Problems & Puzzles:
A follow up to Puzzle
Let's now start two new races. Let's ask for the
best polynomial to produce the best count of primes or semiprimes on
two distinct large tracks.
Let f(x)=Ax^2+Bx+C to be a quadratic
polynomial, and let's count the quantity of primes or semiprimes,
Cps, for x=0 to N
Q1. The small race: Find the
best f(x) for N=500
Q2. The long race: Find the best f(x) for N=10,000.
Please just report your best polynomial
f(x) and its Cps for each race,
only one for each race.
If you want a good starting point to beat, I
will tell you that a good candidate for champion of this race is:
f(x)=x^2+x+247757, because for this polynomial Cps(500)=498 &
Cps(10,000)=9,566. Good luck!
Contributions came from Emanuel Vaniteghem, Luis Rodríguez,
Hakan Summakoğlu & Carlos Rivera.
My 'best' polynomial for the 500-race is : x^2 + x + 21377 with 100%
(for the 1000-race it is x^2+x+72491 with 995 hits, but that was not
a question, sorry)
For the 10000 race, the best is x^2+x+247757, with 9567 hits.
I think this last polynomial remains the best 'for a very long race'
(I took several samples with race lengths of 10^6).
Q1: f(x)=2x^2+2x+67387, Cps(500)=500
Q2: f(x)=2x^2+2x+67387, Cps(10,000)=9,433
Carlos Rivera wrote:
First of all, I let aside the original questions of this puzzle (Q1 &Q2).
Instead of these, I made a search for the polynomials
f(x)=Ax^2+Bx+C producing the largest run of successive primes
or semiprimes from x=x1 to N. My best results were:
f(x) = x^2+x+21377 is prime or semi prime for all x=0 to 535
f(x) = 2.x^2+2.x+53089 is prime or semi prime for all x=0 to 597
f(x) = 2.x^2-94.x+54193 is prime or semi prime for all x=0 to 620
f(x) = 2.x^2-98.x+54289 is prime or semi prime for all x=1 to 622