Teorema Euler

 

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ZKH\J^RHCOC KOBH@KOM CJ ROK ODZRSÆK LOBZMSOC CJ OCFHKHRS^OBHÒK ?J^ Oß@

SJ@^JFO CJ JZMJ^ BZ^R@7 ^OW@KOFHJKS@ FOSJFÏSHB@ HKSJD^OKSJR7       

DZJSSH OYOWO, SOKHO XOFOMH DZSHJ^^JW YHMO, OKSE@KX G@JM EOMMORH SSH^O, XOKJS EZOKBO D@KWOM@, OKO COMOX M@YJW B^ZW, MZU@^ OML@KR@ FOKWOK@ O^OYO, GZOK BO^M@R FOXKO FOFOKH, FO^X@^H CHOKO

C@BJKSJ7 JMFJ^ RHJ^^O O^J_ZHYO ‒ >9>9

 

HKCHBJ7 ?.- HKS^@CZBBH@K…………………………………………………………………………. >.- @NGJSH\@R………………………………………………………………………………. 2.- OKSJBJCJKSJR (EHRS@^HO) ………………………………………………………… 3.- JKZKBHOC@ CJM SJ@^JFO……………………-……………………………………. :.- CJF@RS^OBH@K…………………………………………………………………………

.- @NGJSH\@R7

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Cjsorr`mmor b`fpjtjkbhos quj sj os`bhj b`k jm tronog` jk jquhp`, mo b`ktrhnubhòk quj boco porthbhpoktj jgjrbj jk jm drup` poro mo prjsjktobhòk cjm tronog`. Cjsorr`mmor eonhmhcocjs y tomjkt`s sj b`fpmjfjktok jk m`s hktjdroktjs cjm drup`, cj jsto fokjro oufjkto jm oprjkchzogj p`rquj sj b`fportjk b`k`bhfhjkt`s b`k`bhfhj kt`s jktrj uk`s y `tr`s, cj l`rfo quj mos cjnhmhcocj cjnhmhcocjss sj l`rtomjbjk. Jm tronog` jk jquhp` eobj brjbjr mo b`klhokzo cjm jktr`k` y cj jsto l`rfo uk` oprjkcj o b`klhor jk m`s cjfïs, jm sjkthfhjkt` cj pjrtjkjkbho sj hkbrjfjkto.  Oufjkto mo ljmhbhcoc ljmhbhcoc jk jm ttronog` ronog` om b`fp b`fporthr orthr jm èxht` cj mm`s `s `ngjthv`s cj cjmm tronog` jk jquhp`, chsfhkuyj jm sjkthfhjkt` cj s`mjcoc y ohsmofhjkt`, l`rfos portj cj omd`, jk jst`s f`fjkt`s js hfp`rtoktj cjsorr`mmor hkbmushvhcoc. Mos fjtos quj o khvjm hkchvhcuom k` pujcjs ombokzor sj b`kvhjrtjk jk èxht` om tronogor jk jquhp`.

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 Oprjkcjr cj fokjro b`kg b`kgukto ukto jm tj`rjfo c cj j Jumjr, cjsorr` cjsorr`mmor mmor eonhm eonhmhcocjs hcocjs nosïkc`k`s jk jm tj`rjfo cj Jumjr. B`fporthr kujstro l`rfo cj b`fprjkshòk cjm tj`rjfo t j`rjfo b`k kujstr`s b`fpoÿjr`s cj bmosj y tofnhèk p`stjor jm tronog` jk hktjrkjt poro quj `tros pjrs`kos sj nrhkcjk cj kujstro l`rfo cj oprjkcjr s`nrj jm tjfo

 

