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 xn + yn = zn has no solution for integers x, y, and z not equal to zero, when n >= 3 (Ore 204). Euler later proved this for n=3 in his Elements of Algebra, see the attached proof (Euler 449-50) (omitted in the original). This proof exploits a good understanding of the properties of numbers of the form a2 + 3b22, and therefore there is no reason as to believe that Fermat had not also proven his theorem for n = 3 (Burton 490). In a famous note of Fermat he wrote, "To divide a cube into two other cubes, a fourth power, or in any general power whatever into two powers of the same d enomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it" (Smith SB 213). This makes me wonder whether Euler was actually the first one to settle an important case of Fermat's Last Theorem.

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 y2 - Ax2 = 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 x2 - Ny2= 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 P1 = 6, P2 = 28, P3 = 496, P4 = 8128. Later writers made the following two conjectures:

  1. The nth perfect number Pn contains exactly n digits.
  2. 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 P5 = 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 P6 = 8,589,869,056 was seen to end in a 6, rather than an 8 (Burton 474-5).

Although Euclid's Elements dealt mainly with geometry, it was Euclid in Book IX, Proposition 36, who proved that if the sum 1 + 2 + 22 + 23 + ... + 2 (k-1) = p is a prime number, then 2(k-1)p is a perfect number (Burton 475, Euclid). Euler then made a claim about the occurrence of perfect numbers, he stated "I venture to assert that aside from the cases noted [Euler earlier mentioned 11, 23, 29, 37, 43, 73, 83], every prime less than 50, and indeed than 100, makes 2(n-1) * (2n) -1 a perfect number, whence the eleven values 1, 2, 3, 5, 7, 13, 17, 19, 31, 41 , 47 of n yield perfect numbers" (Burton 480).

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 230 * (231 -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 = x2 + Ny2 for N = 1, +/- 2, 3 with integers x and y. Around 1741, Euler began to study the following two questions:

  1. Given an integer N, describe the primes p = 2 for which p = x2 + Ny2 is solvable with integers x and y;
  2. 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 = x2 + Ny2, with integers x and y
(Chikara 69-70).

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!