Diophantine Equation--2nd Powers

A diagonal quadratic Diophantine equation in two variables and is given by

(1)

where , , and are specified integers and and are unknown integers satisfying the equation whose values are sought, with and not both zero. The slightly more general second-order equation

(2)

is one of the principal topics in Gauss's Disquisitiones arithmeticae. According to Itô (1987), equations of this form with positive nonsquare discriminant can be solved using solutions to Pell equations.

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

(3)

where is a positive integer that is not a square number, is a 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 ; for positive solutions, is one of its convergents. The more complicated equation

(4)

can also be solved for certain values of and , but the procedure is more complicated (Chrystal 1961). However, if a single solution to (4) 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 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 (). For , if zero is allowed and the integers being squared may have either sign, with their order 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

(5)

a quadratic parametrization can be found using the identity

(6)

where

			(7)
			(8)
			(9)

for arbitrary (T. Piezas, pers. comm., Apr. 28, 2006).

In 1769 Euler (1862) noted the identity

(10)

which gives a parametric solution to the equation

(11)

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

(12)

corresponds to finding a Pythagorean triple (, , ) and 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 up to signs and interchange of the two terms. A set of integers satisfying the 2.1.3 equation

(13)

is called a Pythagorean quadruple.

Parametric solutions to the 2.2.2 equation

(14)

are known (Dickson 2005; Guy 1994, p. 140). For a fixed common value , the number of ordered integer quadruples is , where is the sum of squares function.

Solutions to an equation of the form

(15)

are given by the Fibonacci identity

(16)

Iterating the two-square identity also gives

(17)

The Euler four-square identity is

(18)

Degen's eight-square identity is the eight-square case of a bilinear composition identity. Such identities expressing a product of two sums of squares as a sum of squares exist only for , 2, 4, and 8 (Hurwitz 1898). 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

(19)

Ramanujan's square equation

(20)

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

(21)

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

			(22)
			(23)

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 , 4, 5, ... are (1, 1), (1, 3), (1, 5), (3, 1), (1, 11), (5, 9), (7, 13), (3, 31), ... (OEIS A077020 and A077021).