Leonard Euler

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• Leonard Euler

(1707-1783)
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Ge??????e 15 Ap?????? 1707 ?as??e?a, ??ßet?a
???a?e 18 Septeµß???? 1783 (?????a76)???a ?et???p???.
?at????a ???ss?a ??s?a S???d?a
??????t?ta ??ßet??
?ed?a ?a??µat??? F?s??? ?st????µ?a
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???G??F??

• 1707-1727
• Se ?????a 13 et?? ???ft??e st? pa?ep?st?µ?? t??
  ?as??e?a? ?a? se ?????a 16 ?????? p??e t? Master
  t?? µe µ?a d?at??ß? st?? ?p??a s??????e t??
  f???s?f?e? t?? ?a?t?s??? ?a? ?e?t??a. ?a
  ap??e?µata t?? Saßß?t?? ??a?e ?d?a?te?a µa??µata
  µa??µat???? ap? t?? ?????e?e?a?? f??? Bernoulli ?
  ?p???? s??t?µa a?a?????e t? ta???t? t?? ???? t??
  µa??t?.
• Se ?????a 19 et?? te?e??se t? d?da?t?????
  d?at??ß? t?? st? d??d?s? t?? ???? µe t?t?? De
  Sono.
• ??a?e µ?a a?t?s? ??a ?a???es?a st? f?s???
  st? pa?ep?st?µ?? t?? ?as??e?a?, ????? ep?t???a.

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• 1727-1741
• ?et? ap? p??tas? t?? f???? t?? Daniel Bernoulli,
  ???? t?? Bernoulli, ? ?p???? ?d? d?das?e st??
  ??s??? a?ad?µ?a t?? ep?st?µ?? t?? ???a?
  ?et???p????, ta??de?e st?? ??s??? p??te???sa ?a?
  ?at??aße µ?a ??s? t?? F?s??????a?.
• St? s????e?a p?????e? se µ?a ??s? st? tµ?µa
  ?a??µat????.
• ?? 1731 se ?????a 24 et?? ????e ?a????t??
  t?? f?s????, µet? ap? d?? ?????a ?a? ???? t??
  ap?????s?? t?? Daniel ????e ? p??ed??? t??
  tµ?µat?? ?a??µat????.

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• 1741-1766
• ???? t?? p???t???? a?ata?a??? st? ??s?a fe??e?
  ap? t?? ???a ?et???p??? ?a? ?ata?aµß??e? µ?a ??s?
  st?? ??ad?µ?a t?? ?e??????? p?? t?? p??sfe?e ?
  F?e?de????? ? t?? ???ss?a? ?a? a?a?aµß??e? t?
  ??s? t?? d?e????t? t?? tµ?µat?? t?? µa??µat????
  t?? ??ad?µ?a?.
• ????e ?a???st?? ??a t?? e??as?e? t??
• Introductio in analysin infinitorum
• Institutiones calculi differentialis
• ?s? ß??s??ta? st? Ge?µa??a t?f?????e e?te???
  a??? pa???a a?t? de? µe?????e st? e????st? ?
  ap?d?s? t??. ????sta µet? t?? t?f??s? t??,
  pa???a?e
• t? µ?s? ap? t? s??????? ep?st?µ?????
• ???? t?? µe t??µe???? ???µ??? p?? ?fta?a? t?
  1775 se ??a paper ???e eßd?µ?da!

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• 1766-1783
• ? p???t??? ?at?stas? st? ??s?a ????se ?a
  ße?t???eta?
• ap? t? st??µ? p?? st? ????? a??ß??e ?
  ???ate???? ? ?e????. ?et? ap? µ?a p??s???s?
  ep?st?e?e st?? ??ad?µ?a t?? ???a? ?et???p????.
• ???a?e 18 Septeµß???? µet? ap? ??a
  e??efa???? epe?s?d??.

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?????? ?? T?????S??

