1. Leonhard Euler 1707-1784

2. Leonhard Euler was born in Basel, but the family moved to Riehen when he was one year old and it was in Riehen, not far from Basel, that Leonard was brought up. Paul Euler, his father, had some mathematical training and he was able to teach his son elementary mathematics along with other subjects. Leonhard was sent to school in Basel and during this time he lived with his grandmother on his mother's side. This school was a rather poor one, by all accounts, and Euler learned no mathematics at all from the school. However his interest in mathematics had certainly been sparked by his father's teaching, and he read mathematics texts on his own and took some private lessons. Euler's father wanted his son to follow him into the church and sent him to the University of Basel to prepare for the ministry. He entered the University in 1720, at the age of 14, first to obtain a general education before going on to more advanced studies. Johann Bernoulli soon discovered Euler's great potential for mathematics in private tuition that Euler himself engineered. [1] Euler was terrified by the crotchety arrogant Bernoulli, but by the time Euler was coming of age Bernoulli realized that Euler had more insight in mathematics than himself. In 1727 he published another article on reciprocal trajectories and submitted an entry for the 1727 Grand Prize of the Paris Academy on the best arrangement of masts on a ship. This was amazing since Euler had never even seen an ocean going vessel. In 1727 Euler got an appointment to St. Petersburg University in Russia with the help of the Daniel Bernoulli, the son of his mentor. The Russians wanted to build a university to rival those in Paris and Berlin and wanted to attract the best possible talent to the university. Appointments at this time were limited and he was first appointed to medicine and physiology. He served briefly in the Russian navy as a medic. [2]

3. In 1733 Euler was appointed to a mathematical chair at St. Petersburg. Daniel Bernoulli held the senior chair in mathematics at the Academy but when he left St Petersburg to return to Basel it was Euler who was appointed to this senior chair of mathematics. The financial improvement which came from this appointment allowed Euler to marry which he did on 7 January 1734, marrying Katharina Gsell, the daughter of a painter from the St. Petersburg Gymnasium. Katharina, like Euler, was from a Swiss family. They had 13 children altogether although only five survived their infancy. Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing round his feet. [1] In 1741 Euler went to the Berlin Academy in Germany which was under the control of Fredrick the Great at that time. Fredrick did not have a very good impression of Euler, who he thought to be too much a scholar, refined and aloof. Euler did not enjoy his stay in Berlin and returned to St. Petersburg after a very productive 25 years where he lived until his death. [2] Some of Euler's hobbies include growing vegetables and cartography (the making of maps). Cartography was another area that Euler became involved in when he was appointed director of the St Petersburg Academy's geography section in 1735. He had the specific task of helping Delisle prepare a map of the whole of the Russian Empire. The Russian Atlas was the result of this collaboration and it appeared in 1745, consisting of 20 maps. Euler, in Berlin by the time of its publication, proudly remarked that this work put the Russians well ahead of the Germans in the art of cartography. [1] Euler's ability to concentrate on mathematical ideas and calculations were amazing. He was able to carry out calculations to 50 decimal places of accuracy in his head. He memorized the first one hundred prime numbers along with their squares and cubes. He was able to calculate values such as 241 4 and 337 6 with complete accuracy in his head. [2]

4. Euler was also noted for being an extremely good teacher. He would incorporate relevant examples in his work along with his theories and proofs. In his work Introductio in Analysin Infinitorum he published a volume on differential calculus in 1755 and three volumes on integral calculus between 1768 and 1774. Students enjoyed learning from him, it was once remarked that: "He preferred instructing his pupils to the little satisfaction of amazing them." [2] Euler's publication backlog lasted for 47 years after his death. He published in the areas of acoustics, engineering, mechanics, astronomy and a three volume treatise on optical devices such as microscopes. Euler pioneered whole branches of mathematics such as graph theory, complex analysis, calculus of variations and differential equations. [2] [1] Web site: MacTutor History. http://turnbull.mcs.st-and.ac.uk/~history (9/2000).http://turnbull.mcs.st-and.ac.uk/~history [2] Dunham, William. Journey Through Genius The Great Theorems of Mathematics. New York: John Wiley & Sons 1990.

5. Euler is know for many innovations in the area of mathematics. What Euler did spans though generations of mathematics from the ancient Greek Mathematicians to a point that modern day mathematicians are required to know his results. I will first mention his connections with some older works. A good example is what is sometimes known as the Euclid-Euler theorem on perfect numbers. Euclid's theorem on perfect numbers If 2 n -1 is prime and N = 2 n -1 (2 n -1) then N is perfect. An open question for years is if this completely described all perfect numbers. All the perfect numbers that were known followed this pattern, but did they have to? The converse of this statement would be the following: If N is perfect then N = 2 n -1 (2 n -1) where 2 n -1 is prime. If this statement could be proven this would completely classify all perfect numbers. No one has ever been able to prove this even to this day. Notice if this were true it would say that all the perfect numbers were even. Euler was able to prove a weaker version of the converse. If you assumed a number was both even and perfect you could then conclude it could be written in a certain format. This is sometimes known as the Euclid-Euler Theorem.

