1

Plimpton 322: Generating its higher-order reciprocal pairs.

Leon R. Tessier

It is the prevailing view that all the initial values for the construction of the numbers in the Plimpton

322 tablet were generated by a single algorithm, a quotient of commonly used regular numbers.

The present article challenges this theory by demonstrating that the initial values in the partially

completed Plimpton 322 could have been obtained by a simpler algorithm based on simple numbers,

and that its employment for the calculation of a theoretically complete tablet also results in an exact

match with the numbers in the extant Plimpton 322. The origin of the initial values in the parallel text

MS 3971#3 is consistent with this alternative algorithm. It is also shown that an evolved version was

used to generate the higher-order reciprocal pairs in the Seleucid-era tablet AO 6456.

______________________________________________________________________________

Introduction.

The famous Plimpton 322 (P322) tablet is the right-side fragment (13± by 9± cm) of a

“cuneiform tablet written in the ancient Iraqi city of Larsa in the mid-18th century BCE” [Robson

2001, 170]. The obverse contains four columns of numbers, the fourth being a row count. In their

analysis of P322, Neugebauer and Sachs asserted that “simple numbers”, from standard tables of

reciprocals, were the “point of departure” of the three principal columns of P322 [1945, 40].

Friberg asserted a different explanation for the construction of the tablet, but he affirmed that the

initial values for the computation of the first column, the igi and igi.bi, are quotients of those

same simple numbers [2007, 443]. Friberg proposed a reconstruction of P322’s missing part.

Table 1. A numerical transcription of Friberg’s reconstruction of Plimpton 322.

igi igi.bi Width Diagonal Column I Col II Col III IV

2;24 0;25 0;59 30 1;24 30 1;59 00 15 1 59 2 49 ki. 1

2;22 13 20 0;25 18 45 0;58 27 17 30 1;23 46 02 30 1;56 56 58 14 50 06 15 56 07 3 13 ki. 2

2;20 37 30 0;25 36 0;57 30 45 1;23 06 45 1;55 07 41 15 33 45 1 16 41 1 50 49 ki. 3

2;18 53 20 0;25 55 12 0;56 29 04 1;22 24 16 1;53 10 29 32 52 16 3 31 49 5 09 01 ki. 4

2;15 0;26 40 0;54 10 1;20 50 1;48 54 01 40 1 05 1 37 ki. 5

2;13 20 0;27 0;53 10 1;20 10 1;47 06 41 40 5 19 8 01 ki. 6

2;09 36 0;27 46 40 0;50 54 40 1;18 41 20 1;43 11 56 28 26 40 38 11 59 01 ki. 7

2;08 0;27 07 30 0;49 56 15 1;18 03 45 1;41 33 45 14 03 45 13 19 20 49 ki. 8

2;05 0;28 48 0;48 06 1;16 54 1;38 33 36 36 8 01 12 49 ki. 9

2;01 30 0;29 37 46 40 0;45 56 06 40 1;15 33 53 20 1;35 10 02 28 27 24 26 40 1 22 41 2 16 01 ki. 10

2 0;30 0;45 1;15 1;33 45 45 1 15 ki. 11

1;55 12 0;31 15 0;41 58 30 1;13 13 30 1;29 21 54 02 15 27 59 48 49 ki. 12

1;52 30 0;32 0;40 15 1;12 15 1;27 00 03 45 2 41 4 49 ki. 13

1;51 06 40 0;32 24 0;39 21 30 1;11 45 20 1;25 48 51 35 06 40 29 31 53 49 ki. 14

1;48 0;33 20 0;37 20 1;10 40 1;23 13 46 40 56 53 ki. 15

The left portion is Friberg’s reconstruction [2007, 440] with simplified headings. The right portion is the existing

Plimpton 322 tablet without headings. All errors have been corrected per Friberg [2007, 440]. Underlined entries are

remaining errors per Britton et al [2011, 524-525, 536]. Zeroes and sexagesimal points added by the present author.

2

Where the numbers came from.

In a major review and analysis of P322, by Britton, Proust, and Shnider (BPS), the authors asked:

“Where do the numbers come from?” [2011, 531]. Their answer was: “The tablet represents the

unfinished result of an investigation of all ratios, r/s, of regular numbers, where

1 < r/s < 2;25, and 1 ≤ s < 1, 0, arranged in descending order of r/s.” [2011, 532]. That

conclusion affirmed the prevailing understanding of the origin of the initial values in P322.

Referring to Friberg’s P322 reconstruction, the igi is the initial value for the construction of the

numbers in P322 [2007, 440]. But where did the igi come from? The BPS answer is: the igi = r/s

[2011, 542]. The igi is the quotient r/s, obtained by dividing the dividend component r by the

divisor component s. This has essentially been the prevalent understanding of the origin of the

initial values in P322, and that view will be called the igi-component theory.

There is an alternative explanation. The igi is either a primary number, a one- or two-place

regular number such that 1 < igi < 1+√2, or the igi is an igi-fraction sum, a higher-order, three-

or more-place regular number, within the same igi range, that is the sum of a primary number

and its nth part, for n = 5, 8, or 9. “Igi fractions are nth parts, the reciprocal of n, for n = 2, 3, 4,

5…” [Friberg 2007, 374]. This new perspective will be called the igi-fraction theory.

In the extant P322, the igi values 2;24, 2;15, 2;08, 2;05, 2, and 1;48 are primary numbers.

The igi 2;22 13 20 is the sum of 2;08 plus its 9th part (0;14 13 20).

The igi 2;20 37 30 is the sum of 2;05 plus its 8th part (0;15 37 30).

The igi 2;18 53 20 is the sum of 2;05 plus its 9th part (0;13 53 20).

The igi 2;13 20 is the sum of 2;00 plus its 9th part (0;13 20), or 1;40 + its 3rd.

The igi 2;09 36 is the sum of 1;48 plus its 5th part (0;21 36).

The igi 2;01 30 is the sum of 1;48 plus its 8th part (0;13 30), or 1;21 + its 2nd.

The igi 1;55 12 is the sum of 1;36 plus its 5th part (0;19 12).

The igi 1;52 30 is the sum of 1;40 plus its 8th part (0;12 30), or 1;30 + its 4th, or 1;15 + its 2nd.

The igi 1;51 06 40 is the sum of 1;40 plus its 9th part (0;11 06 40).

The 14 P322 primaries are: {1;04, 1;12, 1;15, 1;20, 1;21, 1;30, 1;36, 1;40, 1;48, 2, 2;05, 2;08,

2;15, 2;24}. The extant P322 contains six primaries: 1;48 to 2;24 inclusive. Further, 1;40 is the

primary of 1;51 06 40 and 1;52 30, and 1;36 is the primary of 1;55 12. Other possible primaries

are 1;30, 1;21, and 1;15. At least eight, but possibly 11 of the 14 primaries are listed in, or were

used to construct the extant P322. The extant P322 is about half complete (excepting column

lines, the reverse side is blank), so a completed P322 would presumably contain all 14 primaries,

and the igi-fraction sums derived from them. Since 1/5, 1/8, and 1/9 are the only igi fractions

necessary to generate the igi-fraction sums in the extant P322, only those same three igi fractions

will next be used to generate the igi-fraction sums in a theoretically completed P322.

With 14 primaries and three igi fractions, 42 igi-fraction sums are generated. There are eight

results outside of the 1 < igi < 1+√2 range, and 15 two-place duplicates leaving 19 three- and

four-place results. These 19 igi-fraction sums and the 14 primaries make for a total of 33 igi

entries in a completed P322. Sorted in descending order, the entries are: {2;24}, {2;22 13 20},

{2;20 37 30}, {2;18 53 20}, {2;15}, {2;13 20}, {2;09 36}, {2;08}, {2;05}, {2;01 30}, {2},

3

{1;55 12}, {1;52 30}, {1;51 06 40}, {1;48}, {1;46 40}, {1;41 15}, {1;40}, {1;37 12}, {1;36},

{1;31 07 30}, {1;30}, {1;28 53 20}, {1;26 24}, {1;24 22 30}, {1;23 20}, {1;21},{1;20},

{1;16 48}, {1;15}, {1;12}, {1;11 06 40}, {1;04}.The first 15 entries (in bold style) exactly match

the extant P322, and a total of 33 entries fits the actual size of P322 [cf. BPS 2011, 532].

The numbers in Column I of P322.

