The Best Egyptian Fractons
It is our intention to use some of the recent work of Milo Gardner and others to understand Egyptian Fractions. Below is a list, found on the Rhind (Ahmes) Papyrus, used for 2/n where n is an odd n umber from 3 to 101.
2/3 = 1/2 + 1/6 |
2/5 = 1/3 + 1/15 |
2/7 = 1/4 + 1/28 |
2/9 = 1/6 + 1/18 |
2/11 = 1/6 + 1/66 |
2/13 = 1/8 + 1/52 + 1/104 |
2/15 = 1/10 + 1/30 |
2/17 = 1/12 + 1/51 + 1/68 |
2/19 = 1/12 + 1/76 + 1/114 |
2/21= 1/14 + 1/42 |
2/23 = 1/12 + 1/276 |
2/25 = 1/15 + 1/75 |
2/27 = 1/18 + 1/54 |
2/29 = 1/24 + 1/58 + 1/174 + 1/232 | |
2/31 = 1/20 + 1/124 + 1/155 |
2/33 = 1/22 + 1/66 |
2/35 = 1/25 + 1/30 + 1/42 |
2/37 = 1/24 + 1/111 + 1/296 |
2/39 = 1/26 + 1/78 |
2/41 = 1/24 + 1/246 + 1/328 |
2/43 = 1/42 + 1/86 + 1/129 + 1/301 |
2/45 = 1/30 + 1/90 | |
2/47 = 1/30 + 1/141 + 1/470 |
2/49 = 1/28 + 1/196 |
2/51 = 1/34 + 1/102 |
2/53 = 1/30 + 1/318 + 1/795 |
2/55 = 1/30 + 1/330 |
2/57 = 1/38 + 1/114 |
2/59 = 1/36 + 1/236 + 1/531 |
2/61 = 1/40 + 1/244 + 1/488 + 1/610 | |
2/63 = 1/42 + 1/126 |
2/65 = 1/39 + 1/195 |
2/67 = 1/40 + 1/335 + 1/536 |
2/69 = 1/46 + 1/138 |
2/71 = 1/40 + 1/568 + 1/710 |
2/73 = 1/60 + 1/219 + 1/292 + 1/365 |
2/75 = 1/50 + 1/150 |
2/77 = 1/44 + 1/308 |
2/79 = 1/60 + 1/237 + 1/316 + 1/790 |
2/81 = 1/54 + 1/162 |
2/83 = 1/60 + 1/332 + 1/415 + 1/498 | |
2/85 = 1/39 + 1/195 |
2/87 = 1/58 + 1/174 |
2/89 = 1/60 + 1/356 + 1/534 + 1/890 |
2/91 = 1/70 + 1/130 |
2/93 = 1/62 + 1/186 |
2/95 = 1/60 + 1/380 + 1/570 |
2/97 = 1/56 + 1/679 + 1/776 |
2/99 = 1/66 + 1/198 |
2/101 = 1/101 + 1/202 + 1/303 + 1/606 |
I would like to add a simpler 'Occam's Razor'
view of 2/pq series found in the Rhind
Mathematical Papyrus [RMP]. Rather than 2/pq = (1/q + 1/pq)2/(p
+ 1) as used for all but four series, I suggest that: 2/pq = 2/A
x A/pq where A = (p + 1) and (p + q) covers all but 2/95. As discussed
previously, 2/95 is simple a mod 5 version of 2/p, where 2/p =
1/A + (2A -p)/Ap with
p = 19, A = 12, with aliquot parts of A ,3 + 2, used as follows:
2/19 = 1/12 + (3 + 2)/(12*19) = 1/12 + 1/76 + 1/114
such that: 2/95 = 1/5 x (1/12 + 1/76 + 1/114) = 1/60 + 1/380 +
1/570 as Ahmes may have mentally computed.
In conclusion, I would like to add that Fibonacci
used a parametric version of the 2/p rule, extending Ahmes' range
for A, from p/2 < A , p, with A being a highly composite number,
to p/n < A < 2p for solving n/p series.
Regards,
Milo Gardner 12/16/99
please use the milo.gardner@24stex.com address when responding
URL: http://www.math.buffalo.edu/mad/Ancient-Africa/best-egyptian-fractions.html
©bonvibre&daughters 2/1/99
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