On Tue, 13 Mar 2007 10:15:53 +0100 Thorkil Naur <naur@post11.tele.dk> wrote:
Hello,
On Tuesday 13 March 2007 00:40, Jacques Carette wrote:
And yet Taral would be wrong and Dave Feustel correct: http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
I wouldn't say that a polynomium applies as "using only addition and subtraction".
There is a polynomial (of degree 25) in 26 variables which generates only primes whenever it is positive. Surprising, yes it is. Note that this polynomial is actually rarely positive!
Yes, and as noted in the Prime-GeneratingPolynomial reference, it is really a set of diophantine equations (equations where only integer/natural/rational solutions qualify) in disguise. In fact, if I remember correctly, one of the variables that you have to stick into the polynomium is one less (or some such) than the prime that you get out. So as a prime generating device, it is rather useless.
The idea of defining sets of numbers by diophantine equations was used in 1970, to prove one of the 10th of Hilberts famous 23 problems from 1900 impossible. See
Incidentally, the proof is of a constructive nature, in that it shows how to encode any computable function as Diophantine equations. In particular, the output of the functions are encoded as solutions to the equations. Thus any computable has concrete corresponding equations. However, as has been noted, this does not actually imply the resulting equations are useful for computation. Gen