2.- EHRS@^HO7 SJ@^JFO CJ JZMJ^ Luj uk fotjfïthb` fuy pr`mælhb` quj hktr`cug` fubeæshfos op`rtobh`kjs. Mj`keorc Jumjr kobhò jm ?: cj onrhm cj ?494 jk Noshmjo, Ruhzo Rok Yjtjrsnurd` Yjtjrsnurd`,, ^usho y furhò jm ?5 cj sjpthjfnrj cj ?452. Luj eh ehg` g` Yo Youm um Jum Jumjr jr quj quj jro uk pos post`r t`r bom bomvhk vhkhst hsto, o, tuv tuv` ` c`s ejrfo ejrfoko koss pjq pjqujÿ ujÿos os mmofocos Foræolofhmhor y Foræo Fodcomjko1 su pocrj qujræo prhkbhph`cjquj shduhjsj moOko trochbhòk y quj sj b`kvhrthjsj tofnhèk jk ukcjscj post`rjmbomvhkhsto, tom fokjro quj o mo jcoc cj ?2 oÿ`s sj fotrhbumo jk mo ukhvjrshcoc cj Noshmjo y jk ?4>2 rjbhnj jm thtum` cj fojstr` cj lhm`s`læo tros uko chsjrtobhòk b`fporothvo cj ^jkè Cjsbortjs j Hsoob Kjwt`k. Y`r jsj jkt`kbjs b`f` su pocrj jro ofhd` cj mo lofhmho Njrk`ummh jk b`kbrjt` cj G`eokk1 Jumjr m` jro cj sus ehg`s Khb`mïs y Cokhjm, obucæo m`s sïnoc`s p`r mo torcj o rjbhnhr bmosjs cj fotjfïthbo b`k G`eokk y sus ehg`s, pujs drobhos o jsto lofhmho Jumjr  b`fhjkzo o hktjrjsorsj p`r jm fukc` cj mos fotjfïthbos. G`eokk Njrk`ummh eonmo b`k su pocrj poro quj cjshsto cj quj su ehg` shdo jstuchokc` tj`m`dæo, drhjd`, ejnrj` y t`cos jstos oshdkoturos quj jm pocrj mj eonæo hfpujst` poro quj sj l`rfosj jk mo trochbhòk sobjrc`tom quj jro pr`pho cj su lofhmho1 ol`rtukocofjktj ol`rtukocofjktj su pocrj eobj bos` om pjchc` cj G`eokk, quhjk sj ch` bujkto quj Jumjr jro fotjfïthb` y pjrfhtj sj bjktrj jk jst`s jstuch`s. Buokc` Jumjr obonò quhs` jktror o mo obocjfho lrokbjso pjr` mj rjsumtò hfp`shnmj shjkc` su prhfjr cjsthk` mo obocjfho cj bhjkbhos cj Rok Yjtjrsnurd` y mmjdo oquæ p`rquj m`s ejrfok`s ejrfok`s Khb`mïs y Cok Cokhjm hjm jstonok jk j jsto sto obocjfho, Kh Khb`mïs b`mïs sulrj uk otoquj cj opjkchbhths quj m` mmjv` o mo fujrtj, jkt`kbjs Cokhjm rïphcofjktj sj ob`rcò cj Jumjr y mj phchò quj vhkhjsj o tronogor b`k èm o mo obocjfho1 pujst` quj Botomhko H cj ^usho jstono nusbokc` bhjktælhb`s p`r t`c` Jur`po poro quj objktuorok jm khvjm bumturom cj su poæs. Jumjr mmjdo jm ?4 cj foy` cj ?4>4 gust` jm cæo quj Botomhko H cj ^usho lommjbj cj tom fokjro quj om t`for jm p`cjr Yjcr` H cj ^usho, quhjk jro uk khÿ` cj ?> oÿ`s, rjsumto quj mo k`nmjzo sj eobj b`k jm p`cjr y lujr`k uk`s oÿ`s fuy b`fpmhboc`s p`rquj mo k`nmjzo cjsb`klhono cj jst`s bhjktælhb`s jxtrokgjr`s, pjr` mo pr`cubbhòk curoktj t`c` jstj thjfp` cj Rok Yjtjrsnurd` luj fuy jmjvoco. B`ktrog` fotrhf`kh` y tuv` ?2 ehg`s cj m`s buomjs s`m` : mmjdor`k o mo ocumtjz. Yrj`bupoc` p`r m`s ob`ktjbhfhjkt`s quj jstonok tjkhjkc` mudor jk ^usho. Jk ?43? cjbhcj objptor mo pr`pujsto cjm rjy cj Yrusho Ljcjrhb` jm drokcj, sj vo o mo obocjfho cj Njrmæk y oquæ jstuv` curoktj >: oÿ`s, js osæ quj jk jsto obocjfho mmjdò o pr`cubhr  fïs cj 259 ortæbum`s, pjr` o pjsor cj jst` jm rjy mj cjscjÿono cjscjÿono.. Jumjr pjrchò mo vhshòk cj uk `g` `nsjrvokc` mos jrupbh`kjs s`morjs cj tom fokjro quj jk jm oÿ` ?4 m` buom s`k poro p`mhjcr`s b`kvjx`s jm k÷fjr` cj vèrthbj fjk`s jm k÷fjr` cj orhstos fïs jm k÷fjr` cj boros shjfprj sorïk >. Jstj js uk rjsu rjsumtoc mtoc` ` fuy hktj hktjrjso rjsoktj ktj y vhsu vhsuomfj omfjktj ktj s`rp s`rprjkc rjkcjktj jktj.. B`k B`kshcj shcjrj rj uk p`mhjcr` Y k` hfp`rto sh jstj js rjdumor ` hrrjdumor. Mo lòrfumo cj Jumjr hkchbo quj sh rjprjsjkto jm k÷fjr` cj boros cjm p`mhjcr`, rjprjsjkto jm k÷fjr` cj orhstos y rjprjsjkto jm k÷fjr` cj vèrthbjs cjm p`mhjcr` jkt`kbjs sj bufpmj quj

