Problems & Puzzles:
44 . Twin-primes producing
For sure you already
know about the somewhat popular subject: prime-producing-polynomial-races
(see 1 &
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:
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 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
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?
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.