Problems & Puzzles:
Problems
Problem
44 . Twinprimes producing
polynomials race
For sure you already
know about the somewhat popular subject: primeproducingpolynomialraces
(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:
"n1 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
_____
^{(*)}
The Beiler's claim is only true with a double little help:
a) to take the absolute values of n1 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
twinprimes 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^267725x+603900
and f(x) = 12483x^2187245x+834960.
Both are better than the Beiler's
polynomial by two successive twin primes.
Is the Beiler's
falseclaim (k=20) still affordable?
***
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.
***
