Euler's work in Number Theory

Jodi Dunkelman

History of Mathematics

Rutgers,
Spring 2000

Leonhard Euler made many contributions to the field of mathematics, including his work in number theory. This Swiss mathematician spent most of his working life in Russia, where his number theoretic work was suggested by issues raised by Pierre de Fermat, as well as his own ideas. Euler's work in number theory included topics such as the study of perfect numbers, the quadratic reciprocity law, the so-called Pell equation, and Fermat's Last Theorem, to name just a few. Although Euler did not initiate the study of many of the problems that he worked on, nor did he solve any completely, but he made great contributions for himself and for many other mathematicians. Leonhard Euler was born on April 15, 1707, in Basel, Switzerland, and died on September 18, 1783, in St. Petersburg, Russia. Growing up in Riehen, a town not far from Basel, Euler was taught mathematics by his father because it was not taught in the school that he was attending. Johann Bernoulli had discovered the mathematical talent that young Euler was blessed with. In 1725, at the mere age of 18, Euler was already studying at the University of Basel, and the next year, he was completing his graduate studies there. In the autumn of 1726, Euler was offered a position in the Russian Academy of Sciences in St. Petersburg. It may sound a little strange that such a distinguished foreign academy would be aware of young Euler in Switzerland, but when Bernoulli's sons, Nicholas II and Daniel, had been accepted in 1725, they had made an agreement between themselves that whenever an open seat would come up, Euler would be the first one that they would recommend. As a result Euler arrived in St. Petersburg on May 17, 1727, when he was only 20 years old (Calinger 124-5). It is in this period that the bulk of his work in number theory was undertaken.

Number theory is defined in Encarta Encyclopedia as "a branch of mathematics that deals with the properties and relationships of numbers. The theory of numbers includes much of mathematics, particularly mathematical analysis. Generally, the theory of numbers is confined to the study of integers, or occasionally to some other set of numbers having properties similar to the integers." Number theory contains very basic ideas, but it can also be difficult to prove and understand. There are problems that can be written down easily, but whose answers still amaze the most distinguished mathematicians. All in all, number theory has been of interest to mathematicians since numbers were first found to be curious (Ore 25). On December 1, 1729, conference secretary
Christian Goldbach first asked Euler, in a letter, if he knew
of the conjecture of Fermat (Calinger 130, Winter 10);
this refers to Fermat's statement that the equation
*x ^{n} + y^{n} = z^{n}*
has no solution for
integers

Another one of Fermat's ideas that Euler worked came to be
known, by Euler's own mistake, as Pell's Equation. Pell's equation is
y^{2} - Ax^{2} = 1,
where A is any non-square integer. This problem was
first proposed Fermat as a challenge to English mathem aticians Lord
Brouncker and John Wallis (Smith SB 214). Euler got the impression
from Wallis that Pell was given the acknowledgement of finding the
method to solve this problem, and after he had become aware of his
mistake in 1730, at the age of 23, and then included it in his
Introduction to Algebra that was written in 1770 (Edwards 33). Euler
probably got confused because "Pell's name occurs frequently in
Wallis's Algebra, but never in connection with the equation
*x ^{2}
- Ny^{2}= 1* ... ; since its traditional designation as
`Pell's equation' is unambiguous and convenient, we will go on using
it, even though it is historically wrong" (Weil 174).

A *perfect
number* is a positive integer that is equal to the sum of all its
positive proper divisors, of aliquot parts. For example, 6 = 1 + 2 +
3, and 28 = 1 + 2 + 4 + 7 + 14, which yields 6 and 28 to be the first
two perfect numbers (Burton 474). Around 100 AD
Nicomachus of Gerasa listed the first four perfect numbers as
*P _{1}* = 6,

- The
*n*th perfect number*P*_{n}contains exactly*n*digits. - The even perfect numbers end alternately in the digits 6 or 8.

These conjectures were proven wrong when the 5th perfect number
was correctly given in an anonymous 15th Century manuscript as
*P _{5}* = 33,550,336, which obviously does not have 5 digits,
so the first conjecture can now be discarded.
The second conjecture was refuted
when the sixth perfect number

Although Euclid's *Elements* dealt mainly with geometry, it
was Euclid in Book IX, Proposition 36, who proved that if the sum *1
+ 2 + 2 ^{2} + 2^{3} + ... + 2 ^{(k-1)} =
p* is a prime number, then 2

Although Euler was wrong, he
had found his own mistakes with *n = 41* and *n = 47*, and
corrected them in 1753 (Ore 93). In 1732 Euler also discovered the
8th perfect number, which is 2^{30} * (2^{31} -1) =
2,305,843,008,139,952,128, also know as the Mersenne prime *M31*
(Ore 93, Barlow). Thus perfect numbers have been a topic of interest
for many years; to this day, no mathematician has been able to
determine whether there are finitely or infinitely many perfect
numbers. Mathematicians make empirical conjectures that they believe
to be true, but through counterexamples may find them to be false.
Burton remarks, "Part of the
problem is that in contrast with the single formula for generating
perfect numbers (even), there is no known rule for finding all
amicable pairs of numbers (Burton 483).

The Quadratic Reciprocity
Law was first formulated by Euler and Legendre, and was later
proved by Gauss and
partly by Legendre. Once again, Euler was pushed towards quadratic
forms by the start of Fermat's investigations on primes *p* represented
by *p = x ^{2} + Ny^{2}* for

- Given an integer
*N*, describe the primes p = 2 for which p = x^{2}+ Ny^{2}is solvable with integers*x*and*y*; -
Given an integer
*N*, describe the primes*p = 2*for which*p*divides*m**(p|m)*, where*m*is any integer of the form*m = x*, with integers^{}2 + Ny^{}2*x*and*y*

Another explanation
of the law of quadratic reciprocity is given as follows:

"If *p* and *q*
are two positive odd primes, at least one of which has the form
*4n + 1*, then *q* is a quadratic residue
or nonresidue of *p* according as *p* is
a quadratic residue or nonresidue of *q*. But if both the primes *p*
and *q* have the form *4n + 3*, *q*
is a quadratic residue or nonresidue of *p*
according as *p* is a quadratic nonresidue or residue of *q*"
(Dirichlet 66).

As one example of this take *p =3*, *q = 5 ; then p
is a quadratic
nonresidue of q and at the same time q
is a quadratic nonresidue of p.
(Dirichlet 66). *

These are only a select few of Euler's accomplishments in number theory. He made other contributions to number theory, as well as to other branches of pure and applied mathematics. His end, as quoted by Yushkevich, came as follows. "On September 18, 1783, Euler spent the first half of the day as usual. He gave a mathematics lesson to one of his grandchildren, did some calculations with chalk on two boards on the motion of balloons; then discussed with Lexell and Fuss the recently d iscovered planet Uranus. About five o' clock in the afternoon he suffered a brain haemorrhage and uttered only 'I am dying' before he lost consciousness. He died about eleven o'clock in the evening."

This is how Euler based his life around mathematics - each day to the fullest!

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