HP50g simplifing a root

[Albert Chan]

(10-06-2020 12:06 PM)Albert Chan Wrote:  Simplify equations from find_cbrt(): 4a³ - A = (3a) ³√(A²-R)

This implied ³√(A²-R) must be integer !

There is a much more elegant way to get above relation !

³√(A ± √R) = a ± √r
³√(A +√R) ³√(A -√R) = (a + √r) (a - √r)
³√(A² - R) = a² - r = a² - (A/a - a*a)/3 = (4a² - A/a)/3

Because RHS is integer, LHS must too.

We have a|A, and 3|(4a² - A/a), which simplified to a≡A (mod 3)
I've updated my lua code now to check a in steps of 3

Update: Above assumed a is integer; it might be halves.
Including halves, we have 2a ≡ 2a*(A/a) ≡ 2A (mod 3)