Problems & Puzzles: Problems

Problem 12.- Prime producing polynomials

Ruby found that the absolute value of (36X^2-810X +2753) assumes 45 distinct prime values for the 45 integers X, 0<=X<=44.

This the current record for this kind of polynomials (aX^2 + b.X +c)

Can you get a better polynomial than this?

Notes :

1. Williams noted that the polynomial 36X^2-2358X+36809 produces the same distinct prime values than the Ruby’s polynomial, but appearing in reverse order
2. Ruby and Fung have the second record polynomial, 47X^2 -1701X+10181 which produces 43 distinct prime values for 0<=X<=42

(Ref.1, P. 203)

On November, 2005 Seiji Tomita wrote:

I looked for quadratic Prime-producing polynomials and found the following polynomials.

a,b,c are integers
a>0
-5000<b<0
0<c<100000
f(x)=a*x^2+b*x+c
0<=x<50
d:quantity of distinct primes
d>40

Group 36: Polynomials that coefficient of x^2 is 36

Group 47: Polynomials that coefficient of x^2 is 47

No a b c d
1 36 -522 89 41
2 36 -594 647 42
3 36 -666 1277 43
4 36 -738 1979 44
5 36 -810 2753 45
6 36 -2070 27953 41
7 36 -2142 30059 42
8 36 -2214 32237 43
9 36 -2286 34487 44
10 36 -2358 36809 45

1 47 -1513 6967 41
2 47 -1607 8527 42
3 47 -1701 10181 43
4 47 -2059 17341 41
5 47 -2153 19447 42
6 47 -2247 21647 43

I found some interesting relations between these polynomials.

(1). These polynomials have same Discriminant.
Discriminant=522^2-4*36*89=259668

(2). These polynomials are linked each other.
Let g(x)=36*(x-h)^2-522*(x-h)+89

.Let h=1 then g(x)=36*x^2 -594*x+ 647..........No.2
.Let h=2 then g(x)=36*x^2 -666*x+ 1277..........No.3
.Let h=3 then g(x)=36*x^2 -738*x+ 1979..........No.4
.Let h=4 then g(x)=36*x^2 -810*x+ 2753..........No.5
.Let h=43/2 then g(x)=36*x^2-2070*x+ 27953..........No.6
.Let h=45/2 then g(x)=36*x^2-2142*x+ 30059..........No.7
.Let h=47/2 then g(x)=36*x^2-2214*x+ 32237..........No.8
.Let h=49/2 then g(x)=36*x^2-2286*x+ 34487..........No.9
.Let h=51/2 then g(x)=36*x^2-2358*x+ 36809..........No.10 2. About Group 47

(1). These polynomials have same Discriminant.
Discriminant=1513^2-4*47*6967=979373

(2). These polynomials are linked each other.
Let g(x)=47*(x-h)^2-1513*(x-h)+6967

.Let h=1 then g(x)=47*x^2-1607*x+ 8527..........No.2
.Let h=2 then g(x)=47*x^2-1701*x+10181..........No.3
.Let h=273/47 then g(x)=47*x^2-2059*x+17341..........No.4
.Let h=320/47 then g(x)=47*x^2-2153*x+19447..........No.5
.Let h=367/47 then g(x)=47*x^2-2247*x+21647..........No.6

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