From: rgep@dpmms.cam.ac.uk (Richard Pinch)
Newsgroups: sci.math
Subject: Re: polynomials can't give only primes - sci.math #54025
Date: 6 Sep 1996 11:05:30 GMT

In article <50odl2$ecr@nuscc.nus.sg>, 
sci50090@leonis.nus.sg (VeLaGaMist) writes:
|> In anycase, it has been proved that there is indeed an integral 
|> polynomial that gives primes whenever it takes positive values.
|> 
|> I do not have an idea of the explicit form of the polynomial. Could 
|> anyone enlighten me on this?

Accoring to my notes, it is

(k+2){1-([wz+h+j-q]^2 + [(gk+2g+k+1)(h+j)+h-z]^2   +
         [16(k+1)^3  (k+2) (n+1)^2  +1-f^2]^2   +  [ 2n+p+q+z-e ]^2  +
         [ e^3 (e+2)(a+1)^2 + 1 - o^2]^2  +  [(a^2-1)y^2 + 1 - x^2]^2   +
         [16r^2 y^4 (a^2-1) + 1-u^2]^2   +
         [ ( (a+u^2 (u^2-a))^2  - 1 )  (n+4dy)^2  + 1 - (x+cu)^2]^2   +
         [(a^2-1)l^2  +  1 -  m^2]^2   +   [ai+k+1-l-i]^2   +   [n+l+v-y]^2   +
         [p+l(a-n-1)+b(2an+2a-n^2-2n-2)-m]^2   +
         [q+y(a-p-1)+s(2ap+2a-p^2-2p-2)-x]^2   +
         [z+pl(a-p)+t(2ap-p^2-1)-pm]^2  
        )  }

(the layout may help show why it is not of much practical use!).

See J.Am.Math.Soc., 6/1976, pp449-464.
-- 
Richard Pinch		Queens' College, Cambridge
rgep@cam.ac.uk		http://www.dpmms.cam.ac.uk/~rgep
==============================================================================
For a readable exposition of the development of these ideas see

   Math Reviews v. 54 #2615   10M05 (10B15 02F99)
   Jones, James P.; Sato, Daihachiro; Wada, Hideo; Wiens, Douglas
   Diophantine representation of the set of prime numbers. 
   Amer. Math. Monthly 83 (1976), no. 6, 449--464.

   Since Ju. V. Matijasevic showed that every recursively enumerable set
   is Diophantine, several authors have given polynomials that represent
   the prime numbers. This well-written paper contains two such
   polynomials: one of degree 25 in 26 variables, the other of high
   degree in just 12 variables. It also contains some general results on
   representation of sets of integers. Perhaps the most interesting
   result is: Any algebraic function with only integer values must be a
   polynomial.


   © Copyright American Mathematical Society 1994, 1998