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Re: polynomials can't give only primes  sci.math #54025
Posted:
Sep 6, 1996 7:55 PM


In article <50pmfp$aa2@beyla.ifi.uio.no>, tordro@ifi.uio.no (Tord Kallqvist Romstad) writes: > Richard Pinch (rgep@dpmms.cam.ac.uk) wrote: > : In article <50odl2$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+jq]^2 + [(gk+2g+k+1)(h+j)+hz]^2 + > : [16(k+1)^3 (k+2) (n+1)^2 +1f^2]^2 + [ 2n+p+q+ze ]^2 + > : [ e^3 (e+2)(a+1)^2 + 1  o^2]^2 + [(a^21)y^2 + 1  x^2]^2 + > : [16r^2 y^4 (a^21) + 1u^2]^2 + > : [ ( (a+u^2 (u^2a))^2  1 ) (n+4dy)^2 + 1  (x+cu)^2]^2 + > : [(a^21)l^2 + 1  m^2]^2 + [ai+k+1li]^2 + [n+l+vy]^2 + > : [p+l(an1)+b(2an+2an^22n2)m]^2 + > : [q+y(ap1)+s(2ap+2ap^22p2)x]^2 + > : [z+pl(ap)+t(2app^21)pm]^2 > : ) } > > : (the layout may help show why it is not of much practical use!). > > There is one more reason it is not very practical: > Most of the time, the formula just give you the primes 2 and 3!
No. Most of the time it gives nonpositive values. Note that the polynomial is of the form (k+2){ 1  (sum of squares of quantities in square brackets) } so the value will be either k+2 (if all the quantities in square brackets are 0) or something <= 0. And, as it turns out, the only way all the quantities in square brackets can be 0 is if k+2 is prime.
Robert Israel israel@math.ubc.ca Department of Mathematics (604) 8223629 University of British Columbia fax 8226074 Vancouver, BC, Canada V6T 1Y4



