Diophantine Equation--2nd Powers

A general quadratic Diophantine equation in two variables x and y is given by


where a, c, and k are specified (positive or negative) integers and x and y 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 ax^2+bx+c.

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 D 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 sqrt(D). The more complicated equation


can also be solved for certain values of c and D, 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 p of given forms, where x and y are positive integers. No odd primes other than those indicated share these properties (Nagell 1951, p. 188).

formcongruence for p
x^2+y^2=1 (mod 4)
x^2+2y^2=1,3 (mod 8)
x^2+3y^2=1 (mod 6)
x^2+7y^2=1,9,11 (mod 14)
2x^2+3y^2=5,11 (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 (g(2)=4; Lagrange's four-square theorem), that every "sufficiently large" integer is a sum of no more than 4 positive squares (G(2)=4), and that every integer is a sum of at most 3 signed squares (eg(2)=3). 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 r_2(n) function) is four times the excess of the number of divisors of the form 4x+1 over the number of divisors of the form 4x-1.

Given an initial solution (x,y,z)=(m,n,p) to the equation


a quadratic parametrization can be found using the identity




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

In 1769 Euler (1862) noted the identity


which gives a parametric solution to the equation


for integers A,B,C,x,y with C composite (Dickson 2005, p. 407).

Call a Diophantine equation consisting of finding a sum of m kth powers which is equal to a sum of n kth powers a "k.m.n equation." The 2.1.2 quadratic Diophantine equation


corresponds to finding a Pythagorean triple (A, B, C) has a well-known general solution (Dickson 2005, pp. 165-170). To solve the equation, note that every prime of the form 4x+1 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 r_2(n).

Solutions to an equation of the form


are given by the Fibonacci identity


Another similar identity is the Euler four-square identity


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 n=3, 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 n>=3 in which x and y 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 Q(sqrt(-7)) which exhibits unique factorization (Hickerson 2002). The first few solutions (x,y) for n=1, 2, 3, ... are (1, 1), (1, 3), (1, 5), (3, 1), (1, 11), (5, 9), (7, 13), (3, 31), ... (OEIS A077020 and A077021).

See also

Algebra, Cannonball Problem, Continued Fraction, Diophantine Equation, Euler Four-Square Identity, Genus Theorem, Hilbert Symbol, Lagrange Number, Lebesgue Identity, Pell Equation, Pythagorean Quadruple, Pythagorean Triple, Quadratic Residue, Ramanujan's Square Equation, Square Number, Sum of Squares Function, Waring's Problem

Explore with Wolfram|Alpha


Beiler, A. H. "The Pellian." Ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, pp. 248-268, 1966.Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, p. 159, 1945.Chrystal, G. Textbook of Algebra, 2 vols. New York: Chelsea, 1961.Degan, C. F. Canon Pellianus. Copenhagen, Denmark, 1817.Dickson, L. E. "Number of Representations as a Sum of 5, 6, 7, or 8 Squares." Ch. 13 in Studies in the Theory of Numbers. Chicago, IL: University of Chicago Press, 1930.Dickson, L. E. "Pell Equation; ax^2+bx+c Made a Square" and "Further Single Equations of the Second Degree." Chs. 12-13 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, pp. 341-434, 2005.Engel, A. Problem-Solving Strategies. New York: Springer-Verlag, 1998.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.Hickerson, D. "Re: Diophantine sequence" mailing list. 17 Oct 2002.Itô, K. (Ed.). Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, p. 450, 1987.Lam, T. Y. The Algebraic Theory of Quadratic Forms. Reading, MA: W. A. Benjamin, 1973.Nagell, T. "Diophantine Equations of the Second Degree." Ch. 6 in Introduction to Number Theory. New York: Wiley, pp. 188-226, 1951.Rajwade, A. R. Squares. Cambridge, England: Cambridge University Press, 1993.Scharlau, W. Quadratic and Hermitian Forms. Berlin: Springer-Verlag, 1985.Schroeppel, R. Item 31 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 14, Feb. 1972., D. B. "Products of Sums and Squares." Expo. Math. 2, 235-261, 1984.Sloane, N. J. A. Sequences A060728, A077020, and A077021 in "The On-Line Encyclopedia of Integer Sequences."Smarandache, F. "Un metodo de resolucion de la ecuacion diofantica." Gaz. Math. 1, 151-157, 1988.Smarandache, F. "Method to Solve the Diophantine Equation ax^2-by^2+c=0." In Collected Papers, Vol. 1. Bucharest, Romania: Tempus, 1996.Taussky, O. "Sums of Squares." Amer. Math. Monthly 77, 805-830, 1970.Whitford, E. E. The Pell Equation. New York: Columbia University Press, 1912.

Cite this as:

Weisstein, Eric W. "Diophantine Equation--2nd Powers." From MathWorld--A Wolfram Web Resource.

Subject classifications