MHB Prove 1/x^2+1/xy+1/y^2=1 has no real solution

  • Thread starter Thread starter Albert1
  • Start date Start date
Click For Summary
The equation 1/x^2 + 1/xy + 1/y^2 = 1 is analyzed for natural number solutions x and y. By manipulating the equation, it can be shown that the left-hand side is always less than 1 for positive integers. The terms decrease as x and y increase, indicating that the sum cannot equal 1. Therefore, there are no natural number solutions for the equation. The conclusion is that the equation has no real solutions in the specified domain.
Albert1
Messages
1,221
Reaction score
0
$x,y\in N,\,\, and \,\,\dfrac {1}{x^2}+\dfrac{1}{xy}+\dfrac {1}{y^2}=1----(1)$

prove $(1)$ has no solution
 
Mathematics news on Phys.org
Albert said:
$x,y\in N,\,\, and \,\,\dfrac {1}{x^2}+\dfrac{1}{xy}+\dfrac {1}{y^2}=1----(1)$

prove $(1)$ has no solution

neither x nor y can be 1 as LHS >= 1.

so x , y >= 2 and for x = 2 , y =2 LHS = $\frac{3}{4}$ so LHS < 1 so cannot be 1 higeer x ,y lower is the value
 

Similar threads

MHB Find real solutions to a system
  • · Replies 6 ·
Replies
6
Views
1K
MHB Proving Series Inequality: $\sqrt[3]{\frac{2}{1}}$ to $\frac{1}{8961}$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Solve $2\cos^2(x)=1$: Find $\theta$
  • · Replies 7 ·
Replies
7
Views
2K
MHB Rational Number: Proving $x+\dfrac{1}{x}$ is Rational
  • · Replies 1 ·
Replies
1
Views
985
MHB Is x+1 a Factor of the Polynomial x^3-5x^2+3x+1?
  • · Replies 2 ·
Replies
2
Views
1K
MHB Roots of Polynomial: Find $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Proving $(a)^\dfrac{1}{3}+(b)^\dfrac{1}{3}+(c)^\dfrac{1}{3}=0$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Solving System of Equations: xy, yz, zx
  • · Replies 3 ·
Replies
3
Views
1K
I ##x\lt y \lt z## positive integer numbers and if ##\dfrac 1x+\dfrac 1y+\dfrac 1z=\dfrac 13##
  • · Replies 6 ·
Replies
6
Views
1K
MHB Absolute value of real numbers
  • · Replies 1 ·
Replies
1
Views
1K