The Rhind Papyrus 2/N Table
One of the most puzzling episodes in the history of human thought is the 2000-year reign of Egyptian unit fractions. We can, at least in part, reconstruct the arithmetical manipulations involved, but the underlying reason or motive for expressing fractional quantities as sums of unit fractions remains mysterious. Was it simply a cumbersome style of writing that persisted for so many centuries just out of deference to traditional forms, or did it express an actual way of thinking that has since been forgotten? At the beginning of almost every general history of mathematics we find a description of how the ancient Egyptians operated with fractions almost exclusively in terms of UNIT fractions. For example, instead of saying 2/5 of my land was flooded, they would say 1/3 + 1/15 of my land was flooded. One of the earliest written records from ancient Egypt (transcribed circa 1650 BC from a source believed to date from around 1850 BC or earlier) is known as the Rhind Mathematical Papyrus, and contains a table expressing fractions of the form 2/n as sums of two, three, or four unit fractions with distinct denominators. The table covers 2/n for n up to 101, although the fractions with "even" denominators, e.g., 2/4, 2/6, etc, are omitted, showing that they clearly perceived the obvious equivalence of these with the reduced forms 1/2, 1/3, etc. The first entry in the table is 2/3, to which they assigned the expression 1/2 + 1/6. Every other table entry of the form 2/(3k) is assigned the expression 1/2k + 1/6k, which suggests they consciously treated all denominators divisible by 3 as a single family, just as all denominators divisible by 2 were implicitly treated as a single family. Of the remaining table entries, the next is 2/5, to which they assigned the expression 1/3 + 1/15. All but one of the remaining denominators in the table that are divisible by 5 are assigned a simple multiple of this expression, i.e., for 2/(5k) they used 1/3k + 1/15k. Similarly they assigned 1/4 + 1/28 to the table entry 2/7, and then "seived out" all the remaining denominators divisible by 7 using expressions of the form 1/4k + 1/28k. Finally, they assigned 1/6 + 1/66 to the table entry 2/11 and then used 1/6k + 1/66k for 2/11k with k=5. The prime 11 seems to be where they stoped this procedure, which is consistent with that fact that the table extended only to denominators up to 101, so all the composites are seived out by the primes less than 11. It's remarkable that the Egyptians of 1850 BC (and probably much earlier) had already developed this crude version of the "Sieve of Eratosthenes", and seemed to have a grasp of the difference between prime and composite numbers. Admittedly the seive is not perfect, at least not according to our present understanding. For one thing, the number 55 should have been seived out as a multiuple of 5, but for some reason they chose to treat it as a multiple of 11. Also, the composite numbers 35, 91, and 95 were evidently not treated as composites, but were assigned unique representations. Nevertheless, the overall impression is very strong that they consciously seived out the multiples of the smaller primes up to the square root of the largest denominator in the table, and then treated the remaining primes with unique representations. As we've seen, for each of the small primes 3,5,7,11 the Egyptians expressed 2/p as a sum of two unit fractions using the simple formula 2 1 1 --- = -------- + ---------- (1) p (p+1)/2 p(p+1)/2 (The same formula also applies to the expression they assigned to 2/23, although it may be coincidental.) Once these primes, and their multiples, have been resolved, the table entries for the remaining prime denominators suggest that the Egyptians determined the representations by using the identity 2 1 2a - p --- = --- + ------- (2) p a ap where "a" is just some nice round number a > p/2. To find the remaining terms, you partition the quantity 2a - p into one, two, or three distinct parts such that each part is a divisor of a. (That's why it's good to choose a nice round number for a, so it has lots of divisors.) For example, with n=89 we chose a=60, which gives the difference 31. Thus, we need to express 31 as a sum of three or fewer distinct integers each of which divides 60. One such partition is 31 = 15 + 10 + 6, which leads to the representation that appears in the Rhind Papyrus for 2/89: 2 1 1 1 1 --- = --- + --- + --- + --- 89 60 356 534 890 On this basis, it's possible to summarize the 2/n table in the Rhind Papyrus by giving the values of a,b,(c,(d)) for each prime p such that 2/p = 1/a + 1/b + (1/c + (1/d)). These values are presented in the table below. TABLE 1: Summary of Rhind Papyrus 2/n Representations p 2a-p a b c d Also covers these --- ------ --- --- --- --- ------------------- 3 1 2 6 all multiples of 3 5 1 3 15 25, 65, 85 7 1 4 28 49, 77 11 1 6 66 55 23 1 12 276 13 3 8 52 104 17 7 12 51 68 19 5 12 76 114 31 9 20 124 155 37 11 24 111 296 41 7 24 246 328 47 13 30 141 470 53 7 30 318 795 59 13 36 236 531 67 13 40 335 536 71 9 40 568 710 97 15 56 679 776 29 19 24 58 174 232 43 41 42 86 129 301 61 19 40 244 488 610 73 47 60 219 292 365 79 41 60 237 316 790 83 37 60 332 415 498 89 31 60 356 534 890 exceptional cases: 35 25 30 42 91 49 70 130 95 25 60 380 570 101 1111 606 101 202 303 This table raises two obvious questions. First, assuming the Egyptians