Since all primes can be generated, more than a few combinations result in primes, but it seems to be a (small?) minority.
No, the book contains no proof. Further information can be found on http://primes.utm.edu/glossary/page.php?sort=matijasevicpoly >From that page: Look at the special form of the second part: it is one minus a sum of squares, so the only way for it to be positive is for each of the squared terms to be zero (this is a trick of Putnam's). R.E. Boss -----Oorspronkelijk bericht----- Van: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] Namens Markus Schmidt-Gröttrup Verzonden: zaterdag 15 november 2008 18:07 Aan: Programming forum Onderwerp: Re: [Jprogramming] Minimal prime generators This is really crazy, does the book contain a prove for this? How many pages are needed for the prove? Or is it a trick such that there are only a few combinations can result in a positive (and then prime) number? Greetings, Markus R.E. Boss schrieb: > Perhaps not similar but also remarkable is the following. > On page 200 in 'Music of the primes' by Marcus du Sautoy, there is the > formula (in classic mathematical notation) > > (K+2)* > {1 > - [WZ+H+J-Q]^2 > - [(GK+2G+K+1)(H+J)+H-Z]^2 > - [2N+P+Q+Z-E]^2 > - [16(K+1)^3(K+2)(N+1)^2+1-F^2]^2 > - [E^3(E+2)(A+1)^2+1-O2]^2 > - [(A^2-1)Y^2+1-X^2]^2 > - [16R^2Y^4(A^2-1)+1-U^2]^2 > - [((A+U^2(U^2-A))^2-1)(N+4DY)^2+1-(X+CU)^2]^2 > - [N+L+V-Y]^2 > - [(A^2-1)L^2+1-M^2]^2 > - [AI+K+1-L-I]^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 > } > > which produces a prime for every integer choice of A to Z, unless the > outcome is negative. Moreover, for every prime there is such a choice. > > > R.E. Boss > > > > -----Oorspronkelijk bericht----- > Van: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED] Namens John Randall > Verzonden: vrijdag 14 november 2008 18:24 > Aan: Programming forum > Onderwerp: [Jprogramming] Minimal prime generators > > At the opposite end of the spectrum from industrial strength methods > such as Miller-Rabin are generators which produce primes with minimal > (and apparently insufficient) resources. > > Here are two of my favorites. > > Rowlands' generator > > This is a simple recurrence relation, starting with 7, whose first > differences are either 1 or prime. > > f=:,{: + >:@# +. {: > d=:2 -~ /\ ] > > 1 -.~ d f^:1000 ] 7 > 5 3 11 3 23 3 47 3 5 3 101 3 7 11 3 13 233 3 467 3 5 3 941 3 7 > > Conway's Fourteen Fruitful Fractions > > This is more mysterious. We take a list of fourteen magic fractions. > At each step we have an integer, starting with 2. We multiply the > integer by each of the fractions from left to right until we get > another integer. This is the next iterate. When an iterate is a > power of 2, the exponent is the next prime. > > > num=:17 78 19 23 29 77 95 77 1 11 13 15 15 55x > den=:91 85 51 38 33 29 23 19 17 13 11 14 2 1x > > fff=:num%den NB. The fractions > > h=:{.@:(#~ (=<.))@:(fff&*) > > k=:(#~ (=<.))@:(2&^.) > > k h^:(i.1e4) 2 > 1 2 3 5 7 11 13 17 > > > Does anyone have any similar examples? > > Best wishes, > > John > > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
