Why did Pell’s equation wrongly named?

By a mistake of **Euler**, the
Diophantine equation y^2 - Ax^2 = 1 has been erroneously known as "**Pell's
equation"**; but, in fact, the English mathematician **John Pell**
(1611-1685) did no more than copy it down in his papers, from **Fermat**'s
letters of 1657 and 1658.

For an extensive historical account on
"**Pell’s equation**", see **Sir Thomas L. Heath**, ** Diophantus of
Alexandria : A Study in the History of Greek Algebra** (Dover Pub., New
York, 1931-1963, 552 pages), Supplement, Section II: "Equation y^2 - Ax^2 = 1,
pp. 277-292. Particularly in page 285, after a presentation of the history of
the equation up to Fermat’s time (including citations to Pythagoreans,
Archimedes, Diaphanous, and the Indian solution), one can read that:

" ... **Fermat **rediscovered the problem and was the first to assert that
the equation x^2 - Ay^2 = 1, where A is any integer not a square, always has an
unlimited number of solutions in integers. His statement was made in a letter to
**Frénicle** of February, 1657 (cf. *Oeuvres de Fermat*, II, pp.333-4).
**Fermat** asks **Frénicle** for *a general rule for finding, when any
number not a square is given, squares which, when they are respectively
multiplied by the given number and unity is added to the product, give
squares.* If, says **Fermat**, **Frénicle** cannot give a general rule,
will he give the smallest value of y which will satisfy the equations 61y^2 + 1
= x^2 and 109y^2 + 1 = x^2 ? (Footnote 3: **Fermat** evidently chose these
cases for their difficulty; the smallest values satisfying the first equation
are y=226153980, x=1766319049, and the smallest values satisfying the second are
y=15140424455100, x=158070671986249)." And, after a extensive quotation of
Fermat’s letter, in page 286, one can read that: "The challenge was taken up in
England by William, Viscount **Brouncker**, first President of the Royal
Society, and **Wallis **(Footnote 1: An excellent summary of the whole story
is given in Wertheim’s paper "Pierre Fermat’s Streit mit John Wallis" in
*Abhandlungen zur Gesch. der Math.*, IX. Heft (Cantor-Festschrit), 1899,
pp.557-576). See also H. Konen, *Geschichte der Gleichung t^2-Du^2=1*,
Leipzig (S. Hirzel), 1901). At first, owing apparently to some misunderstanding,
they thought that only rational, and not necessarily integral solutions were
wanted, and found of course no difficulty in solving this easy problem.
**Fermat** was, naturally, not satisfied with this solution, and
**Brouncker**, attacking the problem again, finally succeeded in solving it.
The method is set out in letters of **Wallis** (Footnote 2: *Oeuvres de
Fermat*, III, pp.457-480, 490-503) of 17th December, 1657, and 30th January,
1658, and in chapter XCVIII of **Wallis**’ *Algebra*; **Euler** also
explains it fully in his *Algebra* (Footnote 3: Part II, chap. VII),
wrongly attributing it to Pell (Footnote 4: This was the origin of the erroneous
description of our equation as the "Pellian" equation. Hankel (in *Zur
Geschichte der Math. im Alterthum und Mittlelalter*, p.203) supposed that the
equation was so called because the solution was reproduced by Pell in an English
translation (1668) by Thomas Brancker of **Rahn**’s *Algebra*; but this
is a misapprehension, as the so-called "Pellian" equation is not so much as
mentioned in **Pell**’s additions (Wertheim in *Bibliotheca
Mathematica*, III, 1902, pp.124-6); Konen, pp.33-4 note). The attribution of
the solution to **Pell** as a pure mistake of Euler’s, probably due to a
cursory reading by him of the second volume of **Wallis**’ *Opera* where
the solution of the equation *ax^2 + 1 = y^2* is given as well as
information as to **Pell**’s work in indeterminate analysis. But **Pell**
is not mentioned in connexion with the equation at all (Eneström in
*Bibliotheca Mathematica*, III, 1902, p.206)."

For more information about "**Pell’s
equation**", see **Harold M. Edwards**, ** The Fermat's Last Theorem: A
Genetic Introduction to Algebraic Number Theory** (Springer-Verlag, New
York, 1977, 410 pages), pp. 25-33. Particularly in page 33 one can read that

"This problem of **Fermat** is now known as "**Pell’s equation**" as a
result of a mistake on the part of **Euler**. In some way, perhaps from a
confused recollection of **Wallis**’s ** Algebra**,

See also **André Weil**, *Number
Theory : An approach through history - From Hammurapi to Legendre*
(Birkhäuser, Boston, 1984, xv+375 pages), in many different pages. In
particular, at page 174, one can read that:

"Pell's name occurs frequently in

Wallis'sAlgebra, but never in connection with the equation x^2 - Ny^2 = 1 to which his name, because of Euler's mistaken attribution, has remained attached; since its traditional designation as "Pell's equation"is unambiguous and convenient, we will go on using it, even though it is historically wrong."

Raul Nunes ( raul_nunes@uol.com.br )

NEST Nunes’
Exposition of Scientific Truths

( http://www.geocities.com/raulnunes
)