188 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICSSee the website http://www.euler2007.com/.was given by Euler in 1738. Actually, the proof shows that there can be no positive
integers x, y, æ such that x^4 + y^4 = z^2.
Another area of number theory pioneered by Fermat arises naturally from con-
sideration of quadratic Diophantine equations. The question is: "In how many
ways can an integer be represented as a sum of two squares?" A number of the
form in+ 3 cannot be the sum of two squares. This is an easy result, since when a
square is divided by 4 the remainder is either 0 or 1, and it is impossible to write 3
as the sum of two numbers, each of which is either 0 or 1. But Fermat proved the
much more difficult result that a prime number of the form 4n + 1 can be written
as the sum of two squares in exactly one way. Thus, 73 = 8^2 + 3^2 , for example.
The work of Fermat in number theory was continued by many mathematicians
in the eighteenth century. We shall discuss very briefly the lives and work of three
of them.
1.2. Euler. The Swiss mathematician Leonhard Euler (1707-1783) was one of the
most profound and prolific mathematical writers who ever lived, despite having lost
the sight of one eye early in life and the other later on. His complete works have
only recently been assembled in good order. In 1983, the two hundredth anniver-
sary of his death, many memorial volumes were dedicated to him, including an
entire issue (45, No. 5) of Mathematics Magazine. An even larger celebration is
planned for 2007, the three hundredth anniversary of his birth.^1 He spent most
of the years from 1726 to 1741 and from 1766 until his death in St. Petersburg,
Russia, where he was one of the first members of the Russian Academy of Sciences,
founded by Tsar Peter I (1672-1725) just before his death. From 1741 to 1766 he
was in Berlin, at the Prussian Academy of Sciences of Frederick II (1712-1788).
The exact date at which he made many of his great discoveries is sometimes diffi-
cult to establish, and different dates are sometimes given in the literature. Euler's
contributions to the development of calculus, differential equations, algebra, geom-
etry, and mathematical physics are enormous. The following paragraphs describe
some of his better-known results in number theory.
Fermat primes. Fermat had conjectured that the number F„ = 2^^2 "^ + 1 is always
a prime. This statement is true for ç = 0,1,2,3,4, as the reader can easily check.
For ç = 5 this number is 4,294,967,297, and to prove that it is prime using the
sieve of Eratosthenes, one must attempt to divide it by every prime less than
Fi = 65,537. In 1732 Euler found that this fifth Fermat number is divisible by
- No Fermat number beyond F4 has ever been shown to be prime, and well
over 200 are now known to be composite, including F2478782, discovered by John
Cosgrave and others at St. Patrick's College, Dublin, on October 10, 2003. The
smallest Fermat number not definitely known to be either prime or composite is
F33. The problem of Fermat primes is almost, but not quite, an idle question, that
is, one without connections to anything else in mathematics. The connection in
this case is that the regular polygons that have an odd number of sides and can
be constructed with straightedge and compass are precisely those whose number
of sides is a product of Fermat primes. Thus, until such time as another Fermat
number is proved to be prime, the only Euclidean-constructible regular polygons