It now seems well understood that that the 15 numbers in Column I of P322 are the squares of

the diagonals of unit-length rectangles [BPS 2011, 542, 558]. The numbers in Column I are

sexagesimal fractions. The integer portion is 1; it is equal to the square of the length (the longer

side) of a unit-length rectangle. The fractional portion is equal to the square of the width (the

shorter side). The square root of the entire Column I number is the diagonal, the square root of

the fractional portion is the width, the sum of the diagonal and width is equal to the igi, and the

difference is equal to the igi.bi, the reciprocal of the igi [Friberg 2007, 440]. The diagonal is

equal to ½ (igi + igi.bi), and the width is equal to ½ (igi − igi.bi) [Friberg 2007, 441].

For a unit-length rectangle, the sum of the square of the width and 1 (the square of the length) is

equal to the square of the diagonal. The square of the diagonal is the only number that need be

given to identify or report the squares of all three unit-rectangle sides. For example, for an

igi = 2, the igi.bi = 1/2; the width is ½ (igi − igi.bi) = 3/4, the diagonal is ½ (igi + igi.bi) = 5/4.1

The three sides are 3/4, 1, 5/4 (0;45, 1, 1;15), the width, length, and diagonal of the rectangle in

line 11 of P322. The squares of the sides are 9/16, 1, 1-9/16 (0;33 45, 1, 1;33 45), where 1-9/16

(1;33 45) is the only number required to describe all three squares of all three sides.

In a unit-length rectangle, the shorter side is less than 1. In a unit-width rectangle the shorter side

is equal to 1. The transition from unit-length to unit-width rectangles therefore occurs when

½ (igi − igi.bi) = 1. At this point, the width and length equal 1, so the diagonal is equal to √2.

Since the igi is equal to sum of the width and diagonal, the value of the igi is 1+√2. The igi must

be greater than 1 in order to avoid negative lengths or widths. The igi range of unit-length

rectangles therefore is 1 < igi < 1+√2 [BPS 2011, 542; Friberg 2007, 442].

The P322 igi-fraction sums.

In the Old Babylonian (OB) standard table of reciprocals, there are five sets of sequential regular

numbers: {1, 2, 3, 4, 5, 6}, {8, 9, 10}, {15, 16}, {24, 25}, and {1 20, 1 21} (80, 81). Their

uniqueness provides an opportunity to apply them to igi fractions. A P322 igi-fraction sum is the

sum of a primary and its nth part, for n = 5, 8, or 9. For narrative convenience, a useful modern

substitute, equivalent to an igi-fraction sum, is the product of a primary multiplicand and an igi-

fraction multiplier in the form of (n + 1)/n. An igi-fraction sum must be a regular number,

therefore (n + 1)/n must be regular. The only regular numbers n that satisfy this condition are: 1,

2, 3, 4, 5, 8, 9, 15, 24, and 1 20 (80). The igi-fraction multipliers in the form of (n + 1)/n

therefore are: 2/1, 3/2, 4/3, 5/4, 6/5, 9/8, 10/9, 16/15, 25/24, and 81/80. The equivalent P322 igi-

fraction multipliers therefore are: 6/5 (1+1/5), 9/8 (1+1/8), and 10/9 (1+1/9).

1 Modern fraction notation and terminology is used, and will continue to be employed in the following commentary,

despite its anachronistic nature, because it seems to be helpful in simplifying and clarifying the many arithmetical

operations involving fractional quantities that will be computed or discussed.

4

If P322’s author was aware of all of the igi-fraction multipliers, he clearly did not employ 16/15,

25/24, and 81/80. The igi values that would have been derived from them are not in the extant

P322. For example, 2;08 · (16/15) = {2;16 32}, 2;05 · (25/24) = {2;10 12 30}, and 1;48 · (81/80)

= {1;49 21} are not listed in P322. The extant P322 could have been derived by using only 1/5,

1/8, and 1/9, perhaps chosen because their factors include 5, 2, and 3 [cf. Muroi 2013, under §3].

But if that seems arbitrary, a satisfactory result is also obtained by using 1, 2, 3, 4, 5, 8, and 9.

This seems to be a more logically chosen set because it contains all the sequential single-digit

regular numbers that form regular igi-fraction multipliers in the form of (n + 1)/n (similar in

concept to igi-components where 1 ≤ s ˂ 1 00, are all the one-place regular numbers).

A theoretically complete P322 obtained by using all the single-digit igi-fractions is nearly

identical to a P322 obtained by using only 1/5, 1/8, and 1/9. Only two additional higher-order

igi-fraction sums are generated: 1;25 20, and 1;33 45. The 21 igi-fraction sums and the 14

primaries total 35 igi entries. The first 15 exactly match the extant P322, and 35 igi entries also

fits the actual size of P322 [cf. BPS 2011, 532].2 Either the 5-8-9 version, or the single-digit

version of the igi-fraction theory could have been used to construct P322. (See Table 2.)

Table 2. Generating igi values for Plimpton 322 by employing single-digit igi-fractions.

P P+(1/1)P P+(1/2)P P+(1/3)P P+(1/4)P P+(1/5)P P+(1/8)P P+(1/9)P

1;04 2;08 1;36 1;25 20 1;20 1;16 48 1;12 1;11 06 40

1;12 2;24 1;48 1;36 1;30 1;26 24 1;21 1;20

1;15 1;52 30 1;40 1;33 45 1;30 1;24 22 30 1;23 20

1;20 2;00 1;46 40 1;40 1;36 1;30 1;28 53 20

1;21 2;01 30 1;48 1;41 15 1;37 12 1;31 07 30 1;30

1;30 2;15 2;00 1;52 30 1;48 1;41 15 1;40

1;36 2;24 2;08 2;00 1;55 12 1;48 1;46 40

1;40 2;13 20 2;05 2;00 1;52 30 1;51 06 40

1;48 2;24 2;15 2;09 36 2;01 30 2;00

2;00 2;24 2;15 2;13 20

2;05 2;20 37 30 2;18 53 20

2;08 2;24 2;22 13 20

2;15

2;24 Bold style entries correspond to the extant P322. Underlined entries correspond to a completed P322.

The igi-component theory.

Proust described the igi-component theory. Using only ancient tools, the P322 igi values can be

obtained by dividing each element (r) of the set R of the regular numbers, {1, 2… 2 00, 2 05}, by

each element (s) of the set S of all the one-place regular numbers, {1, 2… 50, 54} [Proust 2011,

664]. Choosing only those results that fall within the range of 1 ˂ r/s ˂ 1+√2, and eliminating

2 The igi-fraction multipliers 9/8, and 10/9 equal 1;07 30, and 1;06 40. If 1;00 is considered to be a primary, then

these two numbers should be considered to be P322 igi entries, increasing the total to a still reasonable 37 entries.

5

duplicates yields 38 igi values, which fits the actual size of P322. The first 15 sorted results are

the 15 igi values in the extant P322 [Proust 2011, 664]. Set R consists of the “numbers more

often used in scribal schools” [Proust 2011, 663], and set S seems logically (not arbitrarily)

chosen which enhances the credibility of the view that P322’s author actually used this method.

Friberg demonstrated how the igi-component theory may be deduced by using the trailing part

algorithm, where one number may be reduced to a ratio of two simpler numbers by successive

multiplications of the last place number by it reciprocal. Friberg reduced all the igi values from

the extant P322 to a ratio of integers (BPS’s r/s) [2007, 442-443]. For example, the igi in line 2

of P322 is 2;22 13 20. It is equal to 1 04/27, where 1 04 is an element r of set R, and 27 is an

element s of set S. Friberg derived the dividends r and the divisors s (the components) for all 15

igi values in the extant P322. The lowest and highest of the dividends r are 2 and 2 05. Set R

therefore is: {2, 3, 4…1 48, 2 00, 2 05} (since r > s, r ≥ 2). The lowest and highest divisors s are

1 and 54. According to Friberg, “the obvious conclusion” is that P322’s author used those 25

values for set S, where 1 ≤ s ˂ 1 00 [2007, 443].

Friberg then demonstrated the calculation of the 38 igi values for a completed P322. Since r > s,

and r/s < s · 2;25 (2;25 ≈ 1+√2), Friberg simply ignores the 700± combinations that are outside

the igi range leaving 200± [2007, 443]. P322’s author could have avoided a complete calculation

of some of the 200± combinations by eliminating some duplicates in advance, but he would still

have had to at least inspect and consider all 200±. For example, with the divisor s = 54, P322’s

author would have had to at least consider those 12 dividends r = 1 00 ≤ r ≤ 2 05, where only

one, 2 05/54, yields a unique result, while all the other r/54 combinations are duplicates.

