (question 3).

a b c d e T N D

3 -312 12813 -244452 1830617 53 0 27

38 -3496 125248 -2062640 13104799 47 0 24

285 -22800 686505 -9220200 46610461 41 0 21

135 -10800 329895 -4555800 23962973 41 0 21

48 -3840 115968 -1566720 7987843 41 0 21

20 -1600 51070 -762800 4437883 41 0 21

4 -320 9846 -137840 741563 41 0 21

1 -80 3881 -91240 822881 41 0 21

135 -10260 303435 -4122810 21585761 39 0 20

12 -912 26588 -351880 1803379 39 0 20

8 -608 17354 -220476 1062311 39 0 20

3 -228 6671 -88882 480133 39 0 20

1 -76 4405 -112518 966751 39 0 20

1 -76 2303 -32642 184837 39 0 20

1 -76 2195 -28538 140891 39 0 20

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On June 21, 06, J. C. Meyrignac reported this:

Following our programming contest http://www.recmath.org/contest/index.php

New records for N=4 (puzzle 355):

45.x^4 - 3416.x^3 + 96738.x^2 - 1212769.x + 5692031
gives 42 positive primes (discovered by Ivan Kazmenko & Vadim Trofimov)
x^4 - 97.x^3 + 3294.x^2 - 45458.x + 213589
abs() of this function gives 49 distinct primes (discovered by Mark Beyleveld)

The polynomial:
1/2.x^4 - 9.x^3 + 655/2.x^2 - 11139.x + 98731
gives 43 positive primes (discovered by Mark Beyleveld)
(starting with 98731 87911 77699 68059 58967...)

The polynomial:
3/4.x^4 - 193/2.x^3 + 14301/4.x^2 - 95759/2.x + 184669
abs() of this function gives 49 discting primes (discovered by
Jaroslaw Wroblewski & Jean-Charles Meyrignac)

BTW, Shyam Sunder Gupta found:
abs(1/4x5 - 133/4x4 + 6729/4x3 - 158379/4x2 + 860147/2x - 1705829)

which provides 57 primes !!!

1705829 1313701 991127 729173 519643 355049 228581 134077 65993 19373 10181 26539 33073 32687 27847 20611 12659 5323 383 3733 4259 1721 3923 12547 23887 37571 53149 70123 87977 106207 124351 142019 158923 174907 189977 204331 218389 232823 248587 266947 289511 318259 355573 404267 467617 549391 653879 785923 950947 1154987 1404721 1707499 2071373 2505127 3018307 3621251 4325119

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