A general quadratic Diophantine equation in two variables and is given by
where , , and are specified (positive or negative) integers and and are unknown integers satisfying the equation whose values are sought. The slightly more general second-order equation
is one of the principal topics in Gauss's Disquisitiones arithmeticae. According to Itô (1987), equation (2) can be solved completely using solutions to the Pell equation. In particular, all solutions of
are among the convergents of the continued fractions of the roots of .
Solution to the general bivariate quadratic Diophantine equation is implemented in the Wolfram Language as Reduce[eqn && Element[x|y, Integers], x, y].
For quadratic Diophantine equations in more than two variables, there exist additional deep results due to C. L. Siegel.
An equation of the form
where is an integer is a very special type of equation called a Pell equation. Pell equations, as well as the analogous equation with a minus sign on the right, can be solved by finding the continued fraction for . The more complicated equation
can also be solved for certain values of and , but the procedure is more complicated (Chrystal 1961). However, if a single solution to (5) is known, other solutions can be found using the standard technique for the Pell equation.
The following table summarizes possible representation of primes of given forms, where and are positive integers. No odd primes other than those indicated share these properties (Nagell 1951, p. 188).
|form||congruence for |
| (mod 4)|
| (mod 8)|
| (mod 6)|
| (mod 14)|
| (mod 24)|
As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 4 positive squares (; Lagrange's four-square theorem), that every "sufficiently large" integer is a sum of no more than 4 positive squares (), and that every integer is a sum of at most 3 signed squares (). If zero is counted as a square, both positive and negative numbers are included, and the order of the two squares is distinguished, Jacobi showed that the number of ways a number can be written as the sum of two squares (the function) is four times the excess of the number of divisors of the form over the number of divisors of the form .
Given an initial solution to the equation
a quadratic parametrization can be found using the identity
for arbitrary (T. Piezas, pers. comm., Apr. 28, 2006).
In 1769 Euler (1862) noted the identity
which gives a parametric solution to the equation
for integers with composite (Dickson 2005, p. 407).
Call a Diophantine equation consisting of finding a sum of th powers which is equal to a sum of th powers a " equation." The 2.1.2 quadratic Diophantine equation
corresponds to finding a Pythagorean triple (, , ) has a well-known general solution (Dickson 2005, pp. 165-170). To solve the equation, note that every prime of the form can be expressed as the sum of two relatively prime squares in exactly one way. A set of integers satisfying the 2.1.3 equation
is called a Pythagorean quadruple.
Parametric solutions to the 2.2.2 equation
are known (Dickson 2005; Guy 1994, p. 140). The number of solutions are given by the sum of squares function .
Solutions to an equation of the form
are given by the Fibonacci identity
Another similar identity is the Euler four-squareidentity
Degen's eight-square identity holds for eight squares, but no other number, as proved by Cayley. The two-square identity underlies much of trigonometry, the four-square identity some of quaternions, and the eight-square identity, the Cayley algebra (a noncommutative nonassociative algebra; Bell 1945).
Chen Shuwen found the 2.6.6 equation
Ramanujan's square equation
has been proved to have only solutions , 4, 5, 7, and 15 (Schroeppel 1972; OEIS A060728). In an unpublished proof, Euler showed that the quadratic Diophantine equation
has a unique solution for every positive in which and are both odd and positive (Engel 1998, p. 126). Rather amazingly, these can be given analytically by
which is related to the norms of elements of the ring of integers in the quadratic field which exhibits unique factorization (Hickerson 2002). The first few solutions for , 2, 3, ... are (1, 1), (1, 3), (1, 5), (3, 1), (1, 11), (5, 9), (7, 13), (3, 31), ... (OEIS A077020 and A077021).