If P322’s author employed the igi-component algorithm, he would have then multiplied 2 05 by

1/54 (1 06 40) to obtain 2 18 53 20. If he employed the igi-fraction algorithm, the author would

have multiplied 2 05 by 1/9 (6 40) = 13 53 20, and added 2 05 to obtain 2 18 53 20, essentially

the same calculation. Using the igi-component algorithm, the author would have multiplied r,

where r is one of 37 different dividends, by the reciprocal of s, where s is one of 25 different

divisors, presumably employing somewhat fewer than 25 different multiplication tables. Using

the igi-fraction algorithm, the author would have multiplied each one of 14 different primaries,

by three different igi fractions, employing only three different multiplication tables. The

igi-component algorithm produces 200± in-range possibilities which require further action by its

user. The single-digit version of the igi-fraction theory produces 60± in-range possibilities, while

the 5-8-9 version produces 34 in-range possibilities requiring further action by the user.

The igi-component theory, in conjunction with several different notions as to how the extant

P322 numbers were determined, has been asserted or affirmed by previous commentators

including Neugebauer and Sachs in 1945, de Solla Price in 1964, Friberg in 1981 and 2007,

Proust in 2011, and BPS in 2011, but it has been challenged by only a few, most notably Robson

in 2001. A significant assumption in the igi-component theory is that P322’s author employed a

single algorithm, r/s, to generate every igi in P322 [Robson 2001, 195]. In particular, every one-

and two-place igi is obtained by dividing the components r and s, in exactly the same way all the

other higher-order three- and four-place igi values are obtained.

6

The origin of the igi values in MS 3971#3.

MS 3971 is a large fragment of a clay tablet from the Martin Schøyen collection, dating from

before 1795 BCE, and containing nine mathematical problems [Friberg 2007, 245]. The third

problem (MS 3971#3) contains five examples of igi-igi.bi problems. Five igi and igi.bi values

are given: {1 04, 56 15}, {1 40, 36}, {1 30, 40}, {1 20, 45}, {1 12, 50} [Friberg 2007, 252].

The five given igi values are in the range 1 < igi < 1+√2, and therefore produce unit-length

rectangles. In each of the five problems, the diagonal and the width are found. The five squares

of those five diagonals would have appeared in Column I in the unfinished portion of P322. The

width can be found directly by solving ½ (igi - igi.bi), as is the case in P322, but the author’s

solution procedure is to first find the diagonal by solving ½ (igi + igi.bi). He then squares the

diagonal result, and then employs the OB Diagonal Rule (Pythagorean Theorem). From the

square of the diagonal, the author subtracts 1, and then finds its square root, which equals the

width [Friberg 2007, 252-253]. That procedure follows the heading of Column I in P322. BPS

translate that heading as: “The number square of the diagonal (from) which 1 is subtracted and

(that of) the width comes up” [2011, 530].

But where did the five igi and igi.bi values come from? Friberg states that they were “borrowed

from the Old Babylonian standard table of reciprocals” [2007, 253], which contains the number

2, and eight two-place numbers between 1 and 1+√2 including the end values: {1;04, 1;12, 1;15,

1;20, 1;21 1;30, 1;40, 2, 2;24}. For MS 3971#3, it is asserted that its igi values were simply

picked from the OB standard table of reciprocals. That is consistent with the igi-fraction theory.

No computation is required since the given values are primary numbers. For P322, the origin of

these same igi values is asserted to be by the computation of the igi-component algorithm.

A comparison of the igi-component and igi-fraction algorithms and an initial conclusion.

It seems likely that P322’s author knew beforehand all the 14 P322 primaries, the one- or two-

place regular numbers within the unit-rectangle igi range. But if he knew only the nine primaries

from the OB standard table of reciprocals, single digit igi-fraction sums would have generated

the other five primaries (See Table 2). P322’s author generated higher-order regular numbers in

between 1 and 1+√2. Rather than doubling or tripling lower-order regular numbers [cf. Robson

2001, 196], the foregoing analysis suggests that OB scribes employed this igi-fraction sum

technique, which consists of multiplying each primary by 0;12 (1/5), 0;07 30 (1/8), and 0;06 40

(1/9), and adding each result to its primary. The igi-fraction sum algorithm generates 19 higher-

order igi values, and 15 duplicates. The igi-component algorithm generates 1 one-place, 13 two-

place igi values, 24 higher-order igi values, and 160± duplicates (174± counting the primaries).

The igi-fraction algorithm is more direct, and the simpler of the two in application and execution.

BPS demonstrated that the numbers in Columns II and III of P322 were obtained by the

application of the trailing part algorithm to the width and diagonal of the unit-length rectangle

[2011, 543]. Friberg concluded that the numbers in Columns II and III are “factor-reduced cores”

obtained by the application of the trailing part algorithm to the width and diagonal of the unit-

length rectangle [2007, 436]. In either case, the r‘s and s‘s of the igi-component theory seem to

play no further role in the construction of the tablet [BPS 2011, 540]. If the igi values in P322 are

indeed igi-fraction sums as proposed herein, then the r‘s and s‘s of the igi-component theory

played no role whatsoever in the construction of the tablet.

7

Igi-fraction sums, differences, and symmetrical pairs.

A higher-order igi can also be obtained by finding the difference between a primary number and

its nth part. A useful modern substitute, equivalent to an igi-fraction difference, is the product of a

primary multiplicand and an igi-fraction multiplier in the form of (n − 1)/n.3 An igi-fraction

difference must be a regular number, therefore (n − 1)/n must be regular. The only regular

numbers n that satisfy this condition are: 2, 3, 4, 5, 6, 9, 10, 16, 25, and 1 21. The igi-fraction

multipliers in the form of (n − 1)/n therefore are: 1/2, 2/3, 3/4, 4/5, 5/6, 8/9, 9/10, 15/16, 24/25,

and 80/81, and they are the reciprocals of the (n + 1)/n multipliers.

A useful application of igi-fraction differences is the calculation of reciprocals. It was previously

shown that the igi 2;09 36 is equal to 1;48 plus its 5th part, the equivalent of 1;48 · (6/5).

The reciprocal of 1;48 is 0;33 20. The reciprocal of 2;09 36 (its igi.bi) is the equivalent of

0;33 20 · (5/6), or 0;33 20 less its 6th part (0;05 33 20) = 0;27 46 40. Another application is the

creation of symmetrical pairs. For the primary 1;15, the (80/81-81/80) symmetrical pair is

1;14 04 26 40 and 1;15 56 15, the (24/25-25/24) symmetrical pair is 1;12 and 1;18 07 30, the

(15/16-16/15) symmetrical pair is 1;10 18 45 and 1;20. These symmetrical pairs fill in the gaps

between low-order primaries. All of these pairs are found in the AO 6456 tablet.

AO 6456.

AO 6456 is a Seleucid-era (358 BC-63 BC) clay tablet (20± by 12± cm) acquired by the Louvre

[Muroi 2013, under §1; Neugebauer [1935] 1973, 15]. It contains 157 reciprocal pairs, the

majority of which are higher-order numbers [Muroi 2013, under §2]. It was published by

Thureau-Dangin in 1922 [plates lv-lviii] and Neugebauer published an analysis in 1935 [14-22].

There are two columns of reciprocal pairs on each side of the AO 6456 tablet. The obverse, and

the first column of the reverse comprise the first section (Section 1), containing 122 reciprocal

pairs in ascending order running from 1;00 to 1;58 39 08 26 15 (corrected). The second column

on the reverse side comprises the second section (Section 2), containing 35 reciprocal pairs

running from 2;00 25 38 14 52 25 29 46 00 29 37 46 40 (corrected) to 3. (See Appendix 1.)

Because of the 1,500±-year age difference between P322 and AO 6456, a comparison of the two

may be said to be inappropriate [cf. Robson 2001, 187,189]. Although separated by a millennium

and a half, they share the same mathematical tradition, and it seems there is a direct corollary

between the two. It is proposed herein that the higher-order igi values in P322 were derived by

igi-fraction sums, and that the higher-order reciprocal pairs in AO 6456 were similarly derived

by igi-fraction sums and differences. The substantial volume of analyzable data available in

AO 6456, 157 reciprocal pairs vs. 15 in P322, makes it impossible to disregard.

Where the numbers came from.