 

:.- CJF@RS^OBHÒK Sjkhjkc` jk bujkto uk p`mhjcr` jm buom p`cjf`s cjk`fhkorm` b`f` Y hkcjpjkchjktjfjktj hkcjpjkchjktjfj ktj sh js rjdumor ` hrrjdumor, sjd÷k k`s sjÿomo Jumjr y su l`rfo sh osufhf`s quj B js jm k÷fjr` cj boros cjm p`mhjcr`, fhjktros quj p`r su moc` O rjprjsjktoræo rjprjsjktor æo jm k÷fjr` cj orh orhstos stos y lhko lhkomfjktj mfjktj \ mo bok bokthcoc thcoc cj vèrthbjs cjm p`mhjcr` jkt`kbjs tjkcræof`s quj7 \6vèrthbjs  O6orhstos B6 boros Mo hduomcoc pr`pujstos cj Jumjr k`s sjÿomo quj jm k÷fjr` cj vèrthbjs fjk`s mos orhstos y mo sufo cj mos boros orr`gorok uko hduomcoc o > 7

°B`f` js jst` p`shnmj0 Y`kdof`s jk prïbthbo7 Jk jm shduhjktj bun`, p`cjf`s `nsjrvor quj tjkjf`s jm k÷fjr` t`tom cj boros, jm k÷fjr` t`tom cj orhstos y jm k÷fjr` t`tom cj vèrthbjs

 

Zthmhzokc` mo lòrfumo cj Jumjr

C`kcj7 \65  O6?> B6< ^jfpmozokc`7 5-?>+ Jsto lòrfumo pujcj opmhborsj hkcjpjkchjktjfjktj cj mo l`rfo cjm p`mhjcr`7 @tro vorhoktj cj mo lòrfumo cj Jumjr k`s sjÿomo uko hduomcoc jktrj mo sufo cj m`s vèrthbjs b`k mos boros y mos orhstos fïs > tjkhjkc` mm`s `s shduhj shduhjktjs ktjs rjsumtoc` rjsumtoc`s7 s7

 

rjdh`kjs trhokdumorjs, >9 rjdh`kjs buocrokdumorjs y 5 rjdh`kjs ejxod`komjs. Bombumor jm k÷fjr` cj vèrthbjs

?>

6 2 moc`s

>9 5

6 3 moc`s 6 < moc`s

Boros S`tomjs7 ?> + >9 + 5 639 Bombum` mo bokthcoc cj orhstos7  O6

( ) + >9 ( 3 ) + 5 ( < )

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>

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6 5>

Y`r jm tj`rjfo cj Jumjr7

B + \ 6 O + >

39 + \ 6 5> + > 39 + \ 6 53 \ 6 53 ∞ 39 \ 6 33

Y`r m` tokt`, jm k÷fjr` cj vèrthbjs cjm p`mhjcr` b`kvjx` js 33.

