Pell's equation is any Diophantine equation of the form In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ...

x² − ny² = 1

where n is a nonsquare integer. In calling it "Diophantine" we are really saying what we intend to do with the equation rather than describing any intrinsic property of the equation: we intend to seek solutions in which both x and y are integers. Infinitely many such solutions of this equation exist. The solutions yield good rational approximations to the square root of the natural number n. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is . ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

The equation is named for Sir John Pell, an English mathematician and father of Pell, who settled much of Westchester County, New York, and whose descendants formed one of the most prominent and progressive political families in the United States. John Pell (March 1, 1610 - December 12, 1685), was an English mathematician. ...

As motivation, consider the square root of two. It is often approximated 1.414..., which some might incorrectly interpret as 1.41414141414..., or 140/99. Likewise, the reciprocal of the square root of two to three decimal places is 0.707, which is suggestive of 0.70707070..., or 70/99. If 70/99 approximates the reciprocal of the square root of two, it follows that 99/70 approximates the square root of two. As it turns out, the square root of two is between 140/99 and 99/70. The arithmetic mean of these two rationals is 19601/13860. That number squared is 384199201/192099600. It turns out that 2 times the denominator 192099600 is 384199200, which differs from the numerator by only one. p = 19601 and q = 13860 satisfies the Diophantine equation 2q² + 1 = p². Any fraction of natural numbers p and q that satisfy this equation will be a reasonably good approximation for the square root of two. The square root of two is the positive real number which, when multiplied by itself, gives a product of two. ... In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ... Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus) to... In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ...

More generally, if n is a given natural number, then any fraction of natural numbers p and q that satisfy the Pell's equation

nq² + 1 = p²

is a reasonably good approximation for the square root of n. The larger the numbers p and q, the better the approximation.

It turns out that if both (a, b) and (c, d) satisfy a Pell's equation, then so do

(bc + ad,bd + nac)

and

(bc − ad,bd − nac).

Fermat proved that p and q can always be found to satisfy a Pell's equation for any natural number n that is not a perfect square. Given a computer with bignum capability, this makes it easy to converge rapidly toward any irrational square root of a n. As an added bonus, a Pell's equation can always be solved in a finite number of steps by calculating the continued fraction representation of the square root of n. Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ... The term perfect square is used in mathematics in two meanings: a positive integer which is the square of some other integer, i. ... A bignum package in a computer or program allows internal representation of arbitrarily large integers, rational numbers, decimal numbers, or floating-point numbers, and provides a set of arithmetic operations on such numbers. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

A general development of solutions of Pell's equation in terms of continued fractions can be presented, as these arise as a special case of quadratic irrationals. Gauss classified such solutions into 64 or 65 sets, with the precise classification of one or the other implying the truth or falsity of the Riemann hypothesis. In mathematics, a quadratic irrational, also known as a quadratic surd or quadratic irrationality, is an irrational number that is the solution to some quadratic equation with rational coefficients. ... In mathematics, the Riemann hypothesis (aka Riemann zeta hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous of all unsolved problems. ...

The relationship to the continued fractions implies that the solutions to Pell's equation form a semigroup subset of the modular group. Thus, for example, if p and q satisfy Pell's equation, then In mathematics, a semigroup is a set with an associative binary operation on it. ... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...

is a matrix of unit determinant. Products of such matrices take exactly the same form, and thus all such products yield solutions to Pell's equation. This can be understood in part to arise from the fact that successive convergents of a continued fraction share the same property: If p_{n − 1} / q_{n − 1} and p_n / q_n are two successive convergents of a continued fraction, then the matrix

has determinant +1 or -1.