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..
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)
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 ?
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