 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
```

from Milo Gardner:

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

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