THE RHIND 2/n TABLE

This table found on the Rhind (Ahmes) Papyrus contains a list of Egyptian fractions used for 2/n where n is an odd n umber from 3 to 101. To learn how or why are the choices made as below, read De-mystifying the Rhind 2/n Table, where the diacritical remarks are explained from my initial understanding. There are several places on the web attempting to (and some which don't) present exact data on the Rhind Mathematical papyrus rolls. I have included these links in a letter, from expert Milo Garner, below. A second 2002 paper by Gardner appears here.

first a general formula for creating the majority of 2/pq series:
2/pq = 2/A x A/pq, where A = (p + 1)

# means factor q=3*k and use (1/2 + 1/6)/k
## means facotr q =5*k and use (1/3 + 1/15)/k
### means factor q = 7*k and use (1/4 + 1/28)/k

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  [this is neither 2/7*5 or 2/5*7]   :::   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 (use 2/11 instead of 2/5)

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 = 1/51 + 1/255  ##   :::   2/87 = 1/58 + 1/174  #

2/89 = 1/60 + 1/356 + 1/534 + 1/890   :::   2/91 = 1/70 + 1/130  (do not use 2/7)

2/93 = 1/62 + 1/186  #   :::   2/95 = 1/60 + 1/380 + 1/570  (do not use 2/5)

2/97 = 1/56 + 1/679 + 1/776   :::   2/99 = 1/66 + 1/198  #

2/101 = 1/101 + 1/202 + 1/303 + 1/606

Calculating 2/51, with p = 3 and q = 17
2/51 = 2/(3 + 1) x (3 + 1)/51 = 1/2 x (1/7 + 1/51) = 1/34 + 1/102

and 2/85 by selecting p = 5 and q = 17,
2/85 = 2/(5 + 1) x (5+ 1)/85 = 1/3 x (1/17 + 1/85) = 1/51 + 1/255

DE-MYSTIFYING THE RHIND 2/n TABLE

First notice for any non-zero number r,

1/(r+1) +1/r(r+1) = r/[r(r+1)] + 1/[r(r+1)] = (r+1)/[r(r+1)] = 1/r.

So that we have the formula

\$). 2/r = 1/[(r+1)/2] + 1/[r(p+1)/2].

For example, if r = 3, then (\$) shows

2/3 = 1/[4/2] + 1/[3*4/2] = 1/2 + 1/6.

The egyptians used (\$) in the table above for the first primes r=3, 5, 7, or 11 (also for r=23). Here is another intriguing obersvation: That the egyptians stopped the use of (\$) at 11 suggests they understood (at least some parts of) Eratosthenes Sieve 2000 years before Eraosthenes "discovered" it.

Now observe for any non-zero a and q, 2/q - 1/a = (2a-q)/aq. Thus, the formula

#). 2/q = 1/a + (2a-q)/aq.

For example 2/13 = 1/8 + (16-13)/104 = 1/8 + 3/104. Since 3 = 2+1,
2/104 = 2/104 + 1/104. Egyptian division yields 2/104 = 1/52. Thus, we have the

2/13 = 1/8 + 1/52 + 1/104.

For any q not a multiple of the first primes, the egyptians seemed to use the formula (#) , more than once if necessary. For example,

2/17 = 1/12 + 7/104 = 1/12 + 4/204 + 2/204 + 1/204 =
= 1/12 + 1/51 + 1/102 + 1/204.

Wait, this differs from the decomposition given in the table above. What happened? We were not careful with the choice for 7/104. Go back,
7/104 = 1/51 + 3/204, and 3/204 = 1/68. So we have the same representation given above:

2/17 = 1/12 + 1/51 + 1/68.

Thus, the egyptians were not blind to other representations, especially if there are choices with a smaller number of unit fractions and/or smaller denominators.

There are nearly 30,000 ways of representing the numbers 2/q for 2< q < 102. For those where the formula (#) is used, how was the number a chosen?

It appears that a was chosen so that 2a-p is the sum of a decreasing sequence of terms each of which, in tur, divides a. For example,

2/29 = 1/24 +19/(24*29), 24 is divisible by 12, 8, 6, 4, 3, 2, and 1. We can write 19 = 12+4+3 or 19/(24*29) = 12/(24*29) + 4/(24*29) + 3/(24*29). So we get:

2/29 = 1/24 +1/58 + 1/174 + 1/232.

By comparison to denominator sizes, this is better than using 19 = 12 + 6 + 1.

There are two more exceptions, 1/35, 1/91, and 1/95. These do not use either (#) or (\$).

NEW!

Before continuing, we should note that Milo Gardner has broken the code; that is, thanks to his work we now believe we understand how the Egyptians constructed the 2/n table above. We discuss Gardner's work on a new (Feb. 1999) web page:

For archival purposes we include Milo's 1997 letters and part of a 2005 letter.