Extracting Square Roots Made Easy: A Little Known Medieval Method - The Fermat Equation (Better Known as the Pell Equation)

In February 1657 (he gave no exact day), the French co-founder of analytic geometry and of probability calculus, pioneer of infinitesimal calculus, and great number-theoretician Pierre de Fermat (1607**-1665) addressed a letter to the French amateur mathematician Bernard Frénicle de Bessy (1605-1675) that is worth translating from French into English:

Every non-square number is of such a nature that one can find an infinity of square numbers, by which you multiply the given number, and when you add the unit [that is, \(1\)] to the product, a square number is coming out.

Example: \(3\) is a non-square number, which multiplied by \(1,\) which is a square number, makes \(3,\) and, taking [adding] the unit, makes \(4,\) which is a square number.

The same \(3,\) multiplied by \(16,\) which is a square number, makes \(48\) and, taking the unit, makes \(49,\) which is a square number.

There are infinite numbers, which, multiplying \(3,\) adding the unit, make likewise a square number. [The next is \(15^2=225,\) for which \( 3\times 225+1=676=26^2. \)]

I demand from you a general rule, when a non-square number is given, to find square numbers, which, multiplied by the given number, adding the unit, produce square numbers.

Which is, for example, the smallest square number, which, multiplying \(61,\) adding the unit, makes a square number?

[The answer is \( 226\: 153\: 980^2.\) Note that \( 61\times 226\: 153\: 980^2 +1 = 1\: 766\: 319\: 049^2.] \)

Equally, which is the smallest square number, which, multiplying \(109,\) and adding the unit, makes a square number?

[The answer is \( 109 \times 15\:140\:424\:455\:100^2+1=158\:070\:671\:986\:249^2.] \)

If you do not send me the general solution, send me the particular one of these two numbers, which I have chosen from the smallest ones in order not to give you too much pains.

After having received your answer I will propose you something else. It is clear, without saying, that my proposition is only for finding integers, which satisfy the question because in the case of fractions the most insignificant arithmetician would manage the problem.

Also in February 1657, Fermat addressed a challenge to English mathematicians about the same mathematical problem, this time in Latin. It was received by the Irish mathematician Viscount William Brouncker (1620-1684), the co-founder and first president of the Royal Society, in March 1657. Brouncker at first delivered solutions in the form of fractions, but, after Fermat's demand for integral solutions, he provided these as well.

The main proposition of the challenge, in Latin and English, was:

Dato quovis numero non quadrato, dantur infiniti quadrati qui, in datum numerum ducti, adscitâ unitate, conficiant quadratum.

Given any non-square number, there are infinite square numbers, which multiplied by the given number, [and] the unit added, yield a square number.

The challenge was to prove this theorem, or to find a square number, which multiplied by \(149,\) \(109,\) or \(433,\) plus \(1,\) produces a square number.

If the non-square number is called \(d,\) the multiplying square number \(y^2,\) and the produced square number \(x^2,\) the resulting equation can be expressed as \(dy^2+1=x^2.\) Today it is usually written in the form \(x^2-dy^2=1.\) (Caution: \(dy^2\) is not a differential!)

Although it would have been logical to call the problem of finding \(x\) and \(y\) for a given \(d\) the Fermat equation, through a misunderstanding it was named instead the Pell equation after the Englishman John Pell (1611-1685), despite Pell's having little to do with it. But it is utterly futile to try to rectify this well-established misnomer.

It was the most prominent mathematician of the 18th century, the Swiss mathematician Leonhard Euler (1707-1783), who wrote in his popular 1770 book Vollständige Anleitung zur Algebra (Complete instruction in algebra; better known in English as Elements of Algebra), in the second section of the second part, “On indeterminate analysis,” in Chapter 7, “About a special method to make the formula \(ann +1\) into a square in integral numbers”: “For this an erudite Englishman by the name of Pell has invented a very ingenious method...” This remark was the origin of the erroneous designation.

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**Most biographies of Fermat still give 1601 as his birth year. However, Klaus Barner, Professor Emeritus, University of Kassel, Germany, has found that Fermat’s father had a son named Piere (one "r"), born in 1601, who died shortly after his birth. A second son named Pierre was born in 1607, and he became the famous mathematician (Barner, 2001; 2007). For more information, see the appendix, "When Was Fermat Born?"