Problems & Puzzles: Problems

Problem 44 . Twin-primes producing polynomials race

For sure you already know about the somewhat popular subject: prime-producing-polynomial-races (see 1 & 2), where you are asked to find better and better polynomials (usually quadratic ones) f(x) producing primes for all the x integer values in a certain range (usually from 0 to k)

What about if we ask the same but in order to produce twin- primes?

This little change was suggested to me by the following phrase, found in the page 225, of the "Recreations in the Theory of Numbers", by A. H. Beiler:

"n-1 and n+1 are both primes for ... n=30(2x - 27)(x - 15) with [x integer] values from 1 to 20"

Evidently this claim is false for x=14 and x=15; but it's a good starting point(*) for our puzzle.

Let's pose formally the target of this problem:

Find better quadratic polynomials than the Bailer's' one, f(x) such that f(x)-1 & f(x)+1 are both primes for x=0 to k

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(*) The Beiler's claim is only true with a double little help: a) to take the absolute values of n-1 and n+1; b) consider that 1 is prime. But with this kind of permission then I would claim that 15x2 -375x +2310 is a better polynomial than the Beiler's one, because produces twin-primes for x=0 to 25. This can be a second race for twins of course... J. K. Andersen reported:

Two solutions for k=15: f(x) = 4515x^2-67725x+603900 and f(x) = 12483x^2-187245x+834960.

Both are better than the Beiler's polynomial by two successive twin primes.

Is the Beiler's false-claim (k=20) still affordable?

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Shyam Sunder Gupta reports:

The computational time can be reduced using the fact that if f(x)=a*X^2+b*X+c then for f(x)+1 and f(x)-1 both to be prime Mod(f(x),10) must be 0,2 or 8. Based on this , combinations of possible values of a, b ,c can be found and tested for primality.

As an historical note he adds:

The origin of this idea I found from the book " Mathematical Diversions" by Madachy and Hunter on page 7. This gives the same polynomial you mentioned from Beiler and is said to be discovered by A. T. Gazsi.

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