(x-1)x(x+1) or x(x+1)(x+2) and their expansions are algebraic solutions to special products. All the solutions posted used one of these 2 forms. (your example uses the 2nd's expansion)
----- Original Message ----- From: Don Guinn <[email protected]> To: Programming forum <[email protected]> Cc: Sent: Saturday, August 8, 2015 9:53 PM Subject: Re: [Jprogramming] Detecting special products Using polynomials to solve this problem is really neat, but why does it work? I can't find anything on the internet explaining why it works. p._720 2 3 1 +-+----------------------------+ |1|_5.5j7.72981 _5.5j_7.72981 8| +-+----------------------------+ p.1;-8 9 10 720 242 27 1 Obviously the polynomial for the roots is quite different from that used above. There has to be a proof somewhere. On Fri, Aug 7, 2015 at 9:47 PM, Don Kelly <[email protected]> wrote: > correction : 'will be the <last> number of the sequence' > in any case for a special number the real root will be an integer. > > > On 06/08/2015 4:06 PM, Don Kelly wrote: > >> also try >> 0=1|( <1; 2) {>p.|.1 3 2,-]720 >> >> 1_ >> >> 0=1|( <1; 2) {>p.|.1 3 2,-]719 >> >> 0 >> >> note that >> (<1;2) {>p._y 2 3 1 >> will return an integer for a special number and this will be the first >> number of the sequence >> based on x*x+1)*(x+2) -y=0 giving an integral root >> p._720 2 3 1 >> >> ┌─┬────────────────────────────┐ >> >> │1│_5.5j7.72981 _5.5j_7.72981 8│ >> >> └─┴────────────────────────────┘ >> >> >> a modification to x(x^2-1) gives an integral real root for the middle root >> p. _720 _1 0 1 >> >> ┌─┬────────────────────────────┐ >> >> │1│9 _4.5j7.72981 _4.5j_7.72981│ >> >> └─┴────────────────────────────┘ >> >> puts the real root first rather than last >> >> As for finding a list of these special numbers-the "stope " function >> works well >> f=:3^!.1~[:i.] >> >> f 10 >> >> 0 6 24 60 120 210 336 504 720 990 >> >> >> Don Kelly >> >> >> >> >> >> On 06/08/2015 1:02 PM, Roger Hui wrote: >> >>> g=: = (^&3 - ])@:(3&(>.@%:)) >>> g */"1 (10^40 60 80x)+/0 1 2 >>> 1 1 1 >>> >>> >>> On Thu, Aug 6, 2015 at 11:23 AM, 'Pascal Jasmin' via Programming < >>> [email protected]> wrote: >>> >>> (] = (^&3 - ])@:(1r3 >.@^~ ])) 990 >>>> >>>> >>>> >>>> ----- Original Message ----- >>>> From: Kip Murray <[email protected]> >>>> To: "[email protected]" <[email protected]> >>>> Cc: >>>> Sent: Thursday, August 6, 2015 1:58 PM >>>> Subject: [Jprogramming] Detecting special products >>>> >>>> The number 720 is special: it is 8*9*10 the product of three >>>> successive non-negative integers. The first few special numbers are >>>> >>>> */"1 [ 0 1 2 +"1 0 i. 10 >>>> 0 6 24 60 120 210 336 504 720 990 >>>> >>>> Write a verb test that tests whether a non-negative integer is >>>> special: >>>> >>>> test 720 >>>> 1 >>>> test 0 >>>> 1 >>>> test 721 >>>> 0 >>>> >>>> --Kip Murray >>>> >>>> >>>> >>>> -- >>>> Sent from Gmail Mobile >>>> ---------------------------------------------------------------------- >>>> 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 >> > > ---------------------------------------------------------------------- > 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
