Exercise · ZW-PELL-S-EQUATION-009 · ★★☆ Intermediate
If \((2,1)\) is the fundamental solution of \(x^2-3y^2=1\), which pair is obtained by computing \((2+\sqrt{3})^2\)?
Choose one option.
Show answer
- A. \((7,4)\)
Explanation
We have \((2+\sqrt{3})^2 = 7+4\sqrt{3}\), and \(7^2-3\cdot 4^2=1\). The pair \((4,2)\) does not satisfy the equation, since it results from squaring the wrong quantity.
Review the concept: Pell's equation
Pell's equation is the quadratic Diophantine equation in integers \(x\), \(y\) that sets equal to 1 the difference between \(x^2\) and \(d\) times \(y^2\), where \(d\) is a positive integer that is not a perfect square.
\[x^2 - d y^2 = 1\]
More practice problems: pell's equation
- ZW-PELL-S-EQUATION-001 Identify which of these equations is a Pell equation.
- ZW-PELL-S-EQUATION-002 State which of these values of d is valid for Pell's equation.
- ZW-PELL-S-EQUATION-003 What is the solution with the smallest positive y of Pell's equation called?
- ZW-PELL-S-EQUATION-004 Identify which property of √d allows solving Pell's equation with continued fractions.
- ZW-PELL-S-EQUATION-005 Which pair (x,y) is a solution of x^2 - 2y^2 = 1?
- ZW-PELL-S-EQUATION-006 Find the fundamental solution of x^2 - 3y^2 = 1 with minimal positive y.
- ZW-PELL-S-EQUATION-007 Compute the convergent p/q of the continued fraction of √2 that gives the solution of x^2-2y^2=1.
- ZW-PELL-S-EQUATION-008 Determine the fundamental solution of x^2 - 7y^2 = 1 knowing that 8^2 - 7· 3^2 = 1.