A general quadratic Diophantine equation in two variables 
and 
is given by
 |
(1)
|
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
 |
(2)
|
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
 |
(3)
|
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
 |
(4)
|
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
 |
(5)
|
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
 |
(6)
|
a quadratic parametrization can be found using the identity
 |
(7)
|
where
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
for arbitrary 
(T. Piezas, pers. comm., Apr. 28, 2006).
In 1769 Euler (1862) noted the identity
 |
(11)
|
which gives a parametric solution to the equation
 |
(12)
|
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
 |
(13)
|
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
 |
(14)
|
is called a Pythagorean quadruple.
Parametric solutions to the 2.2.2 equation
 |
(15)
|
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
 |
(16)
|
are given by the Fibonacci identity
 |
(17)
|
Another similar identity is the Euler four-square identity
 |
(18)
|
 |
(19)
|
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
 |
(20)
|
 |
(21)
|
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
 |
(22)
|
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
 |  | ![(2^(n/2))/(sqrt(7))|sin[ntan^(-1)(sqrt(7))]|](/images/equations/DiophantineEquation2ndPowers/Inline64.svg) |
(23)
|
 |  | ![2^(n/2)|cos[ntan^(-1)(sqrt(7))]|,](/images/equations/DiophantineEquation2ndPowers/Inline67.svg) |
(24)
|
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).
Made a Square" and "Further Single Equations
of the Second Degree." Chs. 12-13 in 
."
In Collected Papers, Vol. 1. Bucharest, Romania: Tempus, 1996.Taussky,
O. "Sums of Squares." Amer. Math. Monthly 77, 805-830, 1970.Whitford,
E. E.