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construct all solutions of the Pell equation Dx^2 ± 1 = y^2 (see Scharlau and Opolka,
1985, pp. 45-56).
The four-squares theorem. In 1770 Lagrange gave a proof that every integer is the
sum of at most four square integers (which Euler also proved a year or so later).
"Wilson's theorem". In 1771 Lagrange proved that an integer ç is prime if and
only if ç divides (n - 1)! + 1. Thus 5 is prime because 4! + 1 = 25, but 6 is not
prime because it does not divide 5! 4-1 = 121. This theorem was attributed to John
Wilson (1741-1793) by the Cambridge professor Edward Waring (1736-1798), who
was apparently unaware that it was first stated by al-Haytham (965-1040). No
proof of it can be found in the work of Wilson, who left mathematics to become a
lawyer.
Quadratic binary forms. The study of quadratic Diophantine equations involves
expressions of the form ax^2 + bxy + cy^2. The integers that can be represented in
this way for given values of a, b, and c were the subject of two memoirs by Lagrange,
amounting to nearly 100 pages of work, during the years 1775-1777.
1.4. Legendre. The volume of work on number theory increased greatly in the
last half of the eighteenth century, and the first treatises devoted specifically to that
subject appeared. One of the prominent figures in this development was Adrien-
Marie Legendre (1752-1833). Like all other mathematicians of the time, Legendre
worked in many areas of mathematics, including calculus (elliptic functions) and
mechanics. He also worked in number theory and produced several profound results
there in an early textbook of the subject, which went through three editions before
his death.
In 1785 he published the paper "Recherches d'analyse indeterminee," in which
he proved the elegant result that there are integers x, y, æ satisfying an equation
ax^2 + by^2 + cz^2 = 0 with a, b, c not all of the same sign if and only if the products
—ab, —be, and -co are quadratic residues modulo |c|, \a\, and |6| respectively. He
also stated the law of quadratic reciprocity, which Euler had been unable to prove,
and gave a flawed proof of it. He invented the still-used Legendre symbol (2) whose
value is 1 if ñ is a quadratic residue modulo q and —1 if not. The law of quadratic
reciprocity can then be elegantly stated as (|) (^) = (-1)û ^^1 • This proof was
improved in his treatise Theorie des nombres, published in 1798, with a subsequent
edition in 1808 and a third in 1830. He also conjectured, but did not prove, that any
arithmetic sequence in which the constant difference is relatively prime to the first
term will contain infinitely many primes. In fact, it was this unproved assumption
that invalidated his proof of quadratic reciprocity (see Weil, 1984, PP- 329-330).
He quoted Fermat's conjecture that every number is the sum of at most ç n-gonal
numbers, noting with regret that either Fermat never completed the treatise he
intended to write or that his executors never found the manuscript. Legendre gave
a proof of this fact for all numbers larger than 50n - 79. Further continuing the
work of Fermat, Euler, and Lagrange, Legendre discovered some important facts in
the theory of quadratic forms.
His most original contribution to number theory, however, lay in a different
direction entirely. Since no general law had been found for describing the nth
prime number or even producing a polynomial whose values are all prime numbers,
Legendre's attempt to estimate the number of primes among the first ç integers,
published in the second (1808) edition of Theorie des nombres, was an important