6. Euclid-Euler Theorem If N is perfect and N is even then N = 2 k -1 (2 k -1) where 2 k -1 is prime. Euler's insight came with the development of a certain function  ( n ). This function is defined as follows:  ( n ) = The sum of all divisors of n. For example:  (2)=1+2=3  (3)=1+3=4  (4)=1+2+4=7 Euler first proved the following results about this function. i. If n >1,  (n)>n+1 ii. p is prime if and only if  ( p )= p +1 iii.  (2 r )=2 r +1 -1 iv. N is perfect if and only if  ( N )=2 N v. If p and q are distinct primes,  ( pq )=  ( p )  ( q ) vi. If a and b are relatively prime (i.e. gcd( a, b )=1), then  ( ab )=  ( a )  ( b ) Proof : Suppose N is even and perfect, then N = 2 k -1 b where b is odd and k >1 (i.e. factor the 2's out of N ). Because N is perfect, we know that  ( N ) = 2 N = 2(2 k -1 b ) = 2 k b. Because 2 k -1 and b are relatively prime, we know that:

7. Equating the expressions for  ( N ) we get: Letting the number we get the following:The inequality c > 1 needs some explanation.

8. With the inequality c > 1 there are two cases to consider. CASE 1 ( c > 1) With some explanation Euler showed that 1, b, c and 2 k -1 are four different divisors of b, Euler deduced that each appears in the summand when calculating  ( b ). From this we get the following: CASE 2 ( c = 1) We know that b = c (2 k -1) = 2 k -1 and substituting we get: It is not possible to have  ( b )>  ( b ), so therefore CASE 1 is impossible. and therefore b is prime. In conclusion the only possible is CASE 2 in which we have if N is an even perfect number, then

9. The previous theorem by Euler completely characterizes all of the even perfect numbers. What about the odd ones? This question is still unanswered to this day. Mathematicians do not even know if an odd perfect number exists. We do know a few things about this: 1. (Sylvester, 1888) An odd perfect number must contain at least 3 different prime factors. 2. No odd perfect number is divisible by 105. 3. The smallest odd perfect number must exceed 10300. 4. The second largest factor of an odd perfect number is greater than 1000. 5. The sum of the reciprocals of an odd perfect number is finite. The development of mathematics is taking a significant turn at this point. Because of the work of Newton and Leibnitz in the development of "The Calculus" problems that seemed unsolvable before could not be done. Two significant areas of study begin to surface. 1. The study of the infinite. Infinite sums are of particular interest because of their ability to estimate values that could not be estimated before. 2. The idea of what a function was in terms of a formula/curve/rule. A shift is made from geometric thinking to algebraic thought especially in light of Desarte's work in analytical geometry. This gave more insight into how you could think of a function.

10. Some preliminary concepts about infinite sums that are already known at this time. Geometric Series It was know through a geometrical argument that : This could also be established algebraically: Let: Subtract the first from the second: This method worked for functions in which this algebraic or geometric relationship could be found. The transcendental functions (i.e. e x, sin ( x ), cos ( x ), ln (1+ x )) had no such relationship. They were related through analytical/geometrical properties. The ability to "calculate" values for these were constrained by certain know facts. For example the sin ( x ) could not be estimated for any value because it was impossible to construct certain angles. Calculus changed all that. It was the computer of its day giving mathematicians calculational abilities never before seen. Take the derivative of sin ( x ) to get cos ( x ):

11. A formula for the - ln (1- x ) can be found by integrating both sides of the formula for the sum of a geometric series. Multiplying both sides by -1 we get a formula for ln (1- x ). If we replace x by - x we get the formula for ln (1+ x ) The concept of an infinite sum was not a new idea. People in previous centuries had looked at various infinite sums and had been able to say something about them (the Geometric series for example). A famous infinite sum is known as the Harmonic Series. This is the sum of the reciprocals of the natural numbers: We will show 3 different arguments for why the harmonic series adds up to be infinity (i.e. we say diverges) to demonstrate how mathematics is developing.

