"숫자 163"의 두 판 사이의 차이

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(사용자 2명의 중간 판 31개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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<h5>간단한 소개</h5>
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==개요==
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* <math>e^{\pi \sqrt{163}}</math>는 정수에 매우 가깝다
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* <math>\large e^{\pi \sqrt{163}}=262537412640768743.9999999999992500725\cdots\approx 262537412640768744</math>
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:<math>e^{\pi \sqrt{163}}=262537412640768743.9999999999992500725\cdots\approx 262537412640768744 \label{ex163}</math>
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* \ref{ex163}으로부터 다음을 얻는다
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:<math>\sqrt[3]{e^{\sqrt{163} \pi }-744}=640319.999999999999999999999999390317352\cdots \label{e163} </math>
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* \ref{e163}으로부터 다음을 얻는다
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:<math>
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\left(\frac{\log \left(640320^3+744\right)}{\pi }\right)^2=163.0000000000000000000000000000232\cdots
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</math>
 
* <math>e^{\pi \sqrt{43}} = 884736743.9997774660349066619374620785\approx 884736744</math>
 
* <math>e^{\pi \sqrt{43}} = 884736743.9997774660349066619374620785\approx 884736744</math>
 
* <math>e^{\pi \sqrt{67}} = 147197952743.9999986624542245068292613\approx 147197952744</math>
 
* <math>e^{\pi \sqrt{67}} = 147197952743.9999986624542245068292613\approx 147197952744</math>
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* 이 숫자들은 정수에 매우 가까우며, 셋 모두 끝 세 자리가 744
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* <math>\sqrt[3]{e^{\sqrt{67} \pi }-744}=5279.999999999999984007382352249\cdots</math>
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* <math>\sqrt[3]{e^{\sqrt{43} \pi }-744}=959.99999999991951173</math>
  
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셋 모두 끝 세 자리가 744
 
  
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==complex multiplication==
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<h5>complex multiplication</h5>
 
  
 
* [[타원곡선]]
 
* [[타원곡선]]
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<h5>j-invariant</h5>
 
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<math>j(\tau) = \frac{1}{{q}} + 744 + 196884{q} + 21493760{q}^2 + 864299970{q}^3 + \cdots</math><br> 이 때, <math>{q} = e^{2\pi i\tau}</math>
 
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<math> j(\tau)= {E_4(\tau)^3\over \Delta(\tau)}= q^{-1}+744+196884q+21493760q^2+\cdots</math>
 
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<math> E_4(\tau)=1+240\sum_{n>0}\sigma_3(n)q^n= 1+240q+2160q^2+\cdots</math>
 
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<math>(\sigma_3(n)=\sum_{d|n}d^3)</math>
 
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<math>\Delta(\tau)= q\prod_{n>0}(1-q^n)^{24}= q-24q+252q^2+\cdots</math>
 
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<math> j(\frac {-1+\sqrt{-43}} {2})=884736744</math><br><br><math> j(\frac {-1+\sqrt{-67}} {2})=147197952744</math><br><br><math> j(\frac {-1+\sqrt{-163}} {2})=262537412640768744</math>
 
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<h5>하위주제들</h5>
 
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==== 하위페이지 ====
 
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* [[1964250|0 토픽용템플릿]]<br>
 
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** [[2060652|0 상위주제템플릿]]<br>
 
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<h5>재미있는 사실</h5>
 
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* Ramanujan observed that <math>\large e^{\pi \sqrt{163}}=262537412640768743.99999999999925\cdots</math> is within <math>10^{-12}</math> of an integer and used this to obtain approximations to <math>\pi</math>. In his Field’s Medal lecture, Richard Borcherds said that every mathematician should see once in his/her life why this should be the case, and this essay is an attempt to do just that.
 
  
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<h5>관련된 단원</h5>
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==j-invariant==
  
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<h5>많이 나오는 질문</h5>
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* [[타원 모듈라 j-함수 (elliptic modular function, j-invariant)|j-invariant]] 항목을 참조
  
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* 네이버 지식인<br>
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** http://kin.search.naver.com/search.naver?where=kin_qna&query=
 
  
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<h5>관련된 고교수학 또는 대학수학</h5>
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==재미있는 사실==
  
