The Akhmin Papyrus
One relatively late document on Egyptian unit fractions is known as the Akhmin Papyrus, apparently written around 400 AD. Considering that the material in the Rhind Papyrus dates from 1850 BC (or earlier), this shows that the use of unit fractions persisted for a remarkably long time. It appears that by the time the Akhmin Papyrus was written there was a fairly sophisticated criterion for the selection of the table entries. To expand N/P, check the smallest solutions where exactly k denominators are divisible by P using the congruences k congruence modulo P ---- ------------------------------ 1 Na = 1 2 Nab = a + b 3 Nabc = ab + ac + bc 4 Nabcd = abc + abd + acd + bcd with 0 < a < b < c < d, and take the one with the smallest maximum value. For example, to find the best expansions of n/17 we have the following choices for (a,b,c,d): n k=1 k=2 k=3 k=4 --- ----- ----- ------- -------- 2 (9) (3,4)* (2,5,6) (1,2,3,6) 3 (6) (4,5) (1,3,4)* (1,2,5,6) 4 (13) (3,8) (1,4,5)* (1,2,3,7) 5 (7) (2,4)* (2,3,5) (1,2,5,7) 6 (3)* (1,7) (1,2,4) (1,2,3,5) 7 (5) (1,3)* (3,4,7) (1,4,5,6) 8 (15) (1,5)* (2,4,6) (2,3,5,6) 9 (2)* (3,6) (3,4,5) (1,2,4,6) 10 (12) (1,2)* (1,3,6) (1,3,4,5) 11 (14) (3,7) (2,3,4)* (1,2,4,7) 12 (10) (2,6) (2,4,5) (1,2,3,4)* 13 (4)* (3,5) (1,2,6) (1,2,4,5) 14 (11) (1,4)* (1,3,5) (2,3,4,6) 15 (8) (2,3)* (1,2,7) (2,4,5,6) 16 (16) (2,5) (1,2,3)* (1,5,6,7) The asterisks mark the solutions with the smallest maximum term. The remarkable thing is that the asterisks also mark the expansions of n/17 appearing in the Akhmin Papyrus. It's a perfect match. Clearly whoever wrote that papyrus was organizing the solutions in a way that is consistent with the method I've described. Applying this same analysis to the n/19 table in the Akhmin Papyrus gives the results n k=1 k=2 k=3 k=4 --- ----- ----- ------- -------- 2 (10)+ (4,6)* (1,5,6) (1,2,3,6) <--- 3 (13) (2,8) (3,4,5)*+ (1,3,6,7) 4 (5)+ (2,3)* (1,2,8) (1,3,4,5) <--- 5 (4)+ (1,5) (1,2,3)* (2,3,5,6) <--- 6 (16) (1,4)*+ (4,5,6) (2,3,4,6) 7 (11) (2,6)*+ (2,4,7) (1,4,5,6) 8 (12) (6,7) (2,3,5)*+ (1,2,4,7) 9 (17) (4,5) (2,3,4)*+ (1,2,3,5) 10 (2)*+ (3,6) (1,4,5) (1,2,3,4) 11 (7) (1,2)*+ (1,3,6) (1,3,4,7) 12 (8) (1,7) (2,4,6)*+ (1,2,5,6) 13 (3)*+ (1,8) (4,6,7) (2,3,4,5) 14 (15) (1,3)*+ (3,5,6) (1,4,6,7) 15 (14) (2,4)*+ (1,2,5) (1,3,5,6) 16 (6) (4,7) (1,2,4)*+ (1,3,4,6) 17 (9) (3,5)*+ (1,4,7) (4,5,6,7) 18 (18) (3,4)*+ (1,3,5) (1,3,7,8) The expansion with the smallest max denominator is indicated by an asterisk, and the one appearing in Akhmin is indicated by a plus sign. In this case the match is nearly perfect, with just the following three exceptions fraction Akhmin Expected ------- --------- ------------------ 2/19 10' 190' 12' 76' 114' (Rhind) 4/19 5' 95' 6' 38' 57' (2*Rhind) 5/19 4' 76' 6' 19' 38' 57' (2*Rhind+ 1/19) In these three cases the Akhmin author selected the expansion with the fewest terms, rather than the expansion with the smallest max denominator. Interestingly, the "expected" series for 2/19 is precisely the one that appears in the Rhind Papyrus, and of course the series for 4/19 is just twice 2/19 (in Akhmin as well as in the expected series), and the expected series for 5/19 is just 1/19 plus 4/19. Even granting that the selection criterion for the Akhmin tables was as described above, this still leaves the question of what algorithm might have been used to compute the results. It occurs to me that the author of the Akhmin Papyrus could have used a "meta-table" to construct his tables. (Maybe he kept the meta-table secret for job security?) Basically there are only a limited number of combinations of coefficients, so you could build a meta-table just once and use it to construct all the individual n/p tables. Meta-Table For Ahkmin Papyrus Unit Fractions A B a b c d A B a b c d --- --- - - - - --- --- - - - - 2 1 2 5 1 5 2 3 2 1 20 9 5 4 3 1 3 15 8 5 3 6 5 3 2 10 7 5 2 3 4 3 1 5 6 5 1 6 11 3 2 1 60 47 5 4 3 4 1 4 40 38 5 4 2 12 7 4 3 20 29 5 4 1 8 6 4 2 30 31 5 3 2 4 5 4 1 15 23 5 3 1 24 26 4 3 2 10 17 5 2 1 12 19 4 3 1 120 154 5 4 3 2 8 14 4 2 1 60 107 5 4 3 1 24 50 4 3 2 1 30 61 5 3 2 1 40 78 5 4 2 1 To find the best unit fraction expansion of n/p, all you need to do is take the first A,B from this table such that nA-B is a multiple of p. For example, to expand 12/17 we try the first entry, A=2,B=1, which gives 12A-B = 23, not a multiple of 17. So we try the next entry, A=2,B=3. This doesn't work either, so we go on to the next. The first entry that works is the 14th: A=24,B=50. Therefore, the optimum expansion of 12/17 is given by [a,b,c,d]=[4,3,2,1]. In this case the solution was given by the 14th entry in the meta- table. If we check all the n/17 expansions in the Akhmin Papyrus we find that the solutions are given by the meta-table entries listed below: n -> 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 entry # -> 8 12 22 9 3 5 19 1 2 11 14 7 10 4 6 Using this method the hardest expansion to find would be 4/17, because you have to check down to the 22nd entry in the meta-table (A=20,B=29), but it isn't particularly laborious. With a little practice you could probably do it in your head. (Notice that you can take A and B modulo p, so the 22nd entry with p=17 is equivalent to A=3,B=12, and obviously 4(3)-12 = 0.) Actually to cover all of the n/19 expansions they would have needed a meta-table going up to the 6's. (I've just shown it up to the 5's.)