• ? Laplace ??e?eltlt ??aß?ste Euler, d?aß?ste Euler
  e??a? ? ?a????t?? ???? µa?gtgt
• ??µ?s?e?se pe?? ta 900 s?????µµata, µe??te? ?a?
  ß?ß??a (148 e??as?e? µ??? ??a t? Te???a ????µ??),
  ta ?p??a ?at? ??a µ???? d?µ?s?e?t??a? 50 ?????a
  µet? t? ???at? t??.
• ?ta? ??a? s????? e??a??µe??? ep?st?µ??a? ?a? ??a?
  ap?? ???s?e??µe??? ?????p??.

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? ???SF??? ??? S?? ??T???????

• ?? e??as?e? t?? ?e?????? µe? ap? t?? ?pe???st???
  ????sµ?, t? Ge?µet??a, t?? ????µ??e???a ?a? t??
  ???eß?a, ?ata?????? de st?? ???a???e???a, t?
  ?e???t??? ?a? efa?µ?sµ??? ???a????,
• t? F?s???, t?? ?st????µ?a ?a??? ?a? t?
  F???s?f?a.
• St?? ???eß?a ? Euler p??s??ese µe????? ??e?
  µe??d??? ep???s?? t?? t??t?ßa?µ??? ?a?
  teta?t?ßa?µ??? e??s?se?? ?a?, pe?a?t???, ??a?e
  t?? esfa?µ???, ?p?? ap?de??e a???te?a ? Abel ,
  ?p??es? p?? ?? ???e? ???e a??eß????? e??s?s??,
  ?p????d?p?te ßa?µ??, µp????? ?a e?f?ast??? µ?s?
  e??? pepe?asµ???? p?????? p??s?afa???se??,
  p???ap?as?asµ??, d?a???se?? ?a? e?a????? ?????.

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• ? Euler ?ta? pepe?sµ???? ?t? ???e p??????µ?
  a?a??eta? se p??t?ß??µ???? pa?????te?, ?p??
  d????e? se µ?a ep?st??? t?? p??? t?? Goldbach, a?
  ?a? a?a??????e ?t? ??p?? e?e? ???ß?ta? ??a p???
  s?µa?t??? ?e???µa, t? ???st? se eµ?? Teµe???de?
  Te???µa.
• ?????te?? s????aµµa t?? Euler ep? t?? ??µat??,
  e?t?? t?? ?p????p?? 166 µ????te??? ?????
  e??as???, e??a? ? ???eß?a , ???? t? ?p???
  ???ft??e ?ta? ?ta? ?d? t?f??? ?a? t? ?p???
  p?a?µate?eta? ??µata ap? t?? a???µ?t???? p???e??
  µeta?? pa?t?e?d?? a???µ?? ?a? t? ?e???a a?a??????
  ?a? p???d?? ?? ta st???e??d? a?apt??µata se
  se???? ?a? t?? ap??sd????st? a????s? p??t?ßa?µ???
  ?a? de?te??ßa?µ??? e??s?se??.

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• ???? t?? ep?d?as?? t?? Bernoulli ? µe??t? t??
  ?pe???st???? ????sµ?? ????e ap? t??? p??t???
  st????? t?? ??e???? t??. S???? ???s?µ?p????se
  se???? d???µe?? ??a ?a e?f??se? t?? s??a?t?se??
  sa? a????sµata ape???? se????.

