CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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

1.1

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 $\sqrt{2}$ .

givenalgebra

1.2

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.

givenalgebra

2.1

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 $3 / 4$ .

F1F2step 1.1step 1.2∎

Depends on

Used by

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9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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