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interests / rec.puzzles / Re: ( x ^ 2 + y ^ 2 + z ^ 2 = xyz ) for positive integers (x, y, z)

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* ( x ^ 2 + y ^ 2 + z ^ 2 = xyz ) for positive integershenh...@gmail.com
`* Re: ( x ^ 2 + y ^ 2 + z ^ 2 = xyz ) for positive integers (x, y, z)Ilan Mayer
 `* Re: ( x ^ 2 + y ^ 2 + z ^ 2 = xyz ) for positive integers (x, y, z)henh...@gmail.com
  `- Re: ( x ^ 2 + y ^ 2 + z ^ 2 = xyz ) for positive integers (x, y, z)Ilan Mayer

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

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Subject: ( x ^ 2 + y ^ 2 + z ^ 2 = xyz ) for positive integers
(x, y, z)
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Mon, 27 Jun 2022 01:13 UTC

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.

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

<ac79adb7-fbed-4200-a356-fbe7e33d1ea4n@googlegroups.com>

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Subject: Re: ( x ^ 2 + y ^ 2 + z ^ 2 = xyz ) for positive integers (x, y, z)
From: ilan_no_...@hotmail.com (Ilan Mayer)
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 by: Ilan Mayer - Thu, 30 Jun 2022 12:34 UTC

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.
..
..
..
..
..
..
..
..
..
..
..
..
..
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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
/__ __\
||

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

<87da54ae-9882-4029-9d46-76b2507919fdn@googlegroups.com>

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Subject: Re: ( x ^ 2 + y ^ 2 + z ^ 2 = xyz ) for positive integers (x, y, z)
From: henha...@gmail.com (henh...@gmail.com)
Injection-Date: Thu, 30 Jun 2022 21:14:16 +0000
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 by: henh...@gmail.com - Thu, 30 Jun 2022 21:14 UTC

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 ?

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

<b1c119ef-e24f-4949-9768-c30d8c0d1fb7n@googlegroups.com>

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Subject: Re: ( x ^ 2 + y ^ 2 + z ^ 2 = xyz ) for positive integers (x, y, z)
From: ilan_no_...@hotmail.com (Ilan Mayer)
Injection-Date: Fri, 01 Jul 2022 13:40:08 +0000
Content-Type: text/plain; charset="UTF-8"
 by: Ilan Mayer - Fri, 1 Jul 2022 13:40 UTC

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
/__ __\
||

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