190 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICSEuler conjectured what we now know as the law of quadratic reciprocity: Given
two primes ñ and q both of which equal 3 modulo 4, exactly one of them is a
quadratic residue modulo the other. In all other cases, either each is a quadratic
residue modulo the other or neither is. For example, 11Î2^2 Î5^2 mod 7, but the
quadratic residues modulo 11 are 0, 1, 4, 3 ( Î 5^2 = 6^2 mod 11), 5 ( = 4^2 Î 7^2
mod 11), and 9; 7 is not among them. That is because both 7 and 11 are equal to
3 modulo 4. On the other hand, since 5 equals 1 modulo 4, we find that 11 Î l^2
mod 5 and 5 = 7^2 mod 11; similarly, neither 5 nor 7 is a quadratic residue modulo
the other. The fact that Euler did not succeed in proving the law of quadratic
reciprocity shows how difficult a result it is.The Goldbach conjecture. A problem of number theory whose fame is second only
to the Fermat conjecture is a conjecture of Christian Goldbach (1690-1764), who
wrote to Euler in 1742 that every integer seemed to be a sum of at most three
prime integers (Struik, 1986, pp. 47-49). Euler wrote back that he believed, but
was unable to prove, the stronger proposition that every even integer larger than
4 is the sum of two odd primes—in other words, that one of the three primes
conjectured by Goldbach can be chosen arbitrarily. Euler's statement is known
as the Goldbach conjecture. In 1937 the Russian mathematician Ivan Matveevich
Vinogradov (1891-1983) proved that every sufficiently large odd integer is the sum
of at most three primes.1.3. Lagrange. The generation after Euler produced the Italian-French mathe-
matician Joseph-Louis Lagrange (1736-1813). His name gives the impression that
he was French, and indeed his ancestry was French and he wrote in French; but
then so did many others, as French was literally the "lingua franca," the common
language of much scientific correspondence during the eighteenth and nineteenth
centuries. Lagrange was born in Turin, however, and lived there for the first 30
years of his life, signing his first name as "Luigi" on his first mathematical paper
in 1754. When the French Revolution came, he narrowly escaped arrest as a for-
eigner; and we have the word of Jean-Joseph Fourier, who heard him lecture, that
he spoke French with a noticeable Italian accent. Thus it appears that the Italians
are correct in claiming him as one of their own, even though his most prominent
works were published in France and he was a member of the Paris Academy of
Sciences for the latter part of his life.
Lagrange's early work impressed Euler, then in Berlin, very favorably, and
attempts were made to bring him to Berlin. But the introverted Lagrange seems
to have been intimidated by Euler's power as a mathematician and refused all such
offers until Euler went back to St. Petersburg in 1766. He then came to Berlin and
remained there until the death of Frederick II in 1788, at which point he accepted
a position at the Paris Academy of Sciences, where he spent the last 15 years of
his life. Lagrange did important work in algebra and mechanics that is discussed
in later chapters. At this point we note only some of his number-theoretic results.The Pell equation. Shortly after arriving in Berlin in 1766, Lagrange gave a defini-
tive discussion of the solutions of the Pell equation x^2 = Dy^2 ± 1, using the theory
of continued fractions. In the course of this work he proved the important fact
that any irrational number satisfying a quadratic equation with integer coefficients
has a periodic continued fraction expansion. The converse of that statement is also
true, and it turns out that the continued-fraction expansion of y/D can be used to