Section 1 of AO 6456 is based on 31 primary multiplicands: 1, 2, and all the two- and three-place

regular numbers in between. Three pairs of igi-fraction symmetrical multipliers were employed:

(15/16)-(16/15), (24/25)-(25/24), and (80/81)-(81/80). Multiplying the 31 primaries by the six

3 Sexagesimal points and zeroes will continue to be added, consistent with the foregoing P322 commentary. Modern

fraction equivalents will continue to be used. The following commentary will characterize both igi-fraction sums

and igi-fraction differences as products. A primary number may therefore continue to be called a multiplicand.

8

multipliers yields 186 primary products.4 Eliminating two- and three-place results, and those that

fall outside the target range, leaves 96 four- or higher-place primary products. Of the 96, 32 are

duplicates or triplicates leaving 64 unique numbers. Section 1 includes 57 of the 64.5 These 57

primary products and the 30 (2 is not listed) primary multiplicands account for 87 of the 122

numbers listed in Section 1 of AO 6456 leaving 35 remainders. (See Appendix 2.)

One primary product is 1;45 28 07 30. It is equal to 1;41 15 · (25/24), or 1;44 10 · (81/80), or

1;52 30 · (15/16). It has six possible successors: the products of each one of the six igi-fraction

multipliers and 1;45 28 07 30. For that 1;45 28 07 30 that is equal to 1;52 30 · (15/16), its

predecessor is 1;52 30, its first sequential successor is 1;45 28 07 30 · (15/16) =

1;38 52 37 01 52 30, its second sequential successor is 1;45 28 07 30 · (15/16) · (15/16) =

1;32 41 49 43 00 28 07 30. Both of these sequential successors are a Section 1 remainder.

Of the 35 remainders, 30 are sequential successors. The five other remainders are non-sequential

successors. Four of these five occur at the very beginning of the text, where it appears that the

author mistakenly or experimentally chose an igi-fraction multiplier different from the one that is

a factor of its predecessor. From that point on, however, the author consistently generated

sequential successors except for one later entry, assumed to be an error. (See Appendix 3.)

Section 2 of AO 6456 contains 35 reciprocal pairs between 2 and 3. The primary multiplicands

2;30, 2;40, 2;42, 2;25 48, 2;50 40, and 2;52 48 are missing. Section 2 is clearly incomplete.6 The

only igi-fraction calculation of primary products necessary in Section 2 is for 2;36 15 and

2;48 45. All the others are the doubles of the Section 1 primary products between 1;00 and 1;30.

Of the 35 Section 2 numbers (including 3), 33 are the doubles of Section 1 numbers. One non-

double, 2;00 25 38 14 52 25 29 46 00 29 37 46 40 (corrected) is the second sequential successor

(the first is missing) to 2;03 27 24 26 40 = 2;05 · (80/81). Its presence is perhaps due to its

simpler (81/80) reciprocal computation path: 0;28 48 - 0;29 09 36 - 0;29 31 28 12 –

0;29 53 36 48 09 [cf. Neugebauer [1935] 1973, 15]. The other is 2;01 04 08 03 00 27. It is the

third sequential successor to 1;56 38 24, but that seems unlikely since the first and second

sequential successors are missing in Section 1. It has an atypical 17-place reciprocal. Perhaps

2;01 55 57 28 33 34 48 53 20 was the intended entry, the reciprocal of 0;29 31 28 12, and the

missing first sequential successor to 2;03 27 24 26 40.

Some Section 2 observations are possible despite its incompleteness. The author did calculate the

three in-range primary products of the primary 3 since 2;48 45 and 2;57 46 40 are listed. The

primary product 2;52 48 = 3 · (24/25) is missing, but its sequential successor 2;45 53 16 48 is

listed. The primary product 2;48 45 is listed; it is one of the two three-place primaries whose

halves are not Section 1 primaries. The other, 2;36 15, is also listed, and so are some of its

successors: 2;34 19 15 33 20, 2;38 12 11 15, and 2;46 40. The largest of the listed two-place

primaries is 2;24. The remaining numbers in Section 2 are all of the two- or three-place primaries

≤ 2;24, some of their primary products, and one sequential successor. (See Appendix 4.)

4 The author knew all the two- and three-place multiplicands, or he derived the three-place from the two-place by

applying the six igi-fraction multipliers. (See the left column of Appendix 2.) 5 Of the missing seven, three are obvious errors. All the igi-difference primary products of 1;49 21 are missing,

1;17 45 36 has a successor. The others are apparently random calculation or compilation omissions. 6 The AO 6456 colophon includes the following: “The first section. ‘1’ is a head number. ‘2’ is a head number. It is

not completed” [Muroi 2013, under §2].

9

But where did 1;29 12 19 26 34 23 19 49 38 08 36 52 20 46 26 40 (corrected) come from? It is

the third sequential successor to the primary product 1;32 35 33 20. Its presence seems likely due

to its primary product’s simpler (81/80) reciprocal path: 0;38 52 48 – 0;39 21 57 36 –

0;39 51 29 04 12 – 0;40 21 22 41 00 09. Its two predecessors are missing, but both are greater

than 1;30, and therefore not candidates for doubling and eventual inclusion in Section 2.

A summary of the AO 6456 igi-fraction calculations and conclusion.

The set of one-place igi-components generates only 38 reciprocal pairs for P322. If the igi-

component theory were applied to AO 6456, a larger set is needed. The next similarly defined

igi-component set is the two-place igi-components. For the AO 6456 Section 1 target range, the

two-place igi-components are all the divisors s, where 1 00 ≤ s ≤ 57 36, and all the dividends r,

where 1 04 ≤ r ≤ 1 52 30. There are 13,800± possible combinations, of which 2,000± in-range

results require further action by its user, and not all the AO 6456 numbers would be generated.

The igi-fraction technique is more direct and simpler. All that AO 6456’s author needed was the

set of two-place numbers between 1 and 3, and six multiplication tables.

For each Section 1 two- and three-place primary, the author added 1/15, 1/24, and 1/80, and

subtracted 1/16, 1/25, and 1/81. By that process, he obtained all the four-place regular

sexagesimals in his target range, many five-place and some six- and seven-place numbers. He

then obtained the reciprocals of the unique primary products by applying the inverse process to

the reciprocals of the primaries. The result included all 25 one- and two-place reciprocals, 21 of

the 26 three-place, and 16 of the 37 four-place Section 1 reciprocals between 0;30 and 1.

The author then calculated sequential successors to the primary products. He did not compute all

the possibilities, but selected only some of the four- and five-place primary products. He clearly

excluded all seven-place and all, but one, of the six-place primary products. A factor in his

selection appears to be the number of places in the reciprocal of the primary product. Those with

two or three places were favored over the higher-place reciprocals. By doing so, he increased the

number of three- and four-place reciprocals in the text. These additional calculations are seen as

supplementary to the principal goal of generating the primary products.

The primary data for the construction of P322 is the set of one- and two-place regular numbers

between 1 and 1+√2. The primary data for the construction of AO 6456 is 1, 2, and 3, and all the

two- and three-place regular numbers in between. In both tablets, the range of the primary data is

the same as the target range. The higher-order reciprocal pairs in P322 were obtained by adding

igi-fraction products to the primary numbers. The higher-order reciprocal pairs in AO 6456 were

obtained by the addition and subtraction of three igi-fraction products to and from the primaries.

The different parameters (5, 8, 9, or single-digit igi-fractions in P322 vs. 15, 24, 1 20, and 16, 25,

1 21 in AO 6456) suggest that P322’s author only partially applied the igi-fraction algorithm, or

that it was only partially developed at the time P322 was constructed, and that it may have been a

precursor of the more fully evolved AO 6456 algorithm.

______________________________________________________________________________

Acknowledgement. I wish to sincerely thank Dr. Gary Devore of the New Hampshire Writers’ Project for his

thoughtful criticisms, and helpful suggestions.

APPENDICES

Notes to Appendix 1

All entries have been corrected. For the original text, see Neugebauer [[1935] 1973, 14-22].

The numbering and the sexagesimal format of the listed entries is that employed by Neugebauer.

That numbering is also consistent with Thureau-Dangin, except for the subscripted numbers.

Notes to Appendix 2

All 96 possible primary products within the target range are rendered in bold style font.

The seven primary products missing from the text are rendered in bold italic font.

The primary products of the two-place primary numbers include all the three-place regular

numbers within the target range. See the left column of Appendix 2.

The primary products of the two- and three-place primary numbers include all the four-place

regular numbers within the target range.