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* 라마누잔은 <math>e^{\pi \sqrt{163}}=262537412640768743.99999999999925\cdots</math> 와 같은 계산을 많이 남겼음
 +
* 이와 유사한 공식들을 <math>\pi</math> 의 근사공식에 사용. [[라마누잔과 파이]] 항목을 참조
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* In his Fields Medallists' lecture, Richard Borcherds said that every mathematician should see once in his/her life why this should be the case (citation needed)
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* <math>x^2+x+41</math>는 정수 <math>-40\leq x\leq 39</math> 에 대하여, 모두 소수가 된다
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* [[겔폰드-슈나이더 정리]] 를 사용하면, <math>e^{\pi \sqrt{163}}=(e^{-i\pi})^{\sqrt{-163}}=(-1)^{\sqrt{-163}}</math> 이므로 초월수임을 알 수 있다
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** http://mathdl.maa.org/images/upload_library/22/Ford/Davis311-320.pdf
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* 마틴 가드너의 만우절 칼럼
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<blockquote>
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\[Ellipsis]when the transcendental number e is raised to the power of \[Pi] times \[Sqrt]163, the result is an integer. The Indian mathematician Srinivasa Ramanujan had conjectured that e to the power of \[Pi]\[Sqrt]163 is integral in a note in the Quarterly Journal of Pure and Applied Mathematics (vol. 45, 1913-1914, p. 350). Working by hand, he found the value to be  262,537,412,640,768,743.999,999,999,999,\[Ellipsis]. The calculations were tedious, and he was unable to verify the next decimal digit. Modern computers extended the 9's much farther; indeed, a French program of 1972 went as far as two million 9's. Unfortunately, no one was able to prove that the sequence of 9's continues forever (which, of course, would make the number integral) or whether the number is irrational or an integral fraction.
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</blockquote>
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<blockquote>
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In May 1974 John Brillo of the University of Arizona found an ingenious way of applying Euler's constant to the calculation and managed to prove that the number exactly equals 262,537,412,640,768,744. How the prime number 163 manages to convert the expression to an integer is not yet fully understood.
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</blockquote>
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* http://math-frolic.blogspot.kr/2014/04/the-joy-of-number-theory.html
  
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<h5>관련된 다른 주제들</h5>
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==관련된 항목들==
  
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*  
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* [[오일러의 소수생성다항식 x\.b2+x+41]]
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* [[오일러의 소수생성다항식 x²+x+41|오일러의 소수생성다항식 x² +x+41]]
 
 
* [[가우스의 class number one 문제]]
 
* [[가우스의 class number one 문제]]
 
* [[라마누잔과 파이]]
 
* [[라마누잔과 파이]]
 
* [[라마누잔의 class invariants]]
 
* [[라마누잔의 class invariants]]
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* [[겔폰드-슈나이더 정리]]
  
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==매스매티카 파일 및 계산 리소스==
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<h5>관련도서 및 추천도서</h5>
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* https://docs.google.com/file/d/0B8XXo8Tve1cxU2lXWVJGdy1ldTg/edit
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*  도서내검색<br>
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==사전형태의 참고자료==
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** http://books.google.com/books?q=
 
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** http://book.daum.net/search/contentSearch.do?query=
 
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*  도서검색<br>
 
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** http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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** http://book.daum.net/search/mainSearch.do?query=
 
  
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<h5>참고할만한 자료</h5>
 
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* [http://www-math.mit.edu/%7Egreen/ramanujanconstant.pdf The Ramanujan Constant. An Essay on Elliptic Curves, Complex. Multiplication and Modular Forms.]<br>
 
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** B.J.Green
 
 
* [http://ko.wikipedia.org/wiki/%ED%9E%88%EA%B7%B8%EB%84%88_%EC%88%98 http://ko.wikipedia.org/wiki/히그너_수]
 
* [http://ko.wikipedia.org/wiki/%ED%9E%88%EA%B7%B8%EB%84%88_%EC%88%98 http://ko.wikipedia.org/wiki/히그너_수]
 
* http://en.wikipedia.org/wiki/Heegner_number
 
* http://en.wikipedia.org/wiki/Heegner_number
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* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
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* http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
 
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* 다음백과사전 http://enc.daum.net/dic100/search.do?q=
 
  
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<h5>관련기사</h5>
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==관련도서==
  
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네이버 뉴스 검색 (키워드 수정)
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* http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
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* http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
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* http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
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* http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
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* http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
  
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==리뷰논문, 에세이, 강의노트==
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* B.J.Green, [http://people.maths.ox.ac.uk/greenbj/papers/ramanujanconstant.pdf The Ramanujan Constant. An Essay on Elliptic Curves, Complex. Multiplication and Modular Forms]
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* I.J. Good, [http://www.pme-math.org/journal/issues/PMEJ.Vol.5.No.7.pdf What Is the Most Amazing Approximate Integer in the Universe?], Pi Mu Epsilon Journal, Vol. 5, Fall 1972, No. 7, pgs. 314-15
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<h5>블로그</h5>
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*  피타고라스의 창[http://bomber0.byus.net/index.php/2007/01/21/336 ]<br>
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==블로그==
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*  피타고라스의 창
 
