HP50g simplifing a root

[Albert Chan]

(10-04-2020 06:05 PM)peacecalc Wrote:  I tried two way, the first was to probe every small a from 1 until third root of A. From your example a has to tested from 1 to 670, what a luck, that a = 99. My second approach is to calculate every divisor of A (with DIVI) as a possible candidate for small a. Then I'm calculate the difference A - third power of a. Negative results can be sorted out. The rest is divided by 3*k*a and be square rooted. If the result is a integer and not a algebraic expression would be a solution for smal b. and we get:

\[ \left( A\pm B\cdot\sqrt{k} \right)^{1/3} = a \pm b\cdot\sqrt{k} \]

Based on above description ... not really.

You are solving ³√(A ± B√k) = a ± √r

r = b²k = (A-a³)/(3a)       // note: no B's !

(I might be wrong, your code may have used B somewhere)

Here is my lua code, that checked b actually get to B
see https://www.hpmuseum.org/forum/thread-15...#pid136957

Code:

cbrt = require'mathx'.cbrt

function simp_cbrt1(A, B, k)         -- simplify cbrt(A + B * sqrt(k))
    local N = abs(A)
    for a = 1+(N-1)%3, cbrt(N), 3 do -- a mod 3 = A mod 3
        if N % a ~= 0 then continue end
        local r = (N/a - a*a)/3
        if r % k ~= 0 then continue end
        local b = B / (3*a*a+r)
        if r == b*b*k then return (A<0 and -a or a), b, k end
    end
end

lua> A, B, k = 1859814842094, -59687820010, 415
lua> simp_cbrt1(A,B,k)
11589      -145      415
lua> simp_cbrt1(A,B+1,k)      -- nothing returned
lua>

Update: added a ≡ A (mod 3), see https://www.hpmuseum.org/forum/thread-15...#pid137252
Update2: fixed a bug if A is negative
Update3: inspired by simp_cbrt2(), calculate b without square root (i.e. not from r = b²k)