Lagrange's algorithm revisited: Solving at2 + btu + cu2 = n in the case of negative discriminant

Keith R. Matthews

Lagrange's algorithm revisited: Solving at² + btu + cu² = n in the case of negative discriminant

Keith R. Matthews^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We make more accessible a neglected continued fraction algorithm of Lagrange for solving the equation at² + btu + cu² = n in relatively prime integers t, u, where a> 0, gcd(a,n) = 1, and D = b² - 4ac < 0. The cases D = -4 and D = -3 present a consecutive convergents phenomenon which aids the search for solutions.

Original language	English
Article number	14.11.1
Journal	Journal of Integer Sequences
Volume	17
Issue number	11
Publication status	Published - 5 Nov 2014

Cite this

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title = "Lagrange's algorithm revisited: Solving at2 + btu + cu2 = n in the case of negative discriminant",

abstract = "We make more accessible a neglected continued fraction algorithm of Lagrange for solving the equation at2 + btu + cu2 = n in relatively prime integers t, u, where a> 0, gcd(a,n) = 1, and D = b2 - 4ac < 0. The cases D = -4 and D = -3 present a consecutive convergents phenomenon which aids the search for solutions.",

keywords = "Continued fraction, Convergent, Diophantine equation, Positive definite binary quadratic form",

author = "Matthews, \{Keith R.\}",

year = "2014",

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language = "English",

volume = "17",

journal = "Journal of Integer Sequences",

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T1 - Lagrange's algorithm revisited

T2 - Solving at2 + btu + cu2 = n in the case of negative discriminant

AU - Matthews, Keith R.

PY - 2014/11/5

Y1 - 2014/11/5

N2 - We make more accessible a neglected continued fraction algorithm of Lagrange for solving the equation at2 + btu + cu2 = n in relatively prime integers t, u, where a> 0, gcd(a,n) = 1, and D = b2 - 4ac < 0. The cases D = -4 and D = -3 present a consecutive convergents phenomenon which aids the search for solutions.

AB - We make more accessible a neglected continued fraction algorithm of Lagrange for solving the equation at2 + btu + cu2 = n in relatively prime integers t, u, where a> 0, gcd(a,n) = 1, and D = b2 - 4ac < 0. The cases D = -4 and D = -3 present a consecutive convergents phenomenon which aids the search for solutions.

KW - Continued fraction

KW - Convergent

KW - Diophantine equation

KW - Positive definite binary quadratic form

UR - https://www.scopus.com/pages/publications/84910048305

M3 - Article

SN - 1530-7638

VL - 17

JO - Journal of Integer Sequences

JF - Journal of Integer Sequences

IS - 11

M1 - 14.11.1

ER -

Lagrange's algorithm revisited: Solving at² + btu + cu² = n in the case of negative discriminant

Abstract

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