12. An Italian by the name of Mengoli was one of the first to formally establish that these numbers will sum to infinity in 1647. If a sum of numbers does not add up to be any particular number in modern times we say the series diverges. Mengoli's proof the harmonic series diverges Proof: He starts by establishing a lemma. or Now consider the following differences between the two numbers at the extremes and the number in the middle. Since a ( a +1) > a ( a -1) the denominator of is larger than the denominator of This means that x < y. The expression is the average of the three numbers. Since the distance y is bigger, the average will be weighted toward it. If the distances were the same the average would be 1/ a, but because y is larger the average is greater than 1/ a.

13. Mengoli completes his proof by applying the inequality to the sum of the harmonic series which he calls H. Since the is no finite number for which H > 1+ H the number H must be infinite. (uses geometry concepts) Bernoulli proves this by making a term by term comparison. (Using algebra concepts.) Since there is no finite number for which H > H it must be the case that H is infinite. Euler proves this by using the power series expansion for the logarithm. This is of course uses calculus ideas.

14. As mathematics progressed many mathematicians saw the importance of studying infinite sums. They were an important tool in how you could represent and then compute the values for some very important functions. Euler was not only able to make great progress on this topic, but showed how the ideas and methods developed in calculus could be applied to do it. A natural question that arises is what the sum of the reciprocals of the numbers that are perfect squares. Bernoulli was able to show this series added up to a finite number and that number was less than 2, but was not able to find the sum expressed as a combination of know numbers. Euler was able to apply the results of calculus and prove it is the number  2 / 6. Euler's Proof of: (Sketch) Look at the function:

15. To understand how Euler viewed this function as a product is more complicated. It depended on two main ideas. 1. The "roots" of f(x) are: , - , 2 , -2 , 3 , -3 ,… 2. The value of f(x) when x = 0 should be 1. A well-known limit from beginning calculus. Euler thought of how f(x) could be factored so that it would have the specified roots and the value at 0 would be 1. The factor has value 1 when x = 0 and a root of  The factor has value 1 when x = 0 and a root of -  The factor has value 1 when x = 0 and a root of 2  The factor has value 1 when x = 0 and a root of 2 

16. Euler then looks at what you would get if you multiplied out ("foiled out") the expression on the right. The first term is of course 1, but there are no x terms and the x 2 terms all come from being multiplied by 1. Equating the coefficients of the x 2 term from the power series and product expansions we get the following:

17. Euler did not just stop here. He went on to find the sum of the reciprocals of the even perfect squares by multiplying both sides by ¼. Euler finds the sum of the reciprocals of the odd perfect squares by subtracting the sum of the reciprocals of even perfect squares from all perfect squares. Euler was also able to illustrate a method that shows how if you know the sum of the reciprocals squares it can be applied to find the reciprocals of the fourth powers. Euler was able to show how this method generalized to find the sum of the reciprocals of any even powers by using the formulas for the even powers smaller than the one you want to find. The sum of the reciprocals of the odd powers is still an open question today.

18. One of the most innovative questions Euler asked and resolved has to do with how prime numbers are distributed. He wondered if the reciprocals of prime numbers added up to be a finite or infinite amount. Euler pioneered the field of analytic number theory. This is an area of mathematics where the techniques of calculus (i.e. analysis) are applied to solve problems about numbers. If it is finite it means the prime numbers occur infrequently like squares for example, if it is infinite they occur often like even numbers. Not only was Euler innovative enough to pose such an important question he was also able to resolve it. Even numbers are infinite. Perfect squares are finite. Prime numbers finite or infinite?

19. Euler begins by proving a theorem that shows a certain product of primes is equal to the harmonic series. Theorem: p is a prime number Proof: Let M be the sum of the harmonic series: Start with the prime 2. Notice no denominator is a multiple of 2. Use the prime 3. Notice no denominator is a multiple of 2 or 3. Do the same for the prime 5 and you have no denominator that is a multiple of 2, 3 or 5.

20. Euler says to carry out this process for every prime number p. You get: QED Euler then goes to prove the following result about the sum of reciprocals of prime numbers. Theorem: Proof:(outline) M is the sum of the harmonic series again. Take the ln and use the properties of ln. Plug each term in the power series expansion

21. Regroup the terms so that you add going down the columns.

22. Euler expresses this sum the following way: Where Euler then utilizes the methods of calculus to get an upper value for the numbers B, C, D, E, … by refining the integral test for series. He then uses this to find an upper estimate for the tail part of the series. The point here is that the terms with B, C, D, E, … add up to be a finite amount.

23. Euler now considers what happens when each side of the series is made the exponent of e. The term on the left is M the sum of the harmonic series which was know to be infinite, but ½ B +⅓ C +¼ D +… is finite and thus e ½ B +⅓ C +¼ D +… is finite as well. Euler says that the "infiniteness" of the right side of the equation must come from somewhere and thus concludes that e A is infinite. He then says that A must be infinite because if A were finite then e A would also be finite. QED