CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-27

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The constant 1/2 in Legendre's criterion cannot be replaced by 3/4

Statement refuted

For every irrational real number $α$ and every reduced rational number $r / s$ with $s > 0$ , $∣ α - \frac{r}{s} ∣ < \frac{3}{4 s^{2}}$ implies that $r / s$ is a convergent of $α$ .

Facts & Assumptions

Given: The irrational number $\sqrt{2}$ and the rational number $10 / 7$ .

[F1]

Legendre's criterion with the sharp constant $1 / 2$ says that $∣ α - r / s ∣ < 1 / (2 s^{2})$ forces $r / s$ to be a convergent (Legendre's criterion for convergents).

[F2]

Convergents of an irrational satisfy the standard error bound $∣ α - p_{n} / q_{n} ∣ < 1 / (q_{n} q_{n + 1})$ (Convergent error bound).

Counterexample

technique · direct

1.1givenalgebra

The first few convergents of $[1; \overline{2}] = \sqrt{2}$ are. [given, algebra] $1, \frac{3}{2}, \frac{7}{5}, \frac{17}{12},$ so $10 / 7$ is not a convergent of $\sqrt{2}$ .

1.2givenalgebra

Nevertheless. [given, algebra] $\frac{10}{7} - \sqrt{2} = \frac{100 - 98}{7 (10 + 7 \sqrt{2})} = \frac{2}{7 (10 + 7 \sqrt{2})} < \frac{3}{4 \cdot 7^{2}} .$ Thus $10 / 7$ satisfies the displayed $3 / (4 s^{2})$ bound.

2.1F1F2step 1.1step 1.2∎

Step 1.1 and step 1.2 together refute the statement. In the light of [F1],. [F1, F2, step 1.1, step 1.2] this shows that the sharp constant $1 / 2$ in Legendre's criterion cannot simply be replaced by the larger constant $3 / 4$ .

Depends on

Used by

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Sources

Peter Hackman, Elementary Number Theory (standard reference, not scraped)
Bruce Ikenaga, Approximation by Rational Numbers (standard reference, not scraped)