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• ????e ??a t??p? ?a ?p??????e? ????????µata µe
  µ??ad??? ???a p??µ?????ta? t?? µe???? p???d? p??
  ?a e??e ? ???ad??? ?????s?.
• ?p?s?? efe??e t? ????sµ? t?? d?af???? ?a?
  ap?de??e t? p??t? sp??da?? ?e???µa st? ?e???a
  ??af?µ?t??. ???µa, a??pt??e e??a?e?a p??
  ?a??st??sa? t? ????sµ? efa?µ?s?µ? se p??ß??µata
  t?? f?s???? ?a? t?? ?a??µe????? p?a?µat???t?ta?.
• ??t?? ap? t?? ep?t??? efa?µ??? t??
  a?a??t???? µe??d?? se p??ß??µata t?? ??ass????
  µ??a????? ? Euler ef??µ?se t?? te?????? a?t?? se
  ast????µ??? p??ß??µata.
• ?? s??e?sf???? t?? st?? ast????µ?a
  pe???aµß????? t?? a???ß? p??sd????sµ? t?? t??????
  t?? ??µ?t??, t?? ?ata???s?? t?? f?s?? t?? ??µ?t??
  ?a? t?? f?se?? t?? ?????.

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• St? Te???a ????µ??, ?p?? ?f?se pe??p?? 148
  e??as?e?, ? Euler ????se t?? ??e??e? t??
  eµp?e?µe??? ap? t? µe??t? t?? ????? t?? ???fa?t??
  ?a? t?? Fermat.
• ?p??te??e t? ????? Te???µa t?? Fermat
  ?ata?????ta? st? p??? ???st? ?e???µa

?p?? f(n) e??a? ? s????t?s?, ? ?p??a d??e? t?
p????? t?? a???µ?? p?? e??a? µ????te??? t?? n ?a?
p??t?? p??? a?t??.
?p?p???? ?de??e, ?p?? a???ß?? e??e ßeßa??se? ?
Fermat, ?t? ???e a???µ?? t?? µ??f?? 4n1 e??a?
??a tet?????? ? ?????sµa d?? tet?a?????, e??
?at?????e t?? ?s????sµ? t?? ?t? ???? ?? a???µ??
t?? µ??f??
e??a? p??t??, ap?de??????ta? ?t? ? a???µ??
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• ???feta? ?? ????µe?? t?? 641x6700417.
• ?????, se µ?a ep?st??? t?? p??? t?? Goldbach
  (1753) ???a?e ?t? e??e a?? ?e??a? t?? ap?de???
  t?? pe??f?µ?? te?e?ta??? ?e???µat?? t?? Fermat,
  ??a t?? e?d???? pe??pt?se?? t?? e??et?? 3 ?a?
  4.
• ?p?p??s?eta, p??? ????te?a t?? Riemann
  µe??t?se t?? ??ta s????t?s?,
• ?a? a?a?????e t? s?et??? s??a?t?s?a??
  e??s?s?

?(s)
(??a s gt1), ?p?? t? pn d?at???e? ????? t???
p??t??? a???µ???.
??t?? ?? a?a?a???e?? ap?d?d??ta? ?a??asµ??a st??
µe???? a?t?? µa??µat???.
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• ?p? t?? ?e???a? t?? tet?a??????? ?p????p??,
  a?a?????e epa?????? t?? ???st? ??µ? t??
  tet?a??????? a?t?st??f??, t? d??s?µ? theorema
  aureum ? ?p???? ??e? ?t? ?p???e? µ?a ?µ??f?
  a?t?st??f? a??µesa st? ?e???? t?? ?s?t?µ???

?p?? p, q p??t?? ?a? ?? d?? ?s?t?µ?e? ep?????ta?
? ?a? ?? d?? de? ep?????ta?, e?t?? ?? a? ?a? ? p
?a? ? q af????? ?p????p? 3 ?ta? d?a??e???? d?a
t?? 4, ?p?te ? µ?a ep???eta? e?? ? ???? ???. ?
Gauss ?ta? ? p??t?? p?? t?? ap?de??e (st??
e?p???t??? ?????a t?? de?ae???a et??, p???µa p??
?a pe?se? ?p????d?p?te p??spa??se? ?a ap?de??e?
t?? ??µ? ?t? ? Gauss ?ta? ??t? pa?ap??? ap?
??a??? sta µa??µat???).
?????, a? ?a? ?f?se st?? Lagrange t?? d??a ?a
ap?de??e? t?? d??at?t?ta a?a??se?? ???e a?e?a???
a???µ?? se ?????sµa tess???? tet?a?????, ? ?d???
ap?de??e t?? ?eµ???? ???? t?? ap?de???? a?t??,
?t? d??ad? t? ????µe?? d?? a???µ??, ??ast?? t??
?p???? e??a? ?????sµa tess???? tet?a?????,
e?f???eta? ?µ????.
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Sudoku Euler

