Leonhard Euler was born in 1707 in Basel, Switzerland, and died in 1783. Over the course of his life he published many articles, some related to fields such as physics and astronomy, but many in the field of mathematics. In mathematics, three contributions he made are Euler’s Identity, his work on Fermat’s Last Theorem, and his solution to the Basel problem. Euler’s Identity is part of the mathematical field of Complex Analysis, which involves the application of various mathematical concepts for complex numbers a + bi, where i = √-1. Fermat’s Last Theorem is part of number theory, a field focused on the relationships between numbers, but primarily the integers. The Basel Problem involves summations, an important part of Calculus where it is used to solve difficult integrals, as well as other areas of Mathematics.
One of Euler’s more profound equations is Euler’s Identity, which states that eiΠ +1 = 0. This is a special case of Euler’s formula, which shows that eix = cos(x) + isin(x). Therefore, in the case of Euler’s Identity, x = Π, so eiΠ = cos(Π) + isin(Π) = -1 + i(0) = -1. By adding 1 to both sides, the standard result of eiΠ +1 = 0 becomes apparent.
This Identity has been considered by many to be “remarkable” and “among the most beautiful formulas in mathematics” for a variety of reasons. First, it includes both the number one and the number zero, the multiplicative and additive identities. Furthermore, it involves the constant Π, the most fundamental constant in geometry, and e, also known as Euler’s number, which is the base of the natural logarithm and can be used in many real world applications. Finally, there is i, the fundamental imaginary unit equal to the square root of negative one.
Euler also worked on Fermat’s Last Theorem. The theorem states that for the equation xn + yn = zn, there are no non-zero integer solutions for n equal to any value greater than two. Euler, in a letter to his friend Christian Goldbach, claimed to have a proof of the theorem for the case where n is equal to three. However, part of his proof was incorrect. He relied upon proving that for any numbers p and q where p2 + 3q2 is a cube, there exist numbers a and b where p2 + 3q2 = (a2 + 3b2)3. He incorrectly attempted to prove this using imaginary numbers, and as such invalidated his proof. Despite this, it could still be argued that he proved the case where n equals three as other mathematicians have used some of his other works to correct his mistakes. Additionally, Euler also proved the case of Fermat’s Last Theorem where n is equal to four.
The Basel problem is finding ∑n=1∞1⁄n2. Pietro Mengoli, an Italian priest, originally constructed the problem in 1644; in later years, both Jakob Bernoulli and Gottfried Leibniz, two prominent mathematicians of the time, were unable to solve it. This led to the Basel problem becoming a sort of challenge for mathematicians at the time. It is unknown when Euler first began work on the Basel problem; however, in 1731, he calculated an approximation of the first one thousand terms 1⁄12 + 1⁄22 + 1⁄32 + … + 1⁄10002. However, we have learned that his approximation of 1.64393 is only accurate to the first two decimal places. Euler continued to develop more and more accurate approximations, eventually ending up with the approximation 1.644934, which is accurate to six decimal places.
In 1734, Euler published his first proof that the Basel problem was in fact equal to Π2⁄6, which he had previously noticed was approximately equal to 1.644934, his most accurate approximation for the Basel problem. However, he made many logical pitfalls in this intial proof, lending it to the criticism of Daniel Bernoulli, a mathematician and one of his contemporaries. In 1741, Euler had completed his final proof, which remedied the problems of the first. I will now show how Euler proved this, using a result from an earlier proof that Π2⁄8 = 1 + 1⁄9 + 1⁄25 + 1⁄49 + …, which is required for this proof to be valid.
First, Euler expanded the sum and split it into fractions with even and odd denominators ∑n=1∞1⁄n2 = (1 + 1⁄9 + 1⁄25 + 1⁄49 + … ) + (1⁄4 + 1⁄16 + 1⁄36 + 1⁄64 + …). He then observed that the sum of fractions with even denominators is equal to one forth of the original sum (1 + 1⁄9 + 1⁄25 + 1⁄49 + … ) + (1⁄4 + 1⁄16 + 1⁄36 + 1⁄64 + …) = (1 + 1⁄9 + 1⁄25 + 1⁄49 + …) + (1⁄4)(1 + 1⁄4 + 1⁄9 + 1⁄16 + …). He already knew that the sum of the fractions with odd denomitors was Π2⁄8. He then substitiued it and the original sum (1 + 1⁄9 + 1⁄25 + 1⁄49 + …) + (1⁄4)(1 + 1⁄4 + 1⁄9 + 1⁄16 + …) = Π2⁄8 + 1⁄4 ∑n=1∞1⁄n2. Euler then subtracted one fourth of the original sum from both sides of the equation, and then multiplied both sides by four thirds to get the final value of Π2⁄6. Π2⁄8 + 1⁄4 ∑n=1∞1⁄n2 –> (3⁄4)∑n=1∞1⁄n2 = Π2⁄8 –> ∑n=1∞1⁄n2 = Π2⁄6
The Basel problem is a special case of the p-series ∑n=1∞1⁄np where p = 2. While we now know that the series diverges for p less than or equal to 1 and converges for p greater than 1. Euler worked on the p = 3 case, but his brute force approximations (as initially used in the Basel problem) did not yield any values he recognized. In 1978, it was proven that the number the p=3 case converges to is irrational, but nothing is known about the odd values of p greater than 3.
These are but three of Leonhard Euler’s many contributions in the field of Mathematics. I wrote about these three because I find all of them to be very interesting. Euler’s Identity, of course, has the many mathematical constants, and it amazes me how so many can be related in one equation. Fermat’s Last Theorem is perhaps the most infamous problem in Number Theory, as Fermat scribbled it in the margin of a journal, claiming to have an incredible proof but it would not fit in the margin. And The Basel Problem is the only p-series where the exact sum of the series is known.