이집트 분수
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.
- http://www.ksmes.net/q/edu_114/story.php?mid=26&r=view&uid=132