** [http://bomber0.byus.net/index.php/2007/01/10/330 숫자 163]
 
** [http://bomber0.byus.net/index.php/2007/01/10/330 숫자 163]
 
** [http://bomber0.byus.net/index.php/2007/01/11/332 숫자 163 (2)]
 
** [http://bomber0.byus.net/index.php/2007/01/11/332 숫자 163 (2)]
 
** [http://bomber0.byus.net/index.php/2007/01/14/333 숫자 163 (3)]
 
** [http://bomber0.byus.net/index.php/2007/01/14/333 숫자 163 (3)]
 
** [http://bomber0.byus.net/index.php/2007/01/21/336 숫자 163 (4)]
 
** [http://bomber0.byus.net/index.php/2007/01/21/336 숫자 163 (4)]
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* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
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[[분류:에세이]]
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* 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=
 
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<h5>이미지 검색</h5>
 
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* http://commons.wikimedia.org/w/index.php?title=Special%3ASearch&search=
 
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* http://images.google.com/images?q=
 
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* [http://www.artchive.com/ http://www.artchive.com]
 
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<h5>동영상</h5>
 
  
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* http://www.youtube.com/results?search_type=&search_query=
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q1322644 Q1322644]
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===Spacy 패턴 목록===
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* [{'LOWER': 'heegner'}, {'LEMMA': 'number'}]

2021년 2월 17일 (수) 04:50 기준 최신판

개요

  • \(e^{\pi \sqrt{163}}\)는 정수에 매우 가깝다

\[e^{\pi \sqrt{163}}=262537412640768743.9999999999992500725\cdots\approx 262537412640768744 \label{ex163}\]

  • \ref{ex163}으로부터 다음을 얻는다

\[\sqrt[3]{e^{\sqrt{163} \pi }-744}=640319.999999999999999999999999390317352\cdots \label{e163} \]

  • \ref{e163}으로부터 다음을 얻는다

\[ \left(\frac{\log \left(640320^3+744\right)}{\pi }\right)^2=163.0000000000000000000000000000232\cdots \]

  • \(e^{\pi \sqrt{43}} = 884736743.9997774660349066619374620785\approx 884736744\)
  • \(e^{\pi \sqrt{67}} = 147197952743.9999986624542245068292613\approx 147197952744\)
  • 이 숫자들은 정수에 매우 가까우며, 셋 모두 끝 세 자리가 744
  • \(\sqrt[3]{e^{\sqrt{67} \pi }-744}=5279.999999999999984007382352249\cdots\)
  • \(\sqrt[3]{e^{\sqrt{43} \pi }-744}=959.99999999991951173\)


complex multiplication



j-invariant



재미있는 사실

  • 라마누잔은 \(e^{\pi \sqrt{163}}=262537412640768743.99999999999925\cdots\) 와 같은 계산을 많이 남겼음
  • 이와 유사한 공식들을 \(\pi\) 의 근사공식에 사용. 라마누잔과 파이 항목을 참조
  • In his Fields Medallists' lecture, Richard Borcherds said that every mathematician should see once in his/her life why this should be the case (citation needed)
  • \(x^2+x+41\)는 정수 \(-40\leq x\leq 39\) 에 대하여, 모두 소수가 된다
  • 겔폰드-슈나이더 정리 를 사용하면, \(e^{\pi \sqrt{163}}=(e^{-i\pi})^{\sqrt{-163}}=(-1)^{\sqrt{-163}}\) 이므로 초월수임을 알 수 있다
  • 마틴 가드너의 만우절 칼럼

\[Ellipsis]when the transcendental number e is raised to the power of \[Pi] times \[Sqrt]163, the result is an integer. The Indian mathematician Srinivasa Ramanujan had conjectured that e to the power of \[Pi]\[Sqrt]163 is integral in a note in the Quarterly Journal of Pure and Applied Mathematics (vol. 45, 1913-1914, p. 350). Working by hand, he found the value to be 262,537,412,640,768,743.999,999,999,999,\[Ellipsis]. The calculations were tedious, and he was unable to verify the next decimal digit. Modern computers extended the 9's much farther; indeed, a French program of 1972 went as far as two million 9's. Unfortunately, no one was able to prove that the sequence of 9's continues forever (which, of course, would make the number integral) or whether the number is irrational or an integral fraction.

In May 1974 John Brillo of the University of Arizona found an ingenious way of applying Euler's constant to the calculation and managed to prove that the number exactly equals 262,537,412,640,768,744. How the prime number 163 manages to convert the expression to an integer is not yet fully understood.


관련된 항목들


매스매티카 파일 및 계산 리소스


사전형태의 참고자료



관련도서

리뷰논문, 에세이, 강의노트



블로그

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'heegner'}, {'LEMMA': 'number'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=숫자_163&oldid=52366"