`fmpq_farey_neighbors`

Synopsis#include <flint/fmpq.h>

void fmpq_farey_neighbors(fmpq_t l, fmpq_t r, const fmpq_t x, const fmpz_t Q)source

Set $l$ and $r$ to the fractions immediately below and above $x$ among the fractions with denominator at most $Q$ (the Farey sequence of order $Q$ , extended to all of $\mathbb{Q}$ ). Throws if the denominator of $x$ exceeds $Q$ . Any aliasing is allowed.

Writing $x = p/q$ , the neighbours are the fractions $p_l/q_l < x < p_r/q_r$ with $p q_l - p_l q = 1 = p_r q - p q_r$ and $Q - q < q_l, q_r \le Q$ [HarWri2008, Chapter III]. The function computes $q_l$ from $p^{-1} \bmod q$ and the rest from these identities.

References#

[HarWri2008] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, sixth edition, revised by D. R. Heath-Brown and J. H. Silverman, Oxford University Press, 2008.

Cite this page

The FLINT team, “fmpq_farey_neighbors (fmpq.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/fmpq/fmpq_farey_neighbors.html (generated 2026-11-02).

@misc{flint-doc-fmpq-fmpq-farey-neighbors,
  author       = {{The FLINT team}},
  title        = {{fmpq\_farey\_neighbors (fmpq.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/fmpq/fmpq_farey_neighbors.html},
  note         = {Generated 2026-11-02}
}

Edit this page · PDF version · Built with fdoc from 0a4634bf95