Notes to Appendix 3

The third column from the left contains the primary products from Appendix 2. The column on

the right contains sequential successors to the primary products unless otherwise noted.

Non-sequential successors are preceded and denoted by (NS - the igi-fraction multiplier).

All underlined entries are those sequential successors that are also primary products.

A primary product or successor in italics is not a listed entry in the AO 6456 text.

A successor in bold style font is a listed entry in the AO 6456 text.

Notes to Appendix 4

Appendix 4 provides the primary and igi-fraction multiplier of the Section 2 primary products.

Section 1 numbers that are halves of Section 2 entries are in brackets and preceded by (½ =).

The underlined Section 2 numbers are the two non-doubled entries.

Any number in Appendix 4 rendered in italics is missing in the AO 6456 text.

Non-sequential successors are preceded and denoted by (NS - the igi-fraction multiplier).

Vs.

I.1

11

Vs.

II.1

1,1

5,5

6,1

54

7,2

4,2

6,4

0

Vs.

I.2

1,

. ,1

6,5

3,5

3,2

05

9,4

3,1

0,5

0,5

2,4

8V

s.II

.21

,16

,48

46

,52

,30

Vs.

I.3

1,

. ,4

0,5

3,2

05

9,1

9,3

4,1

3,7

,30

Vs.

II.3

1,1

6,5

3,1

2,1

1,1

54

6,4

9,1

9,4

0,1

4,4

8,5

3,2

0

Vs.

I.4

1,

. ,4

55

9,1

5,3

3,2

0V

s.II

.41

,17

,09

,37

,46

,40

46

,39

,21

,36

Vs.

I.5

1,1

,2,6

,33

,45

58

,58

,56

,38

,24

Vs.

II.5

1,1

7,4

0,2

0,1

64

6,2

0,5

4,5

1,3

0,1

4,3

,45

Vs.

I.6

1,1

,26

,24

58

,35

,37

,30

Vs.

II.6

1,1

8,7

,30

46

,4,4

8

Vs.

I.7

1,1

,30

,33

,45

58

,31

,39

,35

,18

,31

,6,4

0V

s.II

.71

,18

,43

,55

,12

45

,43

,29

,3,1

2,3

5,3

3,2

0

Vs.

I.8

1,1

,43

,42

,13

,20

58

,19

,12

Vs.

II.8

1,1

9,6

,5,3

7,3

04

5,3

0,4

0

Vs.

I.9

1,2

,12

,28

,48

57

,52

,13

,20

Vs.

II.9

1,1

9,

. ,4

4,2

6,4

04

5,3

3,4

5

Vs.

I.1

01

,2,3

05

7,3

6V

s.II

.9a

1,2

04

5

Vs.

I.1

11

,3,1

2,3

5,3

3,2

05

6,5

7,1

1,1

5V

s.II

.10

1,2

0,2

2,3

1,5

1,6

,40

44

,47

,23

,8,9

,36

Vs.

I.1

21

,3,1

6,5

2,3

05

6,5

3,2

0V

s.II

.11

1,2

0,5

4,3

1,6

,40

44

,29

,40

,39

,50

,37

,30

Vs.

I.1

31

,45

6,1

5V

s.II

.12

1,2

14

4,2

6,4

0

Vs.

I.1

41

,4,1

8,1

,28

,53

,20

55

,59

,13

,55

,12

Vs.

II.1

2a

1,2

1,2

2,4

8,4

54

4,1

4,1

2,2

8,4

8

Vs.

I.1

51

,4,4

85

5,3

3,2

0V

s.II

.13

1,2

1,5

5,1

24

3,5

6,4

3,7

,30

Vs.

I.1

5a

1,5

,6,1

55

5,1

7,4

5,3

6V

s.II

.14

1,2

2,

. ,4

54

3,5

3,4

4,4

1,2

8,5

3,2

0

Vs.

I.1

61

,5,3

2,9

,36

54

,55

,53

,54

,22

,30

Vs.

II.1

51

,22

,18

,16

,17

,46

,40

43

,44

,24

Vs.

I.1

71

,5,3

6,3

65

4,5

2,1

0,5

1,5

1,6

,40

Vs.

II.1

61

,22

,23

,50

,51

,33

,45

43

,41

,26

,24

Vs.

I.1

81

,5,5

0,3

7,2

,13

,20

54

,40

,30

Vs.

II.1

71

,22

,56

,38

,24

43

,24

,10

Vs.

I.1

91

,5,5

5,4

,41

,15

54

,36

,48

Vs.

II.1

81

,23

,20

43

,12

Vs.

I.2

01

,6,2

5,4

8,2

75

4,1

1,3

2,1

2,4

1,3

5,2

8,2

3,4

2,1

3,2

0V

s.II

.19

1,2

3,5

8,5

0,5

2,4

84

2,5

2,

. ,5

9,1

5,3

3,2

0

Vs.

I.2

11

,6,4

05

4V

s.II

.20

1,2

4,2

2,3

04

2,4

0

Vs.

I.2

1a

1,7

,30

53

,20

Vs.

II.2

11

,24

,16

,47

,24

,26

,40

42

,42

,53

,26

,15

Vs.

I.2

21

,8,1

65

2,4

4,3

,45

Vs.

II.2

21

,25

,20

42

,11

,15

Vs.

I.2

31

,8,2

0,3

7,3

05

2,4

0,2

9,3

7,4

6,4

0V

s.II

.23

1,2

5,2

5,4

6,5

2,3

04

2,8

,23

,42

,13

,20

Vs.

I.2

41

,9,7

,12

52

,5V

s.II

.24

1,2

6,2

44

1,4

0

Vs.

I.2

51

,9,2

6,4

05

1,5

0,2

4V

s.II

.25

1,2

6,4

8,2

04

1,2

8,1

9,1

2

Vs.

I.2

61

,10

,18

,45

51

,12

Vs.

II.2

61

,27

,28

,48

41

,9,8

,8,5

3,2

0

Vs.

I.2

71

,11

,6,4

05

0,3

7,3

0V

s.II

.27

1,2

7,4

7,2

9,2

2,5

7,4

6,4

04

1,

. ,2

2,3

0

Vs.

I.2

81

,11

,11

,29

,3,4

55

0,3

4,4

,26

,40

Vs.

II.2

81

,27

,53

,26

,15

40

,57

,36

Vs.

I.2

91

,12

50

Vs.

II.2

91

,28

,28

,24

,57

,36

40

,41

,24

,22

,30

Vs.

I.3

01

,12

,49

,44

9,2

6,1

8,3

0,5

6,1

5V

s.II

.30

1,2

8,5

3,2

04

0,3

0

Vs.

I.3

11

,12

,54

49

,22

,57

,46

,40

Vs.

II.3

11

,28

,59

,21

,19

,41

,15

40

,27

,15

,33

,20

Vs.

I.3

21

,13

,9,3

4,2

9,8

,8,5

3,2

04

9,1

2,2

7V

s.II

.32

1,2

9,1

2,1

9,2

6,3

4,2

3,1

9,-

40

,21

,22

,41

, .

,9

Vs.

I.3

31

,13

,14

,31

,52

,30

49

,9,7

,12

-49

,38

,8,3

6,5

2,2

0,4

4,2

6,4

0

Vs.

I.3

41

,13

,43

,40

,48

48

,49

,41

,15

Vs.

II.3

31

,29

,40

,50

,24

,27

40

,8,3

2,4

4,5

7,2

8,2

9,5

5,2

0,9

,52

,35

,33

,20

Vs.

I.3

51

,13

,48

,40

,30

48

,46

,22

,59

,25

,25

,55

,33

,30

Vs.

II.3

41

,30

40

Vs.

I.3

61

,14

,4,2

6,4

04

8,3

6V

s.II

.34

a1

,31

,7,3

03

9,3

0,2

2,1

3,2

0

Vs.

I.3

6a

1,1

54

8V

s.II

.35

1,3

2,9

,36

39

,3,4

5

Vs.

I.3

71

,15

,51

,6,4

04

7,2

7,3

9,2

2,3

0

Ap

pen

dix

1 -

Nu

mer

ica

l tr

an

scri

pti

on

-A

O 6

456 -

Ob

ver

se

Rs.

I.1

1,3

2,1

5,5

0,3

7,3

03

9,1

,6,2

3,3

2,2

0,4

4,2

6,4

0R

s.II

.12

, .

,25

,38

,14

,52

,25

,-2

9,5

3,3

6,4

8,9

Rs.

