- PLIMPTON 322 161
We should therefore keep in mind that the reciprocals of these numbers may play
a role in the construction of the table.
Notice also that these ten pairs are all relatively prime pairs. Let us now denote
the square root of the average by ñ and the square root of the semidifferenee by
q. Column 2 will then be ñ^2 - q^2 , and column 3 will be p^2 + q^2. Having identified
the pairs (p, q) as important clues, we now ask which pairs of integers occur here
and how they are arranged. The values of q, being smaller, are easily handled. The
smallest q that occurs is 5 and the largest is 54, which also is the largest number less
than 60 whose only prime factors are 2, 3, and 5. Thus, we could try constructing
such a table for all values of q less than 60 having only those prime factors. But
what about the values of p? Again, ignoring the rows for which we do not have
a pair (p, q), we observe that the rows occur in decreasing order of p/q, starting
from 12/5 = 2.4 and decreasing to 50/27 = 1.85185185.... Let us then impose the
following conditions on the numbers ñ and q:
- The integers ñ and q are relatively prime.
- The only prime factors of ñ and q are 2, 3, and 5.
- q < 60.
- 1.8 < p/q < 2.4
Now, following an idea of Price (1964), we ask which possible (p, q) satisfy these
four conditions. We find that every possible pair occurs with only five exceptions:
(2,1), (9,5), (15,8), (25,12), and (64,27). There are precisely five rows in the
table—rows 2, 9, 11, 13, and 15—for which we did not find a pair of perfect squares.
Convincing proof that we are on the right track appears when we arrange these pairs
in decreasing order of the ratio p/q. We find that (2,1) belongs in row 11, (9,5) in
row 15, (15,8) in row 13, (25,12) in row 9, and (64,27) in row 2, precisely the rows
for which we did not previously have a pair p, q. The evidence is overwhelming
that these rows were intended to be constructed using these pairs (p, q). When
we replace the entries that we can read by the corresponding numbers p^2 - q^2 in
column 2 and p^2 + q^2 in column 3, we find the following:
In row 2, the entry 3,12,1 has to be replaced by 1,20,25, that is, 11521 becomes- The other entry in this row, 56,7, is correct.
In row 9, the entry 9,1 needs to be replaced by 8,1, so here the writer simply
inserted an extra unit character.
In row 11, the entries 45 and 75 must be replaced by 3 and 5; that is, both
are divided by 15. It has been remarked that if these numbers were interpreted as
45 · 60 and 75 · 60, then in fact, one would get ñ = 60, q = 30, so that this row was
not actually "out of step" with the others. But of course when that interpretation
is made, ñ and q are no longer relatively prime, in contrast to all the other rows.
In row 13 the entry 7,12,1 must be replaced by 2,41; that is, 25921 becomes - In other words, the table entry is the square of what it should be.
The illegible entries in row 15 now become 56 and 106. The first of these is
consistent with what can be read on the tablet. The second appears to be 53, half
of what it should be.
The final task in determining the mathematical meaning of the tablet is to
explain the numbers in the first column and interpolate the missing pieces of that
column. Notice that the second and third columns in the table are labeled "width"