Saturday, September 23, 2006
JSH: Answer with p-adics
Hey! A Gene Ward Smith made a post in reply to me where he put the following:
"Consider the polynomial x^2 - (3+t)x + 2. The two roots of this can be expanded in series as
r1(t) = 1 - t + 2t^2 - 6t^3 + 22t^4 - 90t^5 + …
r2(t) = 2 + 2t - 2t^2 + 6t^3 - 22t^4 + 90t^5 - …
If t is 2-adically a nonunit, so that |t|_2 < 1, then these converge in the 2-adic integers. One root, r2, is even, the other, r1, is odd. The 2 does not split up. If t is a rational integer, this means that when t is even, the roots split into an even root and an odd root 2-adically."
So I say let t equal a non-rational algebraic integer unit.
Chew on that one for a while, and let's see where this discussion goes.
Needed, (1)verification of Smith's claim. (2)Verification that a non-rational algebraic integer unit will be within the size range.
(3)Some goddamn acceptance from math people when precious ideas that DO NOT FREAKING WORK are proven to be wrong.
Show some damn balls—SOME courage.
Any? Any of you have an ounce of courage in your miserable selves? An ounce of courage for the truth?
"Consider the polynomial x^2 - (3+t)x + 2. The two roots of this can be expanded in series as
r1(t) = 1 - t + 2t^2 - 6t^3 + 22t^4 - 90t^5 + …
r2(t) = 2 + 2t - 2t^2 + 6t^3 - 22t^4 + 90t^5 - …
If t is 2-adically a nonunit, so that |t|_2 < 1, then these converge in the 2-adic integers. One root, r2, is even, the other, r1, is odd. The 2 does not split up. If t is a rational integer, this means that when t is even, the roots split into an even root and an odd root 2-adically."
So I say let t equal a non-rational algebraic integer unit.
Chew on that one for a while, and let's see where this discussion goes.
Needed, (1)verification of Smith's claim. (2)Verification that a non-rational algebraic integer unit will be within the size range.
(3)Some goddamn acceptance from math people when precious ideas that DO NOT FREAKING WORK are proven to be wrong.
Show some damn balls—SOME courage.
Any? Any of you have an ounce of courage in your miserable selves? An ounce of courage for the truth?
# posted by James Harris @ 17:50 