used something like formula (2) to determine their general unit fraction representations for 2/p where p is a "large" prime, how did they select the value of "a" and the partition of 2a - p from the available possibilities? Remarkably, if you examine all the possibilities using a computer, and limit yourself to just the three and four-term representations where the smallest number x in the partition of 2a-p is greater than 1, then in most cases the expression appearing in the Rhind Papyrus is the one for which a/x is minimized. For example, the only possible solutions for p=43 are partition of 2n-p p a 2a-p x y z a/x --- --- ----- ---- ---- ---- ----- 43 24 5 2 3 12 43 28 13 2 4 7 14 43 30 17 2 15 15 43 30 17 2 5 10 15 43 36 29 2 9 18 18 43 42 41 6 14 21 7 and the representation appearing the the Rhind Papyrus is the one with a/x = 7. In all, the Egyptians used the solution with the minimum a/x for the "large" primes 13, 17, 19, 29, 31, 37, 41, 43, 59, 67, 73, 79, 83, 97 whereas they missed it for the primes 47, 53, 61, 71, 89 In these "missed" cases they missed the minimums by 2, 6, 1, 3, and 1 respectively. Another interesting fact that appears from a review of all the possible representations for each prime is that p=29 is the first prime for which there is no three-term representation of 2/p (with the restrictions noted above). Thus, it's not surprising that 2/29 is the first entry in the Rhind Papyrus where a four-term representation is used. The second major question raised by Table 1 is how to explain the four exceptional cases. The first three are the composites 35, 91, and 95, that for some reason were not seived out like the rest of the composites. From out point of view the case 2/95 = 2/(5*19) should have been seived out by the small prime p=5, giving it a representation of 1/3k + 1/15k with k=19. Instead, we find that its representation was evidently based on the "large" prime p=19, i.e., it is of the form 1/12k + 1/76k + 1/114k with k=5. The cases 2/35 and 2/91 are even more unusual, and in a sense these are the most intriguing entries in the table. These are the only two composites whose representations are not simple multiples of the representations of one of their prime factors. Remarkably, in these two cases it appears the Egyptians reverted from the normal multiplicative decomposition to what might be called a "harmonic- airthmetic" decomposition. Recall that the ancient Greeks had definitions for various kinds of "means", including the Arithmetic Mean: A(p,q) = (p+q)/2 Geometric Mean: G(p,q) = sqrt(pq) Harmonic Mean: H(p,q) = 2/(1/p + 1/q) It's believed the Greeks inherited these definitions from the Babylonians, but it's certainly possible they were also known to the Egyptians. In particular, the Harmonic Mean certainly LOOKS Egyptian, given their affinity for unit fractions. In any case, notice that G(p,q) is not only the geometric mean of p and q, it's also the geometric mean of A(p,q) and H(p,q). In other words, for any p,q we have G = sqrt(pq) = sqrt(AH) which follows simply because pq = AH. In other words, AH gives an alternative decomposition of the composite number pq. This leads to the formula 2 2 2 / 1 1 \ --- = -------------- = ----- ( --- + --- ) (3) pq A(p,q) H(p,q) p + q \ p q / where of course the leading factor on the right is a unit fraction because p+q is even. This formula yields the Rhind Papyrus representations 2 1 / 1 1 \ 1 1 --- = --- ( --- + --- ) = ---- + ---- 5*7 6 \ 5 7 / 30 42 and 2 1 / 1 1 \ 1 1 ---- = --- ( --- + ---- ) = ---- + --- 7*13 10 \ 7 13 / 70 130 Thus we can say that every composite entry in the Rhind Papyrus 2/n table is based on a decomposition of n into its prime factors. In most cases the simple geometric decomposition pq was used, but in two cases they used the arithmetic-harmonic decomposition A*H. (As to why the numbers 35, 91, (and 95) might have been singled out for special treatment, see Appendix I ) This leaves only the final entry in the 2/n table, which gives 2 1 1 1 1 --- = --- + --- + --- + --- 101 101 202 303 606 This entry can actually be constructed by formula (2) with a=606 and the partition 1111 = 202 + 303 + 606, but it seems to stand out from the other table entries due to the fact that it's a simple multiple of 1/n. This entry may have been just a formality, suggesting that for any n not covered in the table (i.e., larger than 100), we can use the four-term expansion 2/n = 1/n + 1/2n + 1/3n + 1/6n (4) so this effectively "completes" the table, allowing us to say that it provides a unit fraction representation of 2/n for ALL integers n. Interestingly, formula (4) can be seen as an illustration of the "perfectness" of the number 6, in the sense that the sum of the divisors equals double the number, i.e., 1+2+3+6 = 12 = 2*6. In summary, the 2/n table of the Rhind Papyrus, which dates from more than a thousand years before Pythagoras, seems to show an awareness of prime and composite numbers, a crude version of the "Sieve of Eratosthenes", a knowledge of the arithmetic, geometric, and harmonic means, and of the "perfectness" of the number 6. This all seems to suggest a greater number-theoretic sophistication than is generally credited to the ancient Egyptians. Whether they originated these ideas or borrowed them, perhaps from the Babylonians, is unclear. (We shouldn't overlook the possibility that the Babylonians borrowed them from the Egyptians.)