• ???? ?st???a ??a t? Sudoku...
• ??a ?e????sa? 222 ?????a p???, ?ta? ? ??ßet??
  µa??µat???? Leonard Euler se ?????a 76 et??
  d?µ??????se ta "µa???? tet?????a" (carres
  magiques). ??? ?d?a ?????? p?? "?e??????e" ?
  p??t? µ??f? t?? s?µe????? pa????d??? S???t????, ?
  ??ßet?? µa??µat???? pe?a??e? ?a? t? pa????d?
  pa?aµ??e? st?? af??e?a ??a d?? pe??p?? a???e?.

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??? pa??eta?

• Elementary, p?? ?a ??e?e ?a? ??a? p??????? t??
  ??????. ??s?ast??? ?p???e? ??a? ßas???? ?a???a?
  p?? d??pe? t? pa????d? sta 81 ?e?? ???t???a t??
  tet?a????? ? pa??t?? p??pe? ?a t?p??et?se?
  a???µ??? µe t?t??? t??p?, ?ste ???e se???, st???
  ?a? p?a?s?? 3x3 ?a pe???aµß??e? ??a ta ??f?a ap?
  t? 1 ?? t? 9. ?a tet?????a t?? Sudoku ???????ta?
  se d??f??e? ?at?????e? a?????a µe t? ep?ped?
  d?s????a?.

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?? pa????µe!

•  
• St? pa?a??t? Su Doku - t? ?p??? ?a?a?t????eta?
  p??? e????? - ?e????ste µe t?? a???µ? 7. ?p??
  ß??p??µe, ? a???µ?? a?t?? ?p???e? ?d? st?? p??t?
  se??? t?? tet?a????? (t??t? p?a?s??), a??? ?a?
  st? de?te?? (de?te?? p?a?s??). ??a µp????µe ?a
  t?p??et?s??µe t? 7 µ??? st?? t??t? se??? t??
  p??t?? p?a?s??? (p??? a??ste??). ?a?at????µe ?t?
  ?e??? ????? ??' a?t? ?p???e? µeta?? t?? 9 ?a? t??
  2. S??e?????µe µe t?? a???µ? 9 ??a t? de?te??
  p?a?s?? (st? µ?s? t?? p??? se????). ???p??µe ?t?
  ? a???µ?? ?p???e? ?d? st?? t?ta?t? st??? (st?
  µesa?? p?a?s?? t?? de?te??? se????) ?a? st??
  p?µpt? (p??te?e?ta?? p?a?s??). ??a µp????µe ?a
  t?p??et?s??µe t? 9 µ??? st?? ??t? st???. ?d?
  ?p?????? d?? ?p???f?e? ??se?? µ?a st?? p??t?
  se??? ?a? µ?a st?? t??t?. H t??t? ?µ?? ?d?
  pe????e? t?? a???µ? 9, ?p?te ap????pteta?.

8 1 7 3
6 7 8
9 2 3 1 6
4 7 5 6
7 9 1 2
6 3 4 9
4 9 2 1 6
6 5 4
7 8 5 9
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? ??S?
8 1 6 4 5 9 7 2 3
4 3 5 6 2 7 9 1 8
9 7 2 3 1 8 6 4 5
2 4 9 8 7 3 5 6 1
5 8 7 9 6 1 2 3 4
1 6 3 2 4 5 8 9 7
3 5 4 7 9 2 1 8 6
6 9 1 5 8 4 3 7 2
7 2 8 1 3 6 4 5 9