I.2

1,3

2,3

5,3

3,2

03

8,5

2,4

8-2

9,4

6,

. ,2

9,3

7,4

6,4

0R

s.I.

31

,32

,41

,49

,43

, .

,28

,7,3

03

8,5

0,1

0,8

Rs.

II.2

2,1

,4,8

,3,

. ,2

72

9,4

4,6

,28

,51

,27

,46

,36

,32

,42

,52

,17

,26

,54

,48

,53

,20

Rs.

I.4

1,3

3,1

8,4

3,1

23

8,3

4,4

8,5

3,2

0R

s.II

.32

,1,3

02

9,3

7,4

6,4

0R

s.I.

51

,33

,45

38

,24

Rs.

II.4

2,3

,1,7

,30

29

,15

,49

,47

,39

,15

,33

,20

Rs.

I.6

1,3

4,4

8,5

3,2

03

7,5

8,7

,30

Rs.

II.5

2,3

,27

,24

,26

,40

29

,9,3

6R

s.I.

71

,34

,55

,18

.45

37

,55

,33

,20

Rs.

II.6

2,4

,24

,57

,36

28

,56

,6,4

0R

s.I.

81

,36

37

,30

Rs.

II.7

2,5

28

,48

Rs.

I.8

a1

,36

,27

,2,1

3,2

03

7,1

9,2

9,1

6,4

8R

s.II

.7a

2,6

,33

,45

28

,26

,40

Rs.

I.9

1,3

7,1

23

7,2

,13

,20

Rs.

II.8

2,8

28

,7,3

0R

s.I.

9a

1,3

7,3

9,2

2,3

03

6,5

1,5

0,2

4R

s.II

.8a

2,9

,36

27

,46

,40

Rs.

I.1

01

,37

,32

,45

,58

,50

,51

,51

,6,4

03

6,5

4,2

0,1

5R

s.II

.92

,10

,12

,30

27

,38

,52

,48

Rs.

I.1

11

,38

,18

,14

,24

36

,37

,15

,56

,15

Rs.

II.1

02

,11

,41

,14

,4,2

6,4

02

7,2

0,1

5R

s.I.

12

1,3

8,4

5,5

5,3

3,2

03

6,2

7R

s.II

.11

2,1

3,2

02

7R

s.I.

13

1,3

8,5

2,3

7,1

,52

,30

36

,24

,32

Rs.

II.1

1a

2,1

52

6,4

0R

s.I.

14

1,4

03

6R

s.II

.12

2,1

6,4

1,1

52

6,2

0,1

4,4

8,5

3,2

0R

s.I.

14

a1

,41

,8,8

,53

,20

35

,35

,44

,31

,52

,30

Rs.

II.1

32

,16

,32

26

,22

,1,5

2,3

0R

s.I.

15

1,4

1,1

53

5,3

3,2

0R

s.II

.14

2,1

8,5

3,2

02

5,5

5,1

2R

s.I.

15

a1

,42

,24

35

,9,2

2,3

0R

s.II

.15

2,2

0,3

7,3

02

5,3

6R

s.I.

16

1,4

2,5

2,5

0,2

2,1

3,2

03

4,5

9,3

1,1

2R

s.II

.16

2,2

2,2

2,5

8,7

,30

25

,17

,2,1

3,2

0R

s.I.

17

1,4

3,4

0,4

83

4,4

3,2

0R

s.II

.17

2,2

42

5R

s.I.

17

a1

,44

,10

34

,33

,36

Rs.

II.1

82

,25

,38

,82

4,4

3,9

,15

,28

,7,3

0R

s.I.

18

1,4

5,2

0,5

9,1

5,3

3,2

03

4,1

0,1

8,4

5R

s.II

.19

2,2

7,3

7,2

12

4,2

3,1

1,2

9,4

2,4

2,5

7,4

6,4

0R

s.I.

19

1,4

5,2

8,7

,30

34

,8R

s.II

.20

2,2

8,8

,53

,20

24

,18

Rs.

I.1

9a

1,4

6,4

03

3,4

5R

s.II

.21

2,3

1,4

2,1

3,2

02

3,4

3,4

9,4

1,1

5R

s.I.

20

1,4

83

3,2

0R

s.II

.22

2,3

3,3

62

3,2

6,1

5R

s.I.

20

a1

,48

,30

,25

33

,10

,39

,21

,36

Rs.

II.2

32

,34

,19

,15

,33

,20

23

,19

,40

,48

Rs.

I.2

11

,49

,13

,36

32

,57

,32

,20

,37

,30

Rs.

II.2

42

,35

,20

,40

,32

23

,10

,27

,25

,45

,7,1

,52

,30

Rs.

I.2

21

,49

,21

32

,55

,18

,31

,6,4

0R

s.II

.25

2,3

6,1

52

3,2

,24

Rs.

I.2

2a

1,5

0,3

5,3

1,1

23

2,3

3,7

,30

Rs.

II.2

62

,38

,12

,11

,15

22

,45

,20

Rs.

I.2

31

,50

,43

, .

,45

32

,30

,55

,19

,36

,57

,17

,2,1

3,2

0R

s.II

.27

2,4

5,5

3,1

6,4

82

1,4

2,5

Rs.

I.2

41

,51

.6,4

03

2,2

4R

s.II

.28

2,4

6,4

02

1,3

6R

s.I.

25

1,5

1,1

4,1

1,3

9,3

6,3

3,4

53

2,2

1,4

8,2

6,4

0R

s.II

.29

2,4

8,4

52

1,2

0R

s.I.

26

1,5

2,3

03

2R

s.II

.30

2,5

5,4

6,5

2,3

02

0,2

8,4

8R

s.I.

26

a1

,53

,46

,40

31

,38

,26

,15

Rs.

II.3

12

,57

,46

,40

20

,15

Rs.

I.2

71

,53

,54

,22

,30

32

,36

,17

,46

,40

Rs.

II.3

23

20

Rs.

I.2

81

,55

,12

31

,15

Rs.

I.2

91

,55

,44

,26

,40

31

,6,1

4,2

4R

s.I.

30

1,

56

,38

,24

30

,51

,51

,6,4

0R

s.I.

31

1,5

7,3

,19

,10

,37

,2,1

3,2

03

0,4

5,1

6,5

2,3

0R

s.I.

32

1,5

7,1

1,1

53

0,4

3,1

2R

s.I.

33

1,5

7,5

7,5

3,1

6,4

83

0,3

1,3

,16

,52

,30

Rs.

I.3

41

;58

,31

,6,4

03

0,2

2,3

0R

s.I.

