Problems & Puzzles: Puzzles

Puzzle 597. A follow up to Puzzle 596

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.


Emmanuel wrote:

My 'best' polynomial for the  500-race is : x^2 + x + 21377  with 100% 'hits'.
(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).


Luis wrote:

My results for x^2+x+A are:
A                  Hits in x < =500         Hits in  x< = 1000        Hits in x < = 10000
19421                  499                             984                          9025
21377                  500                             992                          9141        (C. Rivera) 
27941                  499                             993                          9234
72491                  498                             994                          9306
247757                498                             991                          9566
601037                492                             988                          9323


Hakan wrote:

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



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