20030327, 13:47  #1 
Mar 2003
121_{8} Posts 
Smooth polynomial formulas to produce all primes
I have just downloaded the polynomial formula
to produce all primes. But I cannot compel it to work. What's the matter? f(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z)= (k + 2)*(1  (w*z + h + j  q)^2  (2*n + p + q + z  e)^2  (a^2*y^2  y^2 + 1  x^2)^2  ((e^4 + 2*e^3)*(a + 1)^2  o^2)^2  (16*(k + 1)^3*(k + 2)*(n + 1)^2 + 1  f^2)^2  (((a + u^4  u^2*a)^2  1)*(n + 4*d*y)^2 + 1  (x + c*u)^2)^2  (a*i + k + 1  l  i)^2  ((g*k + 2*g + k + 1)*(h + j) + h  z)^2  (16*r^2*y^4*(a^2  1) + 1  u^2)^2  (p  m + l*(a  n  1) + b*(2*a*n + 2*a  n^2  2*n  2))^2  (z  p*m + p*l*a  p^2*l + t*(2*a*p  p^2  1))^2  (q  x + y*(a  p  1) + s*(2*a*p + 2*a  p^2  2*p  2))^2  (a^2*l^2  l^2 + 1  m^2)^2  (n + l + v  y)^2) 
20030327, 15:36  #2 
Aug 2002
3×83 Posts 
Maybe you should talk to the person you got this "formula" from?
I'd note that trying to produce all the primes without testing individual numbers is something people have been trying to do for centuries, without success. I hope you forgive us for being slightly pessimistic. 
20030327, 15:43  #3 
"Mike"
Aug 2002
2×19×199 Posts 
I sure wish this forum had a formula editor...
Maybe I'll toss that into Maple 7 and see what we get... http://www.teamprimerib.com/gif/equation.gif Hmm... I told Maple to "solve" it and I got a really long weirdlooking answer... 
20030327, 16:28  #4 
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
Spacing down a few lines ...
.
. Spacing down a few lines to get out from under that last blurb ... there, that's better. 
20030327, 16:32  #5 
Dec 2002
Frederick County, MD
562_{8} Posts 
What a coincidence!
Gee, I wish I had maple :( , that would be cool.
A polynomial function of 26 variables, which by amazing coincidence, is also the number of letters in the English language! Very convenient, we don't need to go to Greek letters for the function! I guess this is believable, since now we have physicists becoming experts in number theory because they have studied a limited sequence of primes :? . http://www.nature.com/nsu/030317/03031713.html 
20030327, 16:44  #6  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}·3·641 Posts 
Re: Smooth polynomial formulas to produce all primes
Quote:
f(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z)= Quote:
It turns out that whenever the formula produces a positive value, that value is a prime. Warwick Harvey wrote, Quote:
Quote:
Quote:
Quote:


20030327, 17:52  #7  
Dec 2002
Frederick County, MD
562_{8} Posts 
Re: Smooth polynomial formulas to produce all primes
Quote:
In the first equation, the fifth term is ((e^4 + 2*e^3)*(a + 1)^2  o^2)^2 The sixth term in the second equation is the one that should be equivalent as far as I can tell, but that term is [e³(e+2)(a+1)²+1o²]² Of course, there is a e³ factored out in the second term, but that's not the problem. The problem is the +1 in the second term that is not in the first term in any form. But other than that, the equations are equivalent. The +1 is probably from one of the equations not being copied properly. Anyway, thanks for the clarification of the function Cheesehead, and sorry if I messed up your blurb spacing ;) . 

20030328, 03:53  #8 
Sep 2002
2^{2}×3×5 Posts 
Is there a name for this function? Sorry if I missed it in a previous post.
And are there any internet sites where I can find it? asdf 
20030328, 11:35  #9 
Mar 2003
3^{4} Posts 
I see !!!
That's not a formula at all !!! That's a great joke and kinda sieve. Almost time it's negative, but if not, it's equal to k+2 and other factor is 1. As for me, I've tried a loop from 1 to 1000000 with all vars being random from 1 till 1000000, but this stupid polynomial was always negative ! :D 
20030328, 17:28  #10 
∂^{2}ω=0
Sep 2002
República de California
3^{3}·419 Posts 
Hans Riesel's book, "Prime Numbers and Computer Methods for Factorization"
has a nice discussion of these kinds of "primeproducing" polynomials. One of the earliest of these is Euler's famous x^2x+41, which produces primes for x=0,1,2,...,40. This polynomial clearly cannot produce a prime for x=41, since in that case the x+41 part is zero, leaving a perfect square. There is a more general proven theorem along these lines which applies to (nonconstant) polynomials in arbitrarily many variables, namely that NO POLYNOMIAL CAN PRODUCE ONLY PRIMES. However, it has also been proven that there do exist (nonpolynomial) formulae which DO produce only primes. For example, Mills showed in 1947 that there exists a real number theta in (1,2), such that for every positive integer n, the number floor(theta^3^n) is prime. (Cf. the book by Crandall and Pomerance for more details and references). However, none of the known formulae of this type is practical for actually finding primes, in particular recordsize primes. 
20030329, 07:08  #11  
Mar 2003
3^{4} Posts 
Quote:


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