On Sat, Dec 20, 2008 at 12:56 PM, <rwg@sdf.lonestar.org> wrote:
On Fri, Dec 19, 2008 at 12:20 PM, <rwg@sdf.lonestar.org> wrote:
Amen. I've now tried ~ 10^8 cases of (a^(1/5)-b^(1/5))^(1/3) without further success, and for sqrt, only
3/5 4/5 3/5 2/5 2/5 4/5 1/5 1/5 1/5 1/5 - 2 3 + 3 + 2 2 3 - 2 3 + 2 sqrt(4 - 3 ) = ---------------------------------------------------. 5
And, of course, these can only give pentanomials. --rwg ALGORISMIC MICROGLIAS
I have no technical knowledge on denesting, but here is how I look at the problem:
Since sqrt( a^(1/5) + b^(1/5) ) = a^(-2/5) * sqrt( a + (a^4*b)^(1/5) ), I'll consider only the form: sqrt( a + b^(1/5) ). If sqrt( a + b^(1/5) ) could be denested, I'd expect it to be denested into the form:
x0 + x1*b^(1/5) + x2*b^(2/5) + x3*b^(3/5) + x4*b^(4/5).
Notice that in the (comparatively easy) 4th root case, your conjecture catches
1/4 3/4 1/4 sqrt(161 - 12 5 ) = 2 5 - 3 sqrt(5) + 4 5 + 6
but not 1/4 1/4 1/4 3/4 3/4 1/4 sqrt(2) sqrt(31 - 4 3 5 ) = 3 5 - 2 sqrt(5) + 3 5 + 2 sqrt(3).
That's why I don't think you could find anything longer than pentanomial. OTOH, sqrt( a^(1/7) + b^(1/7) ) is a better bet for finding hexanomials or longer.
Indeed, but the only one I've found so far is perversely pentanomial:
1/7 Sqrt(7 (2 - 2 )) 1/7 3/7 5/7 6/7 ------------------ = -1 + 2 2 + 2 + 2 - 2 . 1/14 2
Just my 2 cents.
Warut
You didn't propose 6th roots. Did you have reason to suspect that they are as infertile as they seem to be? --rwg ALGORISMIC MICROGLIAS
Oops, found this in my "notes": 1/3 5/6 1/6 5/6 1/6 1/6 2/3 1/6 5 5 2 sqrt(5) 2 5 2 sqrt(4 - sqrt(3) 5 ) = ------- - --------- + ------------ - --------- + ------- sqrt(6) 3 sqrt(2) 3 3 sqrt(3)
In fact, I didn't mean x0, x1, ... to be rational numbers. It could be sqrt(2) if it could disappear after squaring. Of course, for the fifth root case, sqrt(2) could appear in x0 iff it appears in x1, ..., x4, too.
I used to think that 7th roots would have more chance than 6th roots due to more variables (i.e., more flexible), but now I believe I was wrong and that pentanomial may be the limit.
It seems so incidental that one or two coeffs keep vanishing. It's like a carnival sucker game. It looks so solvable, but something is secretly forbidding it.
Later, Warut