Observations: This is equivalent to solving the equation x=x^2 mod m, or the equation 0=(x^2)-x mod m , or the equation 0=(x^2)+(-x)+(k*m) where k is an integer. The last equation has roots 2 %~ 1 (+,-) %: 1-4*k*m . We want integer roots, so the problem is to finding positive integers c such that %:1+4*c*m is an odd integer.
For m=: 10^10x, c=:319375992x fits the bill, and the roots are: m=: 10^10x c=: 319375992x 2 %~ 1 (+,-) %:1+4*c*m 1787109376 _1787109375 2 %~ 1 (+,-) %:1+4*c*m 1787109376 _1787109375 m | 2 %~ 1 (+,-) %:1+4*c*m 1787109376 8212890625 x m&|@^ 1 2 1787109376 1787109376 y m&|@^ 1 2 8212890625 8212890625 ----- Original Message ----- From: "Mike Day" <[EMAIL PROTECTED]> To: "Programming forum" <[email protected]> Sent: Monday, February 27, 2006 2:18 AM Subject: Re: [Jprogramming] If Maple can, why can't J? Incidentally (or is it?) I've just noticed that not only, as Roger said: 2 (10^10x)&|@^ 10^100x 1787109376 ,but also (discovered while wondering about factoring powers): 2 (10^10x)&|@^ 10^10x 1787109376 which led me to observe this! : 1787109376 (10^10x)&|@^ 1 2x 1787109376 1787109376 and so : 1787109376 (10^10x)&|@^ i.10x 1 1787109376 1787109376 1787109376 1787109376 1787109376 1787109376 1787109376 1787109376 1787109376 I suppose there are many such numbers but it's a bit surprising that this arises for a power of two - or is it? All truncated versions of this number have similar properties, eg 376, and I've found one single digit LH extension, 81787109376x(10^11x)&|@^1 2x 81787109376 81787109376 ... ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
