"이집트 분수"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) (section '관련논문' added) |
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| 135번째 줄: | 135번째 줄: | ||
[[분류:초등정수론]] | [[분류:초등정수론]] | ||
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| + | == 관련논문 == | ||
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| + | * Christian Elsholtz, Egyptian Fractions with odd denominators, arXiv:1606.02117 [math.NT], June 07 2016, http://arxiv.org/abs/1606.02117 | ||
2016년 6월 7일 (화) 21:13 판
개요
- 분자가 1인 분수를 단위분수라 한다
- 유리수를 단위분수의 합으로 표현하는 것을 이집트 분수라 함
예
\begin{array}{|rcl|} \hline \frac{4}{5} & = & \frac{1}{2}+\frac{1}{4}+\frac{1}{20} \\ \hline \frac{2}{7} & = & \frac{1}{4}+\frac{1}{28} \\ \hline \frac{5}{121} & = & \frac{1}{25}+\frac{1}{757}+\frac{1}{763309}+\frac{1}{873960180913}+\frac{1}{1527612795642093418846225} \\ \hline \end{array}
1의 이집트 분수 표현
- 1을 서로 다른 단위분수로 표현하는 문제를 생각하자
- $r$은 자연수
- 다음의 조건을 만족하는 자연수 $s_1, \cdots ,s_r$을 모두 찾아라
- $s_1<\cdots <s_r$
- $\sum_{i=1}^r \frac{1}{s_i}=1$
- 주어진 $r$에 대하여 해는 유한개이다
- 1, 0, 1, 6, 72, 2320, 245765, 151182379,...
\begin{array}{c|c} 1 & \{1\} \\ \hline 0 & \cdot \\ \hline 1 & \{2,3,6\} \\ \hline 1 & \{2,3,7,42\} \\ 2 & \{2,3,8,24\} \\ 3 & \{2,3,9,18\} \\ 4 & \{2,3,10,15\} \\ 5 & \{2,4,5,20\} \\ 6 & \{2,4,6,12\} \\ \hline 1 & \{2,3,7,43,1806\} \\ 2 & \{2,3,7,44,924\} \\ 3 & \{2,3,7,45,630\} \\ 4 & \{2,3,7,46,483\} \\ 5 & \{2,3,7,48,336\} \\ 6 & \{2,3,7,49,294\} \\ 7 & \{2,3,7,51,238\} \\ 8 & \{2,3,7,54,189\} \\ 9 & \{2,3,7,56,168\} \\ 10 & \{2,3,7,60,140\} \\ 11 & \{2,3,7,63,126\} \\ 12 & \{2,3,7,70,105\} \\ 13 & \{2,3,7,78,91\} \\ 14 & \{2,3,8,25,600\} \\ 15 & \{2,3,8,26,312\} \\ 16 & \{2,3,8,27,216\} \\ 17 & \{2,3,8,28,168\} \\ 18 & \{2,3,8,30,120\} \\ 19 & \{2,3,8,32,96\} \\ 20 & \{2,3,8,33,88\} \\ 21 & \{2,3,8,36,72\} \\ 22 & \{2,3,8,40,60\} \\ 23 & \{2,3,8,42,56\} \\ 24 & \{2,3,9,19,342\} \\ 25 & \{2,3,9,20,180\} \\ 26 & \{2,3,9,21,126\} \\ 27 & \{2,3,9,22,99\} \\ 28 & \{2,3,9,24,72\} \\ 29 & \{2,3,9,27,54\} \\ 30 & \{2,3,9,30,45\} \\ 31 & \{2,3,10,16,240\} \\ 32 & \{2,3,10,18,90\} \\ 33 & \{2,3,10,20,60\} \\ 34 & \{2,3,10,24,40\} \\ 35 & \{2,3,11,14,231\} \\ 36 & \{2,3,11,15,110\} \\ 37 & \{2,3,11,22,33\} \\ 38 & \{2,3,12,13,156\} \\ 39 & \{2,3,12,14,84\} \\ 40 & \{2,3,12,15,60\} \\ 41 & \{2,3,12,16,48\} \\ 42 & \{2,3,12,18,36\} \\ 43 & \{2,3,12,20,30\} \\ 44 & \{2,3,12,21,28\} \\ 45 & \{2,3,14,15,35\} \\ 46 & \{2,4,5,21,420\} \\ 47 & \{2,4,5,22,220\} \\ 48 & \{2,4,5,24,120\} \\ 49 & \{2,4,5,25,100\} \\ 50 & \{2,4,5,28,70\} \\ 51 & \{2,4,5,30,60\} \\ 52 & \{2,4,5,36,45\} \\ 53 & \{2,4,6,13,156\} \\ 54 & \{2,4,6,14,84\} \\ 55 & \{2,4,6,15,60\} \\ 56 & \{2,4,6,16,48\} \\ 57 & \{2,4,6,18,36\} \\ 58 & \{2,4,6,20,30\} \\ 59 & \{2,4,6,21,28\} \\ 60 & \{2,4,7,10,140\} \\ 61 & \{2,4,7,12,42\} \\ 62 & \{2,4,7,14,28\} \\ 63 & \{2,4,8,9,72\} \\ 64 & \{2,4,8,10,40\} \\ 65 & \{2,4,8,12,24\} \\ 66 & \{2,4,9,12,18\} \\ 67 & \{2,4,10,12,15\} \\ 68 & \{2,5,6,8,120\} \\ 69 & \{2,5,6,9,45\} \\ 70 & \{2,5,6,10,30\} \\ 71 & \{2,5,6,12,20\} \\ 72 & \{3,4,5,6,20\} \\ \end{array}
관련된 항목들
매스매티카 파일 및 계산 리소스
- https://drive.google.com/file/d/0B8XXo8Tve1cxUmxaTm9CSWg0bjA/view
- http://oeis.org/A002966
- http://oeis.org/A006585
리뷰, 에세이, 강의노트
- Graham, Ronald L. “Paul Erdős and Egyptian Fractions.” In Erdős Centennial, edited by László Lovász, Imre Z. Ruzsa, and Vera T. Sós, 289–309. Bolyai Society Mathematical Studies 25. Springer Berlin Heidelberg, 2013. http://link.springer.com/chapter/10.1007/978-3-642-39286-3_9.
- Mackenzie, Dana. “Fractions to Make an Egyptian Scribe Blanch.” Science 278, no. 5336 (October 10, 1997): 224–224. doi:10.1126/science.278.5336.224.
- http://www.ksmes.net/q/edu_114/story.php?mid=26&r=view&uid=132
사전 형태의 자료
관련논문
- Christian Elsholtz, Egyptian Fractions with odd denominators, arXiv:1606.02117 [math.NT], June 07 2016, http://arxiv.org/abs/1606.02117