Leonard Euler (1707-1783)

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Leonard Euler (1707-1783)

• 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.
• 1726 completes his studies at the University.
• 1726 First paper in print.
• 1727 submitted an entry for the 1727 Grand Prize
  of the Paris Academy on the best arrangement of
  masts on a ship awarded 2nd place.
• 1726 awarded post in St. Petersburg also serving
  as a medical lieutenant in the Russian navy.
• 1730 became professor of physics at the St.
  Petersburg Academy of Science.
• 1733 promoted to senior chair of mathematics,
  when Bernoulli left.
• 1738 and 1740 won the Grand Prize of the Paris
  Academy.
• 1741 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.
• 1771 Became totally blind. He produced almost
  half his work totally blind.
• 1783 Died from a brain hemorrhage.

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Leonard Euler (1707-1783)

• Studied number theory stimulated by Goldbach and
  the Bernoullis had in that topic.
• 1729, Goldbach asked Euler about Fermat's
  conjecture that the numbers 2n 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 232 1 4294967297 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), p for pi, S
  for summation (1755), the notation for finite
  differences Dy and D2y and many others.
• Found a closed form for the sum of the infinite
  series
• z(2) S (1/n2). Moreover showed that z(4)
  p4/90, z(6) p6/945, z(8) p8/9450, z(10)
  p10/93555 and z(12) 691p12/638512875. And z
  (s) S (1/ns) P (1 - p-s)-1.
• Euler also gave the formula eix cos x i sin x
  and ln(-1) pi
• In 1736 Euler published Mechanica which provided
  a major advance in mechanics
• Also gave cases of n3 (and n4) 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

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Leonard Euler (1707-1783)
4
Eulers Solution for Exponent Four

• He actually proves a little more
• a4b4c2
• has no solutions.
• His proof from Elements of Algebra.
• 202. Thm There are neither solutions to x4y4z2
  nor to x4-y4z2 except if x0 or y0.
• 203. We may assume x and y are relatively prime.
• 204. Outline of the strategy. Use infinite
  descent. I.e. from any solution produce a smaller
  solution.
• But there are no solutions for small numbers.

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Leonard Eulers Elements of Algebra

• 205. The case x4y4z2 ()
• If x,y relatively prime, then (A) either both odd
  or (B) one is odd and the other is even.
• (A) is not possible An odd square is of the
  form 4n1 (If k2m1,k24(m2m)1). So the sum of
  two odd squares is of the form 4n2. 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 4th
  powers are also squares, so the equation cannot
  hold.
• (B) If (x,y,z) is a solution (x2)2(y2)2z2,
  then there are p,q such that x2p2-q2 and
  y22pq.
• Moreover, y is even and x is odd. Then p is odd
  and q is even. Proof First since x2p2-q2,
  either p or q has to be odd and the other even.
  Also p cannot be even, since then p2-q2 would be
  of the form 4n-1 or 4n3 and cannot be a square.
  (If p2k and q24m1, the p2-q24(k2-m)-1).
• Now (x,q,p) is another Pythagorean triple
  x2p2-q2, so there are r, s s.t. pr2s2,
  q2rs, xr2-s2.
• 2pq4rs(r2s2)y2 so 4rs(r2s2) must be a square.
  Also r, s, and r2s2 have no common prime factors
  (why?).
• If the statement of VI holds all the factors
  -r,s, r2s2- must be squares (why?). So
  there are t, u such that rt2, s u2. Also
  r2s2t4u4 is a square and hence (t,u,
  (t4u4)1/2) is a solution of (). Since
  x2p2-q2(t4-u4)2, yt2u2(t4-u4) it follows that
  x,ygtt,u. So that if (x,y) yields a
  solution we have another solution (t,u) which is
  smaller.
• Repeat step VII to obtain smaller and smaller.
  But there are no small solutions (except the ones
  with zeros).
• But the smaller solutions from above are also
  non-zero. This yields a contradiction.

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Euler and the case n3

• Consider x3y3z3 with relative prime (x,y,z)
• Exactly one of the integers is even.
• Say z is even and so x,y odd. Then xy2p and
  x-y2q are both even.
• Factorize x3y3(xy)(x2-xyy2).
• Then by inserting xpq, yp-q one finds that p,q
  have opposite parity are relatively prime and
  2p(p23q2) has to be a cube.
• The same conclusion is also true if z is odd.
• (A) 2p and (p23q2) are relatively prime then
  they have to be cubes and Euler argues that there
  exist (a,b) such that pa3-9ab2, q3a2b-3b3.
• Factor to obtain 2p2a(a-3b)(a3b) is a cube and
  show that the factors are relatively prime making
  them all cubes. Say 2aa3,a-3bb3,a3bg3. Then
  (b,g,a) is a new smaller solution.
• This also works in a variation if 2p and
  (p23q2) are not relatively prime.
• The problem is in step 6. He does not show that
  this is the only way for
• (p23q2) to be a cube.