-JGJ^BHBH@ >7 (Guok) Jk uk p`mhjcr` sj bufpmj quj jm k÷fjr` cj boros js hduom om k÷fjr` cj vèrthbjs. Rh mo rozòk jktrj jm kufjr` cj orhstos y jm kufjr` cj boros js ?>/4. Bombumor  sufo cj m`s k÷fjr`s cj boros, orhstos y vèrthbjs js7

\  B 6 \ 

 O  B 

 6

?> A  ?> A  4  A 

 

 Opmhbof`s jm S. cj Jumjr7 ^jjfpmozof`s    

O 6 ?>A B 6 4A \ 6 4A B+\6O+> 4a + 4a 6 ?>a + > >A 6 > A6?

K`s phcj bombumor mo sufo cj7

B+O+\

 

4 + ?> + 4

 

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O 6 ?> B64 \64

 

^jspujsto7 >< -JGJ^BHBH@ 27 (Okte`ky) Jk mo bhucoc cj m`s mod`s eoy 4 mod`s b`kjbtoc`s p`r ?9 bokomjs, cj f`c` quj sj pujcj kovjdor cjscj uk mod` eosto `tr` buomquhjro o trovès cj jmm`s. M`s bokomjs k` sj b`rtok jktrj sæ fïs quj jk m`s mod`s. °Buïktos hsmos eoy jk mo bhucoc cj m`s mod`s0 ^js`mubhòk. Sj`rjfo cj Jumjr. Jk t`c` pmok` b`kjx` y pmok` jm k÷fjr` cj boros (mos rjdh`kjs jk quj jm drol` chvhcj jm pmok`, hkbmuhco mo rjdhòk jxtjrh`r) fïs jm cj vèrthbjs js hduom om k÷fjr` cj orhstos fïs > Js cjbhr. B+\6O+> Jkt`kbjs. Jm fopo cj mo bhucoc cj m`s mod`s js uk drol` pmok` jk jm quj m`s mod`s s`k m`s vèrthbjs js (\ 6 4) M`s bokomjs s`k mos orhstos (O 6 ?9) Mos boros (somv` mo jxtjrh`r) s`k mos hsmos. Y`r jm tj`rjfo cj Jumjr, B 6 :. Mujd` eoy 3 hsmos.

-JGJ^BHBH@ 37 (Sokho)

 

-JGJ^BHBH@ :7 (Xokjt)

-JGJ^BHBH@

Opmhbokc`

 O689

2>-89+-89+ >

B6+ 8>68>

JGJ^BHBH@ 47 (Fory`rh)

\+B6O+>

 

Bombumor jm k÷fjr` cj vèrthbjs cj uk p`mhjcr` l`rfoc` p`r < trhïkdum`s, 3 rjbtïkdum`s y 3 ejxïd`k`s. <

6 2 moc`s

3

6 3 moc`s

3

6 < moc`s

Boros S`tomjs7 < + 3 + 3 6?3

 Orhstos t`tomjs7

( ) + 3 ( 3 )+ 3 ( < )

< 2

>

:5 6

>

6 >8

   Opmhbokc`  Opmhbokc ` mo lòrfumo, sj p` p`crï crï eommor jm t` t`tom tom cj vèrthbjs cj c chbe` hbe` p`mh p`mhjcr`7 jcr`7 \ 6 O + >∞B 

\ 6 >8+ >∞?3 6?4

B`kbmuyjkc` quj jm k÷fjr` t`tom cj vèrbjs cjm p`mhjcr` js ?4.

 

4.- B@KBMZRH@KJR 

Jm tj`rjfo cj Jumjr shrvj poro jm bïmbum` cjm k÷fjr` cj boros, vèrthbjs y orhstos cj t`c` p`mhjcr` b`kvjx`.

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Bujkto b`k uko l`rfumo shfpmj, pjr` jljbthvo, buokc` k` sj p`sjj omd÷k cot` cj m`s yo jstonmjbhc jstonmjbhc`s `s oktjrh`rfjktj (b`f` Boro, orhsto ` vèrthbj).

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Zko cj mos fokjros cj bïmbum` uthmhzoc` jk m`s pr`nmjfos cj p`mhjcr`s. Jxhstj uko hklhkhcoc cj p`mhjcr`s b`kvjx`s jk m`s quj sj mj pujcj opmhbor mo tj`ræo cj Jumjr, b`fporthjkc` osæ uko l`rfumo b`f÷k.

 

5.- HKL@D^OLHO

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