35

1,5

8,3

9,8

,26

,15

30

,20

,26

,40

Ap

pen

dix

1 -

Nu

meri

cal

tra

nsc

rip

tion

-A

O 6

456 -

Rev

erse

1;00 81/80 1;00 45 1;00 45 15/16 < 1 1;26 24 15/16 1;21

1;00 25/24 1;02 30 1;00 45 24/25 < 1 1;26 24 24/25 1;22 56 38 24

1;00 16/15 1;04 1;00 45 80/81 1;00 1;26 24 80/81 1;25 20

1;04 15/16 1;00 1;00 45 81/80 1;01 30 33 45 1;26 24 81/80 1;27 28 48

1;04 24/25 1;01 26 24 1;00 45 25/24 1;03 16 52 30 1;26 24 25/24 1;30

1;04 80/81 1;03 12 35 33 20 1;00 45 16/15 1;04 1;26 24 16/15 1;32 09 36

1;04 81/80 1;04 48 1;02 30 15/16 < 1 1;33 45 15/16 1;27 53 26 15

1;04 25/24 1;06 40 1;02 30 24/25 1;00 1;33 45 24/25 1;30

1;04 16/15 1;08 16 1;02 30 80/81 1;01 43 42 13 20 1;33 45 80/81 1;32 35 33 20

1;12 15/16 1;07 30 1;02 30 81/80 1;03 16 52 30 1;33 45 81/80 1;34 55 18 45

1;12 24/25 1;09 07 12 1;02 30 25/24 1;05 06 15 1;33 45 25/24 1;37 39 22 30

1;12 80/81 1;11 06 40 1;02 30 16/15 1;06 40 1;33 45 16/15 1;40

1;12 81/80 1;12 54 1;04 48 15/16 1;00 45 1;37 12 15/16 1;31 07 30

1;12 25/24 1;15 1;04 48 24/25 1;02 12 28 48 1;37 12 24/25 1;33 18 43 12

1;12 16/15 1;16 48 1;04 48 80/81 1;04 1;37 12 80/81 1;36

1;15 15/16 1;10 18 45 1;04 48 81/80 1;05 36 36 1;37 12 81/80 1;38 24 54

1;15 24/25 1;12 1;04 48 25/24 1;07 30 1;37 12 25/24 1;41 15

1;15 80/81 1;14 04 26 40 1;04 48 16/15 1;09 07 12 1;37 12 16/15 1;43 40 48

1;15 81/80 1;15 56 15 1;06 40 15/16 1;02 30 1;41 15 15/16 1;34 55 18 45

1;15 25/24 1;18 07 30 1;06 40 24/25 1;04 1;41 15 24/25 1;37 12

1;15 16/15 1;20 1;06 40 80/81 1;05 50 37 02 12 30 1;41 15 80/81 1;40

1;20 15/16 1;15 1;06 40 81/80 1;07 30 1;41 15 81/80 1;42 30 56 15

1;20 24/25 1;16 48 1;06 40 25/24 1;09 26 40 1;41 15 25/24 1;45 28 07 30

1;20 80/81 1;19 00 44 26 40 1;06 40 16/15 1;11 06 40 1;41 15 16/15 1;48

1;20 81/80 1;21 1;07 30 15/16 1;03 16 52 30 1;42 24 15/16 1;36

1;20 25/24 1;23 20 1;07 30 24/25 1;04 48 1;42 24 24/25 1;38 18 14 24

1;20 16/15 1;25 20 1;07 30 80/81 1;06 40 1;42 24 80/81 1;41 08 08 53 20

1;21 15/16 1;15 56 15 1;07 30 81/80 1;08 20 37 30 1;42 24 81/80 1;43 40 48

1;21 24/25 1;17 45 36 1;07 30 25/24 1;10 18 45 1;42 24 25/24 1;46 40

1;21 80/81 1;20 1;07 30 16/15 1;12 1;42 24 16/15 1;49 13 36

1;21 81/80 1;22 00 45 1;08 16 15/16 1;04 1;44 10 15/16 1;37 39 22 30

1;21 25/24 1;24 22 30 1;08 16 24/25 1;05 32 09 36 1;44 10 24/25 1;40

1;21 16/15 1;26 24 1;08 16 80/81 1;07 25 25 55 33 20 1;44 10 80/81 1;42 52 50 22 13 20

1;30 15/16 1;24 22 30 1;08 16 81/80 1;09 07 12 1;44 10 81/80 1;45 28 07 30

1;30 24/25 1;26 24 1;08 16 25/24 1;11 06 40 1;44 10 25/24 1;48 30 25

1;30 80/81 1;28 53 20 1;08 16 16/15 1;12 49 04 1;44 10 16/15 1;51 06 40

1;30 81/80 1;31 07 30 1;12 54 15/16 1;08 20 37 30 1;46 40 15/16 1;40

1;30 25/24 1;33 45 1;12 54 24/25 1;09 59 02 24 1;46 40 24/25 1;42 24

1;30 16/15 1;36 1;12 54 80/81 1;12 1;46 40 80/81 1;45 20 59 15 33 20

1;36 15/16 1;30 1;12 54 81/80 1;13 48 40 30 1;46 40 81/80 1;48

1;36 24/25 1;32 09 36 1;12 54 25/24 1;15 56 15 1;46 40 25/24 1;51 06 40

1;36 80/81 1;34 48 53 20 1;12 54 16/15 1;17 45 36 1;46 40 16/15 1;53 46 40

1;36 81/80 1;37 12 1;16 48 15/16 1;12 1;49 21 15/16 1;42 30 56 15

1;36 25/24 1;40 1;16 48 24/25 1;13 43 40 48 1;49 21 24/25 1;44 58 33 36

1;36 16/15 1;42 24 1;16 48 80/81 1;15 51 06 40 1;49 21 80/81 1;48

1;40 15/16 1;33 45 1;16 48 81/80 1;17 45 36 1;49 21 81/80 1;50 43 00 45

1;40 24/25 1;36 1;16 48 25/24 1;20 1;49 21 25/24 1;53 54 22 30

1;40 80/81 1;38 45 55 33 20 1;16 48 16/15 1;21 55 12 1;49 21 16/15 1;56 38 24

1;40 81/80 1;41 15 1;23 20 15/16 1;18 07 30 1;52 30 15/16 1;45 28 07 30

1;40 25/24 1;44 10 1;23 20 24/25 1;20 1;52 30 24/25 1;48

1;40 16/15 1;46 40 1;23 20 80/81 1;22 18 16 17 46 40 1;52 30 80/81 1;51 06 40

1;48 15/16 1;41 15 1;23 20 81/80 1;24 22 30 1;52 30 81/80 1;53 54 22 30

1;48 24/25 1;43 40 48 1;23 20 25/24 1;26 48 20 1;52 30 25/24 1;57 11 15

1;48 80/81 1;46 40 1;23 20 16/15 1;28 53 20 1;52 30 16/15 2;00

1;48 81/80 1;49 21 1;25 20 15/16 1;20 1;55 12 15/16 1;48

1;48 25/24 1;52 30 1;25 20 24/25 1;21 55 12 1;55 12 24/25 1;50 35 31 12

1;48 16/15 1;55 12 1;25 20 80/81 1;24 16 47 24 26 40 1;55 12 80/81 1;53 46 40

2;00 15/16 1;52 30 1;25 20 81/80 1;26 24 1;55 12 81/80 1;56 38 24

2;00 24/25 1;55 12 1;25 20 25/24 1;28 53 20 1;55 12 25/24 2;00

2;00 80/81 1;58 31 06 40 1;25 20 16/15 1;31 01 20 1;55 12 16/15 > 2

Appendix 2 - AO 6456 - Section 1 Primary Products

1;00 45 81/80 1;01 30 33 45 1;02 16 41 40 18 45

1;00 45 25/24 1;03 16 52 30 1;05 55 04 41 15 [See 1;15 (15/16)]

1;02 30 80/81 1;01 43 42 13 20 1;00 57 58 44 16 47 24 26 40

1;02 30 81/80 1;03 16 52 30 1;04 04 20 09 22 30

1;02 30 25/24 1;05 06 15 1;07 49 00 37 30

1;05 06 15 (NS-15/16)-1;01 02 06 33 45 - (NS-80/81)-1;00 16 53 53 20

1;05 06 15 (NS-80/81)-1;04 18 01 28 53 20 - (NS-15/16)-1;00 16 53 53 20

1;01 26 24 (NS-80/81)-1;00 40 53 20

1;04 24/25 1;01 26 24 0;58 58 56 38 24

1;04 80/81 1;03 12 35 33 20 1;02 25 46 13 39 45 11 06 40

1;04 48 24/25 1;02 12 28 48 0;59 43 10 50 52 48

1;04 48 81/80 1;05 36 36 1;06 25 48 27 - 1;07 15 37 48 20 15

1;04 48 16/15 1;09 07 12 1;13 43 40 48 - 1;18 38 35 31 12

1;06 40 80/81 1;05 50 37 02 12 30 1;05 01 50 39 13 54 34 04 06 40

1;06 40 25/24 1;09 26 40 1;12 20 16 40

1;06 40 16/15 1;11 06 40 1;15 51 06 40 - 1;20 54 31 06 40

1;07 30 15/16 1;03 16 52 30 0;59 19 34 13 07 30

1;07 30 81/80 1;08 20 37 30 1;09 11 52 58 07 30

1;07 30 25/24 1;10 18 45 1;13 14 31 52 30 [See 1;23 20 (15/16)]

1;08 16 24/25 1;05 32 09 36 1;02 54 52 24 57 36

1;08 16 80/81 1;07 25 25 55 33 20 1;06 35 29 18 34 24 11 51 06 40

1;08 16 81/80 1;09 07 12 1;09 59 02 24

1;08 16 25/24 1;11 06 40 1;14 04 26 40 - 1;17 09 37 46 40 - 1;20 22 31 51 06 40

1;08 16 16/15 1;12 49 04 1;17 40 20 16

1;12 24/25 1;09 07 12 1;06 21 18 43 12

1;12 80/81 1;11 06 40 1;10 13 59 30 22 13 20

1;12 54 15/16 1;08 20 37 30 1;04 04 20 09 22 30

1;12 54 24/25 1;09 59 02 24 1;07 11 04 42 14 24

1;12 54 81/80 1;13 48 40 30 1;14 44 02 00 22 30

1;12 54 25/24 1;15 56 15 1;19 06 05 37 30 - 1;22 23 50 51 33 45 [See 1;30, 1;33 45]

