Leonard Euler ( ) 1723 Euler obtains Master's degree in philosophy for the University of Basel having compared and contrasted the philosophical ideas of Descartes and Newton. Begins his study of theology completes his studies at the University First paper in print submitted an entry for the 1727 Grand Prize of the Paris Academy on the best arrangement of masts on a ship awarded 2 nd place awarded post in St. Petersburg also serving as a medical lieutenant in the Russian navy became professor of physics at the St. Petersburg Academy of Science promoted to senior chair of mathematics, when Bernoulli left and 1740 won the Grand Prize of the Paris Academy moved to the Berlin Academy of sciences on the invitation of Frederic the Great. During the twenty-five years spent in Berlin, Euler wrote around 380 articles 1766 Euler returned to St Petersburg after disputes with Frederic the Great Became totally blind. He produced almost half his work totally blind Died from a brain hemorrhage.

Leonard Euler ( ) Studied number theory stimulated by Goldbach and the Bernoullis had in that topic. 1729, Goldbach asked Euler about Fermat's conjecture that the numbers 2 n + 1 were always prime. if n is a power of 2. Euler verified this for n = 1, 2, 4, 8 and 16 and, by 1732 that the next case = is divisible by 641 and so is not prime. We owe to Euler the notation f(x) for a function (1734), e for the base of natural logs (1727), i for the square root of -1 (1777),  for pi,  for summation (1755), the notation for finite differences  y and  2 y and many others. Found a closed form for the sum of the infinite series  (2) =  (1/n 2 ). Moreover showed that  (4) =  4 /90,  (6) =  6 /945,  (8) =  8 /9450,  (10) =  10 /93555 and  (12) = 691  12 / And  (s) =  (1/ns) =  (1 - p -s )-1. Euler also gave the formula e ix = cos x + i sin x and ln(-1) =  i In 1736 Euler published Mechanica which provided a major advance in mechanics Also gave cases of n=3 (and n=4) for Fermat (with little mistakes), the Euler equation for of an inviscid incompressible fluid, investigating the theory of surfaces and curvature of surfaces, and much, much more …

Leonard Euler ( )

Euler’s Solution for Exponent Four He actually proves a little more: a 4 +b 4 =c 2 has no solutions. His proof from Elements of Algebra Thm: There are neither solutions to x 4 +y 4 =z 2 nor to x 4- y 4 =z 2 except if x=0 or y= We may assume x and y are relatively prime Outline of the strategy. Use infinite descent. I.e. from any solution produce a smaller solution. But there are no solutions for small numbers.

Leonard Euler’s :Elements of Algebra 205. The case x 4 +y 4 =z 2 (*) I.If x,y relatively prime, then (A) either both odd or (B) one is odd and the other is even. II. (A) is not possible: An odd square is of the form 4n+1 (If k=2m+1,k 2 =4(m 2 +m)+1). So the sum of two odd squares is of the form 4n+2. This means it is divisible by 2 and not by 4, so it is not a square. So the sum cannot be a square. But 4 th powers are also squares, so the equation cannot hold. III. (B) If (x,y,z) is a solution (x 2 ) 2 +(y 2 ) 2 =z 2, then there are p,q such that x 2 =p 2 -q 2 and y 2 =2pq. IV. Moreover, y is even and x is odd. Then p is odd and q is even. Proof: First since x 2 =p 2 -q 2, either p or q has to be odd and the other even. Also p cannot be even, since then p 2 -q 2 would be of the form 4n-1 or 4n+3 and cannot be a square. (If p=2k and q 2 =4m+1, the p 2 -q 2 =4(k 2 -m)-1). V.Now (x,q,p) is another Pythagorean triple x 2 =p 2 -q 2, so there are r, s s.t. p=r 2 +s 2, q=2rs, x=r 2 -s 2. VI.2pq=4rs(r 2 +s 2 )=y 2 so 4rs(r 2 +s 2 ) must be a square. Also r, s, and r 2 +s 2 have no common prime factors (why?). VII.If the statement of VI holds all the factors -r,s, r 2 +s 2 - must be squares (why?). So there are t, u such that r=t 2, s = u 2. Also r 2 +s 2 =t 4 +u 4 is a square and hence (t,u, (t 4 +u 4 ) 1/2 ) is a solution of (*). Since x 2 =p 2 -q 2 =(t 4 -u 4 ) 2, y=t 2 u 2 (t 4 -u 4 ) it follows that x,y>t,u. So that if (x,y) yields a solution we have another solution (t,u) which is smaller. VIII. Repeat step VII to obtain smaller and smaller. But there are no small solutions (except the ones with zeros). IX.But the smaller solutions from above are also non-zero. This yields a contradiction.

Euler and the case n=3 Consider x 3 +y 3 =z 3 with relative prime (x,y,z) 1.Exactly one of the integers is even. 2.Say z is even and so x,y odd. Then x+y=2p and x-y=2q are both even. 3.Factorize x 3 +y 3 =(x+y)(x 2 -xy+y 2 ). 4.Then by inserting x=p+q, y=p-q one finds that p,q have opposite parity are relatively prime and 2p(p 2 +3q 2 ) has to be a cube. 5.The same conclusion is also true if z is odd. 6.(A) 2p and (p 2 +3q 2 ) are relatively prime then they have to be cubes and Euler argues that there exist (a,b) such that p=a 3 -9ab 2, q=3a 2 b-3b 3. 7.Factor to obtain 2p=2a(a-3b)(a+3b) is a cube and show that the factors are relatively prime making them all cubes. Say 2a=  3,a-3b=  3,a+3b=  3. Then (  ) is a new smaller solution. 8.This also works in a variation if 2p and (p 2 +3q 2 ) are not relatively prime. The problem is in step 6. He does not show that this is the only way for (p 2 +3q 2 ) to be a cube.