A quadratic surd is a number of the form 
, where 
is rational and 
is a positive rational number
that is not the square of a rational number. It
is pure when 
and is sometimes called mixed when 
(Hardy 1967, p. 20).
Quadratic surds are sometimes also called quadratic irrationals.
In 1770, Lagrange proved that any quadratic surd has a regular continued fraction which is periodic after some point. This result is known as Lagrange's continued fraction theorem.
Pisani (2026) classified pairs of positive integer exponents 
for which there is a real number 
such that both 
and 
are rational,
for a fixed positive nonsquare integer 
. The pairs are exactly 
and 
with 
a positive odd integer. For distinct
exponents, 
must be negative. Pisani (2026) credits AI with suggesting the reduction and Lucas
sequence ideas used in the proof.
