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  • Factorize: 2016 and ( 81 ^ 3 + 4 )

    From henhanna@gmail.com@21:1/5 to All on Wed Jun 29 15:23:40 2022
    1. Factorize: 2016

    2. Factorize: ( 81 ^ 3 + 4 )


    pls wait a few days before posting answers or hints.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From henhanna@gmail.com@21:1/5 to Edward Murphy on Wed Jun 29 21:04:02 2022
    On Wednesday, June 29, 2022 at 8:44:19 PM UTC-7, Edward Murphy wrote:
    On 6/29/2022 3:23 PM, henh...@gmail.com wrote:

    1. Factorize: 2016

    2. Factorize: ( 81 ^ 3 + 4 )


    is the second one SO routine ????



    for the 1st one, i meant to write :

    1b. Factorize: 2009




    pls wait a few days before posting answers or hints.
    How is this not just routine arithmetic? Do you have some
    particular shortcut in mind?

    rot13(
    1. gjb rkc svir gvzrf guerr rkc gjb gvzrf frira
    2. svir rkc bar uhaqerq svsgl frira rkc fvk uhaqerq friragl frira
    )

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Edward Murphy@21:1/5 to henh...@gmail.com on Wed Jun 29 20:44:16 2022
    On 6/29/2022 3:23 PM, henh...@gmail.com wrote:

    1. Factorize: 2016

    2. Factorize: ( 81 ^ 3 + 4 )


    pls wait a few days before posting answers or hints.

    How is this not just routine arithmetic? Do you have some
    particular shortcut in mind?

    rot13(
    1. gjb rkc svir gvzrf guerr rkc gjb gvzrf frira
    2. svir rkc bar uhaqerq svsgl frira rkc fvk uhaqerq friragl frira
    )

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From henhanna@gmail.com@21:1/5 to Edward Murphy on Sun Jul 3 13:31:44 2022
    On Sunday, July 3, 2022 at 1:16:50 PM UTC-7, Edward Murphy wrote:
    On 6/29/2022 9:04 PM, henh...@gmail.com wrote:

    On Wednesday, June 29, 2022 at 8:44:19 PM UTC-7, Edward Murphy wrote:
    On 6/29/2022 3:23 PM, henh...@gmail.com wrote:

    1. Factorize: 2016

    2. Factorize: ( 81 ^ 3 + 4 )


    is the second one SO routine ????


    It works out to 531445, which obviously has a factor of 5. That leaves 106289, whose factors (157 and 677) are less obvious.


    thakns... the formula i had in mind was ...

    The [ Sophie Germain Identity ]

    a^4 + 4 b^4 = (a^2 + 2b^2 + 2ab) (a^2 + 2b^2 - 2ab)

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Edward Murphy@21:1/5 to henh...@gmail.com on Sun Jul 3 13:16:43 2022
    On 6/29/2022 9:04 PM, henh...@gmail.com wrote:

    On Wednesday, June 29, 2022 at 8:44:19 PM UTC-7, Edward Murphy wrote:
    On 6/29/2022 3:23 PM, henh...@gmail.com wrote:

    1. Factorize: 2016

    2. Factorize: ( 81 ^ 3 + 4 )


    is the second one SO routine ????

    It works out to 531445, which obviously has a factor of 5. That leaves
    106289, whose factors (157 and 677) are less obvious.

    In light of factorizing 2009, we can also try to express 531445 as a
    difference of squares:

    81 = 3^4, s 81^3 = 3^12 = (3^6)^2 = 729^2. The number in question
    equals:

    * 729^2 + 4 (given), so for a difference of squares, we need to
    start with the square of an integer > 729.

    * 730^2 - 1455 (not a square)

    * 731^2 - 2916
    = 731^2 - 54^2
    = (731 + 54) * (731 - 54)
    = 785 * 677

    From there, by inspection 785 has a factor of 5, and then a bit of
    trial and error confirms that 5 * 157 * 677 can't be broken down
    any further.

    for the 1st one, i meant to write :

    1b. Factorize: 2009

    Trial and error quickly leads to 7 * 7 * 41

    If you notice that it's close to 2025 = 45^2, and then notice that the difference is 16 = 4^2, then you can do
    2009 = 45^2 - 4^2
    = (45 + 4) * (45 - 4)
    = 49 * 41

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    * Origin: fsxNet Usenet Gateway (21:1/5)