A Source Book in Mathematics, 1200-1800

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Author: Pierre Fermat

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Mathematics

7 FERMAT.

# The "Pell" Equation

Given any number not a square, then there are an infinite number of squares which, when multiplied by the given number, make a square when unity is added.

Example.—Given 3, a nonsquare number; this number multiplied by the square number 1, and 1 being added, produces 4, which is a square.

Moreover, the same 3 multiplied by the square 16, with 1 added makes 49, which is a square.

And instead of 1 and 16, an infinite number of squares may be found showing the same property; I demand, however, a general rule, any number being given which is not a square.

It is sought, for example, to find a square which when multiplied into 149, 109, 433, etc., becomes a square when unity is added.

Every nonsquare is of such a nature that one can find an infinite number of squares by which if you multiply the number given and if you add unity to the product, it becomes a square.

Example.—3 is a nonsquare number, which multiplied by 1, which is a square, makes 3, and by adding unity makes 4, which is a square.

The same 3, multiplied by 16, which is a square, makes 48, and with unity added makes 49, which is a square.

There is an infinity of such squares which when multiplied by 3 with unity added likewise make a square number.

I demand a general rule,—given a nonsquare number, find squares which multiplied by the given number, and with unity added, make squares.

What is for example the smallest square which, multiplied by 61 with unity added, makes a square?

Moreover, what is the smallest square which, when multiplied by 109 and with unity added, makes a square?

If you do not give me the general solution, then give the particular solution for these two numbers, which I have chosen small in order not to give too much difficulty.

After I have received your reply, I will propose another matter. It goes without saying that my proposition is to find integers which satisfy the question, for in the case of fractions the lowest type of arithmetician could find the solution.

On the procedure for obtaining an infinite number of solutions which give square or cubic values to expressions in which enter more than three terms of different degrees.

1. I shall discuss here in particular expressions which contain the five terms in x4, z3, x2, x, and the constant, but I also wish to discuss expressions with four terms which may be all positive [true], or mixed with negative [false] terms. We wish to give these expressions square values (in the case of five terms), or cubic ones (in the case of four terms), and this in an infinity of ways. In general we must say that for the square value at least the coefficient of the term in x4 or the constant term must be a square; as to the cubic values, the coefficient of x3 or the constant term must be a cube.

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