Exercise · ZW-PELL-S-EQUATION-007 · ★★☆ Intermediate
Compute the convergent \(p/q\) of the continued fraction of \(\sqrt{2}\) that gives the solution of \(x^2-2y^2=1\).
Choose one option.
Show answer
- C. \(3/2\)
Explanation
The continued fraction of \(\sqrt{2}\) is \([1;\overline{2}]\), with convergents \(1, 3/2, 7/5,\dots\). The convergent \(3/2\) gives \(3^2-2\cdot 2^2=1\). The \(1/1\) is just the first term.
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-008 Determine the fundamental solution of x^2 - 7y^2 = 1 knowing that 8^2 - 7· 3^2 = 1.
- ZW-PELL-S-EQUATION-009 If (2,1) is the fundamental solution of x^2-3y^2=1, which pair is obtained by computing (2+√3)^2?