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March 11th, 2011, 11:45 AM   #1
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Conjecture: Decimal Representation of root

The Conjecture

- Treat Each digit as a coordinate in a vector
- Consider the space of numbers of the form w=q1*r1^(1/N1)++ qv*rv^(1/Nv)
Where q is rational, N is an integer and no factors of r have an Nth root.
-denote the digits of w as w0w1wm
And the inner product between two number w and x as:
Lim (m->inf)(1/(2m))sum_1..m (wi(xi-wi)+(wi-xi)wi)

Then if r1..rv and N1..Nv are known then qi is the projection of w on ri^(1/Ni)

The rational

Since irational numbers are random one expects their components to be orthogonal. I'm sure their are simple tests one could try to verify this. I will think of some later.
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March 11th, 2011, 12:54 PM   #2
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Re: Conjecture: Decimal Representation of root

wi isn't well-defined; consider q1 = q2 = 1, r1 = 2, r2 = -2, N1 = N2 = 3.

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March 14th, 2011, 11:03 AM   #3
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Re: Conjecture: Decimal Representation of root

Quote:
Originally Posted by CRGreathouse
wi isn't well-defined; consider q1 = q2 = 1, r1 = 2, r2 = -2, N1 = N2 = 3.

opps. My miskate. r1 and r2 should be whole numbers. Anyway, I'll try to test it soon but I wish to make another post first.
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