Message: 7 Date: Sat, 2 Nov 2013 16:41:51 -0700 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] bizarre algebraic trivium
I am using Maple. Denote the polynomial in that Root by p, the given root by x0. Using Sturm sequences one sees, that p has exactly 2 real roots, both are positive and 1 < x0 is confirmed by the system (the other one is smaller than 1). Now substitute x = 1/X^8 (ok, a bit strange) and assuming 0 < X that becomes 0 = -X^(31/4)*sqrt(sqrt(X+1)+sqrt(X))+1 Maple finds those roots, 2 of them are Reals, one is -1 and the other one is X0:= RootOf(_Z^30-_Z^29+_Z^28-_Z^27+_Z^26-_Z^25+_Z^24-_Z^23+_Z^22-_Z^21+_Z^20 -_Z^19+_Z^18-_Z^17+_Z^16+_Z^15-_Z^14+_Z^13-_Z^12+_Z^11-_Z^10 +_Z^9-_Z^8+_Z^7-_Z^6+_Z^5-_Z^4+_Z^3-_Z^2+_Z-1, index = 1); Feeding ln(x)/ln(x^2-2) with x = (1/X0)^8 is simplified to -8/15 by Maple.