A drag racer Drag Racing Prime Numbers!

What Is Drag Racing?

Drag racing prime numbers is all about finding a sequence of numbers, related by a simple equation, which has an unusually high prime density. You race them by finding several sequences, and either finding the one of them produces the largest number of primes in the smallest number of terms.

There are two ways of running the race:

  1. Fix the number of terms in the sequence - the winner is the sequence which produces the most primes.
  2. Fix the number of primes required - the winner is the sequence which produces that number of primes in the fewest terms.

What Sequences Can You Race?

The two most common forms to race are the simple polynomial, and the exponential form.

Simple polynomials would include arithmetic progressions, the so-called Euler Trinomials fA(x)=x^2+x+A for constant A, or arbitrary quadratics f(x)=A*x^2+B*x+C.

Exponential forms would include Proth forms fk(n)=k*2^n+1 and Riesel forms fk(x)=k*2^n-1, where k is fixed in both cases.

However, you're limited by your imagination, and even recently people have discovered new interesting, and more importantly challenging, forms to race, or variations on previous forms (in particular look out for the Proth/Riesel Dual forms, invented by Payam Samidoost, in the following "proth racing" pages.)

Famous Historic Races

Possibly the most famous prime drag race was that of Euler's quadratic polynomials f(x)=x^2+x+A, for a constant A. Euler noted that for A=2,3,5,11,17,41 the polynomial took prime values for every x in 0..(A-2).

A f(x) for x in 0..(A-2)
2 2
3 3, 5
5 5, 7, 11, 17
11 11, 13, 17, 23, 31, 41, 53, 67, 83, 101
17 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257
41 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601
Record prime-yielding Euler Trinomials

It is a mathematically proven fact that no Euler trinomial can have a longer constant run of primes than A=41 [Ribenboim 1995], and therefore it will win any drag race up to 41 primes. People are still racing these and other forms.

Another famous pair of races, in the opposite direction - attempting to find as few primes as possible, are still being fought. These are the races to find Sierpinski and Riesel numbers. Actual Sierpinski and Riesel numbers form expenential Proth and Riesel series that have no primes at all on them. Some such series are provably always composite, but racing these others is like a holding-your-breath competition. It's actually a great relief to finally find a prime (these can be very large primes, some of the largest in the world), but once you've found one that series is out of the competition. Currently "Seventeen or Bust" project is racing all by itself on all of the remaining Sierpinski candidates (there were 17 when it started, hence the name, but now are only a dozen), but many individuals are still underwater looking for (or avoiding) primes in the Riesel numbers sequences.

Another curious racing form was that of the Fermat Numbers 2^2^n+1. Fermat conjectured that these might always be prime, but it falls far short of that target. This sequence is prime for n=0,1,2,3,4, but for no other known term. When generalised to fb(n)=b^2^n+1 these Generalised Fermat Numbers (GFNs) again provide an infinite set of functions to race against each other.

Let's Race!

Another hastily constructed page by Phil Carmody
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