1. Forum
  2. Usenet
  3. REC.PUZZLES

  • ( x ^ 2 + y ^ 2 + z ^ 2 = xyz ) for positive integers (x, y, z)

    From henhanna@gmail.com@21:1/5 to All on Sun Jun 26 18:13:51 2022
    x ^ 2 + y ^ 2 + z ^ 2 = xyz

    1. Show some positive integers (x, y, z) that satisfy this equation

    2. How many positive-integer triples (x, y, z) satisfy this equation ?




    pls wait a few days before posting answers or hints.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Ilan Mayer@21:1/5 to henh...@gmail.com on Thu Jun 30 05:34:42 2022
    On Sunday, June 26, 2022 at 9:13:53 PM UTC-4, henh...@gmail.com wrote:
    x ^ 2 + y ^ 2 + z ^ 2 = xyz

    1. Show some positive integers (x, y, z) that satisfy this equation

    2. How many positive-integer triples (x, y, z) satisfy this equation ?




    pls wait a few days before posting answers or hints.
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .

    Some solutions:
    3 3 3
    3 3 6
    3 6 15
    3 15 39
    3 39 102
    3 102 267
    3 267 699
    6 15 87
    6 87 507
    15 39 582

    There is an infinite number of solutions.
    If a,b,c is a solution (a^2+b^2+c^2 = a*b*c), then a,c,a*c-b is also a solution Proof: a^2+c^2+(a*c-b)^2 = a^2+c^2+a^2*c^2+b^2-2*a*b*c = a^2*c^2-a*b*c = a*c*(a*c-b)

    Please reply to ilanlmayer at gmail dot com

    __/\__
    \ /
    __/\\ //\__ Ilan Mayer
    \ /
    /__ __\ Toronto, Canada
    /__ __\
    ||

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From henhanna@gmail.com@21:1/5 to Ilan Mayer on Thu Jun 30 14:14:15 2022
    On Thursday, June 30, 2022 at 5:34:45 AM UTC-7, Ilan Mayer wrote:
    On Sunday, June 26, 2022 at 9:13:53 PM UTC-4, henh...@gmail.com wrote:
    x ^ 2 + y ^ 2 + z ^ 2 = xyz

    1. Show some positive integers (x, y, z) that satisfy this equation

    2. How many positive-integer triples (x, y, z) satisfy this equation ?




    pls wait a few days before posting answers or hints.
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .

    Some solutions:
    3 3 3
    3 3 6
    3 6 15
    3 15 39
    3 39 102
    3 102 267
    3 267 699
    6 15 87
    6 87 507
    15 39 582

    There is an infinite number of solutions.
    If a,b,c is a solution (a^2+b^2+c^2 = a*b*c), then a,c,a*c-b is also a solution
    Proof: a^2+c^2+(a*c-b)^2 = a^2+c^2+a^2*c^2+b^2-2*a*b*c = a^2*c^2-a*b*c = a*c*(a*c-b)


    all multiples of 3 ... but never 9 ? or 12 ?

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Ilan Mayer@21:1/5 to henh...@gmail.com on Fri Jul 1 06:40:08 2022
    On Thursday, June 30, 2022 at 5:14:18 PM UTC-4, henh...@gmail.com wrote:
    On Thursday, June 30, 2022 at 5:34:45 AM UTC-7, Ilan Mayer wrote:
    On Sunday, June 26, 2022 at 9:13:53 PM UTC-4, henh...@gmail.com wrote:
    x ^ 2 + y ^ 2 + z ^ 2 = xyz

    1. Show some positive integers (x, y, z) that satisfy this equation

    2. How many positive-integer triples (x, y, z) satisfy this equation ?




    pls wait a few days before posting answers or hints.
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .
    .

    Some solutions:
    3 3 3
    3 3 6
    3 6 15
    3 15 39
    3 39 102
    3 102 267
    3 267 699
    6 15 87
    6 87 507
    15 39 582

    There is an infinite number of solutions.
    If a,b,c is a solution (a^2+b^2+c^2 = a*b*c), then a,c,a*c-b is also a solution
    Proof: a^2+c^2+(a*c-b)^2 = a^2+c^2+a^2*c^2+b^2-2*a*b*c = a^2*c^2-a*b*c = a*c*(a*c-b)

    all multiples of 3 ... but never 9 ? or 12 ?

    As mentioned above, if a,b,c is a solution (a^2+b^2+c^2 = a*b*c), then a,c,a*c-b is also a solution

    In the above solutions all numbers are divisible by 3, but not by 9.
    If a = 3*x, b = 3*y and c = 3*z, where x, y, and z are not divisible by 3, then a*c-b = 3*(3*x*z-y).
    3*x*z-y in not divisible by 3 as well, and so a*c-b is divisible by 3 but not divisible by 9.
    This means that in all solutions derived from the ones above the numbers are divisible by 3 but not by 9.

    In the above solution odd numbers have the form 4*n+3 and even numbers have the form 4*n+2.
    If a, b, c are all odd, then a*c has the form 4*n+1 ((4*x+3)*(4*y+3) = 4*(4*x*y+3*x+3*y+2)+1).
    Then a*c-b has the form 4*n+2 ((4*x+1)-(4*y+3) = 4*(x-y-1)+2).
    If b is even, then a*c-b has the form 4*n+3 ((4*x+1)-(4*y+2) = 4*(x-y-1)+3).
    If a or c is even, then a*c has the form 4*n+2 ((4*x+2)*(4*y+3) = 4*(4*x*y+3*x+2*y+1)+2).
    Then a*c-b has the form 4*n+3 ((4*x+2)-(4*y+3) = 4*(x-y-1)+3).
    This means that in all solutions derived from the ones above even numbers have the form 4*n+2, and therefore are not divisible by 12.

    Please reply to ilanlmayer at gmail dot com

    __/\__
    \ /
    __/\\ //\__ Ilan Mayer
    \ /
    /__ __\ Toronto, Canada
    /__ __\
    ||

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)