HP50g simplifing a root
10-05-2020, 05:01 PM
Post: #13
 Albert Chan Senior Member Posts: 2,705 Joined: Jul 2018
RE: HP50g simplifing a root
(10-05-2020 11:36 AM)peacecalc Wrote:  thank you for critics. Of course I forgot to implement the testing procedure being shure that I found a correct solution!
The checking is absolut necessary.

This checking is the reason my previous posts had different arguments requirement.

find_ar(n, m) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ // note, no k's

It returned simplified a + b√k, but the user *must* check √k matches.

find_cbrt(n, m, k) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ // we required k to build the cubic

It solved the cubic, checked a is integer, but assumed b is also integer.
Here is a proof that shows b is indeed an integer.

(09-30-2020 02:22 AM)Albert Chan Wrote:  For general case, to solve for all a, b:

﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ³√(n + m√k) = a + b√k

(a + b√k)³ = a³ + 3a²b√k + 3ab²k + b³k√k

n = a³ + 3ab²k ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → n/a - a² = 3b²k
m = 3a²b + b³k ﻿ ﻿ ﻿ ﻿ ﻿ → 3m/b - 9a² = 3b²k

Since a|n, n/a - a² = 3b²k = integer

Assume b ≠ integer, but b = c/3, where c = integer.

m = 3a²b + b³k = a²c + c³k/27 = integer

Assuming √k is fully reduced, k can not have factor of 27 = 3³

(c³k/27 = integer) ⇒ (3|c) ⇒ (b = c/3 = integer)

Assumption were wrong, b must be integer. ﻿ ﻿ ﻿ ﻿ ﻿ QED
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 Messages In This Thread HP50g simplifing a root - peacecalc - 09-29-2020, 09:22 PM RE: HP50g simplifing a root - Albert Chan - 09-29-2020, 11:47 PM RE: HP50g simplifing a root - Albert Chan - 09-30-2020, 02:22 AM RE: HP50g simplifing a root - Albert Chan - 09-30-2020, 10:50 PM RE: HP50g simplifing a root - Albert Chan - 10-01-2020, 07:31 AM RE: HP50g simplifing a root - peacecalc - 09-30-2020, 05:33 AM RE: HP50g simplifing a root - peacecalc - 10-01-2020, 02:20 PM RE: HP50g simplifing a root - Albert Chan - 10-01-2020, 05:22 PM RE: HP50g simplifing a root - peacecalc - 10-04-2020, 06:05 PM RE: HP50g simplifing a root - Albert Chan - 10-04-2020, 11:48 PM RE: HP50g simplifing a root - peacecalc - 10-04-2020, 07:36 PM RE: HP50g simplifing a root - peacecalc - 10-05-2020, 11:36 AM RE: HP50g simplifing a root - Albert Chan - 10-05-2020 05:01 PM RE: HP50g simplifing a root - peacecalc - 10-06-2020, 05:25 AM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 09:40 AM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 12:06 PM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 04:13 PM RE: HP50g simplifing a root - Albert Chan - 10-07-2020, 06:12 PM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 12:20 AM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 02:31 PM RE: HP50g simplifing a root - Albert Chan - 10-11-2020, 06:28 PM RE: HP50g simplifing a root - Albert Chan - 10-12-2020, 03:17 AM RE: HP50g simplifing a root - Albert Chan - 10-24-2020, 02:19 PM RE: HP50g simplifing a root - Albert Chan - 10-12-2020, 10:54 PM RE: HP50g simplifing a root - CMarangon - 10-12-2020, 11:45 PM RE: HP50g simplifing a root - grsbanks - 10-13-2020, 06:46 AM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 05:21 PM RE: HP50g simplifing a root - Albert Chan - 10-10-2020, 03:58 PM RE: HP50g simplifing a root - Albert Chan - 10-10-2020, 04:49 PM RE: HP50g simplifing a root - peacecalc - 10-12-2020, 08:49 PM RE: HP50g simplifing a root - peacecalc - 10-13-2020, 06:30 AM RE: HP50g simplifing a root - peacecalc - 10-13-2020, 06:36 AM

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