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A Source Book in Mathematics, 1200-1800
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Historical SummaryPierre de FermatIn a letter of February 1657 (Oeuvres, II, 333–335; III, 312–313) Fermat challenged all mathematicians (thinking probably in the first place of John Wallis in England) to find an infinity of integer solutions of the equation where A is any nonsquare integer. He may have been led to this by his study of Diophantus, who set the problem of finding, for example, a number x such that both and are squares. If these squares are called u2 and v2 respectively, then and a solution is The problem was taken up by De Billy (see below) and later by Euler, who in his "De solutione problematum Diophanteorum per numeros integros," Commentarii Academiae Scientiarum Petropolitanae 6 (1732/33, publ. 1738), 175–188, Opera omnia, ser. I, vol. 2, 6–17, referred to the problem as that of Pell and Fermat. John Pell (1611–1685), an English mathematician, had little to do with the problem, but the problem of Fermat has since been known as that of the Pell equation. It had already been studied by Indian mathematicians, and even in the Cattle Problem, attributed to Archimedes, which leads to a "Pell" equation with see T. L. Heath, A manual of Greek mathematics (Clarendon Press, Oxford, 1931), 337.Fermat, after observing that "Arithmetic has a domain of its own, the theory of integral numbers," defines his problem as follows:In the same month (February 1657) Fermat, in a letter to Frénicle, suggests the same problem, and expressly states the condition, implied in the foregoing, that the solution be in integers:Connected with this problem are a number of others, assembled by Fermat’s friend Jacques de Billy (1602–1669), a Jesuit teacher of mathematics in Dijon, in his Doctrinae analyticae inventum novum (ed. S. Fermat; Toulouse 1670), translated in Fermat, Oeuvres, III, 325–398. They begin with the Diophantine problem (called a double equation), to make both and squares Part III (p. 376) begins (we change to modern notation):Applied to making a square, De Billy writes which, set equal to the given form, gives In the case of De Billy equates this to and gets then he equates it to and gets and so on.Then, by substituting for x the value where x is a "primitive" solution, for example or and repeating the process, he obtains new solutions. For he requires that be a square, which gives hence is a solution of the original equation. Here he turned a "false" solution into a "true" one. This process can be repeated.It was from these problems by Fermat that Euler, in the paper of 1732/33, started his research on the "Pell" equation.
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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Chicago: Pierre Fermat, "The Pell Equation," A Source Book in Mathematics, 1200-1800 in A Source Book in Mathematics, 1200-1800, ed. D. J. Struik (Princeton: Princeton University Press, 1969, 1986), 29–31. Original Sources, accessed April 25, 2024, http://www.originalsources.com/Document.aspx?DocID=DD6Y6UFDVDJY6XY.
MLA: Fermat, Pierre. "The "Pell" Equation." A Source Book in Mathematics, 1200-1800, Vol. III, in A Source Book in Mathematics, 1200-1800, edited by D. J. Struik, Princeton, Princeton University Press, 1969, 1986, pp. 29–31. Original Sources. 25 Apr. 2024. http://www.originalsources.com/Document.aspx?DocID=DD6Y6UFDVDJY6XY.
Harvard: Fermat, P, 'The "Pell" Equation' in A Source Book in Mathematics, 1200-1800. cited in 1969, 1986, A Source Book in Mathematics, 1200-1800, ed. , Princeton University Press, Princeton, pp.29–31. Original Sources, retrieved 25 April 2024, from http://www.originalsources.com/Document.aspx?DocID=DD6Y6UFDVDJY6XY.
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