Leonhard Euler (1707-1783)   Historical sketch

It is in tones of awe that mathematicians speak of Euler.

Leonhard Euler was the most prolific writer of mathematics ever. He made major contributions to geometry, calculus, and number theory; and he developed mathematical analysis, building on Leibniz's differential calculus and Newton's method of fluxions. He is credited with introducing the notation f(x) for a function (1734), establishing the letter e for the base of natural logarithms (1727), the letter i for the square root of -1 (1777), the Greek letter p for the ratio of the diameter of a circle to its circumference, and the Greek letter S for summation (1755).

Euler was born in Basel, Switzerland. His father had been an undergraduate student with Johann Bernoulli at the University of Basel, though his father was studying for the ministry. But his father did attend some mathematics lectures by Jacob Bernoulli, and the young Leonhard benefitted from his father's teaching thereof.

To indulge his father's wishes, Leonhard entered the University of Basel at age 14 to study for the ministry. Professor Johann Bernoulli at Basel was able to see and widen the mathematical possibilities for the youngster, providing him with advice and occasional weekend assistance. Such familial connections also afforded the young Euler to associate with two of Bernoulli's sons, Nicolaus and Daniel, who themselves became mathematicians. At age 16, Euler was awarded a Masters Degree in Philosophy and undertook studies in theology. But with Johann Bernoulli's help, he persuaded his father to let him pursue mathematics instead, obtaining his doctorate under Johann Bernoulli at age 19.

The next year, he arrived in St. Petersburg, Russia, to assume a post at the mathematical-physical division of the St. Petersburg Academy of Sciences, becoming Professor of Physics at age 23, then progressing to become the Academy's chief mathematician three years later. Shortly afterward, he married the daughter of a local painter, who was from a Swiss family. They eventually had 13 children. The St. Petersburg Academy established a research journal, and it was soon replete with research papers by Euler, many of which it is said that he wrote while he played with his children.

At the young age of 28, Euler lost the sight in his right eye, but his research output continued unabated in spite of this loss. Indeed, the very next year, he summed an infinite series that had dumbfounded his predecessors: the sum of the infinite series

The mathematician Henry Oldenburg posed this problem (called the Basel problem) to Gottfried Leibniz in 1673, but evidently Leibniz was unable to answer. Nor was Jacquez Bernoulli able to find the sum in 1689. But Euler, in about 1736, finally found its sum by a remarkable route, proving that

   

Euler was a genius at both calculations and in noticing connections, among which is the remarkable equation

   

connecting the transcendental numbers e and p with the imaginary unit i and the most basic numbers 0 and 1.

In 1741, at age 34 and after having won the Grand Prize of the Paris Academy for the second time, Euler was persuaded, partly by salary and partly by political unrest in Russia, to join the Berlin Academy of Science, where he spent 25 years and continued with his huge production of mathematical research.

In 1766, Euler returned to St. Petersburg, and through an illness, became almost blind. A few years later, a fire destroyed his home, but he was able to save his mathematical papers. Soon thereafter, he became completely blind. But that did not stop him, and he produced about half his lifetime production of mathematics after his blindness. He died in 1873 of a brain hemorrhage. After his death, it took the St. Petersburg Academy some 50 years to complete the publication of his unpublished works, totalling some 70 volumes.

1. How many terms of the series V(2) does it take to approximate p˛ to the nearest hundred thousandth?

2. Euler also proved that

  

Using this and the sum for p˛/6, show that

   

3. What other sums can you deduce from Euler's result and the previous problem?

4. Fermat conjectured that all the integers of the form 2²ⁿ + 1 are prime, where n is a positive integer. Try some values of n and see what you think of this conjecture. Euler proved it false for n = 5, and you can do the same with the help of a computer algebra system, but imagine doing this with pencil and paper.

5. Do some reading on what is called the Königsburg Bridge Problem, and tell about Euler's solution.

6. A common function for those who study number theory is the Euler phi-function. It is defined for each positive integer n to be the number of integers less than or equal to n that are relatively prime to n. (Two integers are relatively prime if they have no prime factor in common.) Thus, f(6) = 2 because the integers less than or equal to 6 and relatively prime to 6 are 1 and 5. And f(8) = 4 because the only positive integers less than or equal to 8 that are relatively prime to 8 are 1,3,5, and 7.

a. What is f(p) where p is a prime number?

b. What is f(pq) where p and q are distinct primes?

c. Investigate this phi-function for a bit and see if you can notice any other properties of it.