1;12 54 16/15 1;17 45 36 1;22 56 38 24 - 1;28 28 24 57 36

1;15 15/16 1;10 18 45 1;05 55 04 41 15 [See 1;00 45 (25/24)]

1;15 80/81 1;14 04 26 40 1;13 09 34 29 08 08 53 20

1;15 81/80 1;15 56 15 1;16 53 12 11 15

1;15 25/24 1;18 07 30 1;21 22 48 45

1;16 48 24/25 1;13 43 40 48 1;10 46 43 58 04 48

1;16 48 80/81 1;15 51 06 40 1;14 54 55 28 23 42 13 20

1;16 48 81/80 1;17 45 36 1;18 43 55 12

1;16 48 16/15 1;21 55 12 1;27 22 52 48

1;20 80/81 1;19 00 44 26 40 1;18 02 12 47 04 41 28 53 20

1;21 15/16 1;15 56 15 1;11 11 29 03 45

1;21 24/25 1;17 45 36 1;14 38 58 33 36

1;21 81/80 1;22 00 45 1;23 02 15 33 45

1;21 25/24 1;24 22 30 1;27 53 26 15 - 1;31 33 09 50 37 30

1;23 20 15/16 1;18 07 30 1;13 14 31 52 30 [See 1;07 30 (25/24)]

1;23 20 80/81 1;22 18 16 17 46 40 1;21 17 18 19 02 23 12 35 33 20

1;23 20 81/80 1;24 22 30 1;25 25 46 52 30

1;23 20 25/24 1;26 48 20 1;30 25 20 50

1;23 20 16/15 1;28 53 20 1;34 48 53 20 - 1;41 08 08 53 20 - 1;47 52 41 28 53 20

1;25 20 24/25 1;21 55 12 1;18 38 35 31 12

1;25 20 80/81 1;24 16 47 24 26 40 1;23 14 21 38 13 00 14 48 53 20

1;25 20 25/24 1;28 53 20 1;32 35 33 20 - 1;36 27 02 13 20

Appendix 3 - Section 1 Sequential Successors

1

1;25 20 16/15 1;31 01 20 1;37 05 25 20

1;26 24 24/25 1;22 56 38 24 1;19 37 34 27 50 24

1;22 56 38 24 (NS-81/80) - 1;23 58 50 52 48

1;26 24 81/80 1;27 28 48 1;28 34 24 36 - 1;29 40 50 24 27

1;26 24 16/15 1;32 09 36 1;38 18 14 24 - 1;44 51 27 21 36

1;30 15/16 1;24 22 30 1;19 06 05 37 30 [See 1;12 54 (25/24)]

1;30 80/81 1;28 53 20 1;27 47 29 22 57 46 40

1;30 81/80 1;31 07 30 1;32 15 50 37 30

1;33 45 15/16 1;27 53 26 15 1;22 23 50 51 33 45 [See 1;12 54 (25/24)]

1;33 45 80/81 1;32 35 33 20 1;31 26 58 06 25 11 06 40-1;30 19 13 41 09 19 07 30 22 13 20-

1;29 12 19 26 34 23 19 49 38 08 36 52 20 44 26 40

1;33 45 81/80 1;34 55 18 45 1;36 06 30 14 03 45

1;33 45 25/24 1;37 39 22 30 1;41 43 30 56 15

1;36 24/25 1;32 09 36 1;28 28 24 57 36

1;36 80/81 1;34 48 53 20 1;33 38 39 20 29 37 46 40

1;37 12 15/16 1;31 07 30 1;25 25 46 52 30

1;37 12 24/25 1;33 18 43 12 1;29 34 46 16 19 12

1;37 12 81/80 1;38 24 54 1;39 38 04 16 03

1;37 12 16/15 1;43 40 48 1;50 35 31 12 - 1;57 57 53 16 48

1;40 80/81 1;38 45 55 33 20 1;37 32 45 58 50 51 51 06 40

1;41 15 15/16 1;34 55 18 45 1;28 59 21 19 41 15

1;41 15 81/80 1;42 30 56 15 1;43 47 49 27 11 15

1;41 15 25/24 1;45 28 07 30 1;49 51 47 48 45

1;42 24 24/25 1;38 18 14 24 1;34 22 18 37 26 24

1;42 24 80/81 1;41 08 08 53 20 1;39 53 13 57 51 36 17 46 40

1;42 24 81/80 1;43 40 48 1;44 58 33 36 - 1;46 17 17 31 12

1;42 24 16/15 1;49 13 36 1;56 30 30 24

1;44 10 15/16 1;37 39 22 30 1;31 33 09 50 37 30

1;44 10 80/81 1;42 52 50 22 13 20 1;41 36 37 53 47 59 00 44 26 40

1;44 10 81/80 1;45 28 07 30 1;46 47 13 35 37 30

1;44 10 25/24 1;48 30 25 1;53 01 41 02 30

1;44 10 16/15 1;51 06 40 1;58 31 06 40

1;46 40 80/81 1;45 20 59 15 33 20 1;44 02 57 02 46 15 18 32 06 40

1;46 40 25/24 1;51 06 40 1;55 44 26 40

1;46 40 16/15 1;53 46 40 2;01 21 46 40

1;48 24/25 1;43 40 48 1;39 31 58 04 48

1;49 21 15/16 1;42 30 56 15 1;36 06 30 14 03 45

1;49 21 24/25 1;44 58 33 36 1;40 46 37 03 21 36

1;49 21 81/80 1;50 43 00 45 1;52 06 03 00 33 45

1;49 21 25/24 1;53 54 22 30 1;58 39 08 26 15 - (NS-15/16)-1;51 14 11 39 36 33 45 [See 2;15]

1;49 21 16/15 1;56 38 24 2;04 24 57 36

1;52 30 15/16 1;45 28 07 30 1;38 52 37 01 52 30 - 1;32 41 49 43 00 28 07 30

1;52 30 80/81 1;51 06 40 1;49 44 21 43 42 13 20

1;52 30 81/80 1;53 54 22 30 1;55 19 48 16 52 30

1;52 30 25/24 1;57 11 15 2;02 04 13 07 30

1;55 12 24/25 1;50 35 31 12 1;46 10 05 57 07 12

1;55 12 80/81 1;53 46 40 1;52 22 23 12 35 33 20

1;55 12 81/80 1;56 38 24 1;58 05 52 48

2;00 80/81 1;58 31 06 40 1;57 03 19 10 37 02 13 20

2;15 15/16 2;06 33 45 1;58 39 08 26 15 - 1;51 14 11 39 36 33 45

2

Appendix 4 – Details of the Section 2 listed numbers.

Listed Primary numbers: 2;01 30, 2;05, 2;08, 2;09 36, 2;13 20, 2,15, 2;16 32, 2;24, 2;33 36,

2;36 15, 2;46,40, 2;48 45, 3;00.

Listed Primary Products and Successors

[1;55 12 (81/80)] 1;56 38 24 – 1;58 05 52 48 – 1;59 34 27 12 36 – 2;01 04 08 03 00 27

[2;01 30 (81/80)] 2;03 01 07 30 [(½ =) 1;01 30 33 45]

[2;05 (80/81)] 2;03 27 24 26 40 [(½ =) 1;01 43 42 13 20]

2;01 55 57 28 33 34 48 53 20 – 2;00 25 38 14 52 25 29 46 00 29 37 46 40

[2;09 36 (24/25)] 2;04 24 57 36 [(½ =) 1;02 12 28 48]

[2;01 30 (25/24)] 2;06 33 45 [(½ =) 1;03 16 52 30]

[2;05 (25/24)] 2;10 12 30 [(½ =) 1;05 06 15]

[2;13 20 (80/81)] 2;11 41 14 04 26 40 [(½ =) 1;05 50 37 02 12 30]

[2;15 (81/80)] 2;16 41 15 [(½ =) 1;08 20 37 30]

[2;13 20 (25/24)] 2;18 53 20 [(½ =) 1;09 26 40]

[2;15 (25/24)] 2;20 37 30 [(½ =) 1;10 18 45]

– NS(81/80) – 2;22 22 58 07 30 [(½ =) 1;11 11 29 03 45

[2;16 32 (25/24)] 2;22 13 20 [(½ =) 1;11 06 40] – 2;28 08 53 20 [(½ =) 1;11 06 40]

[2;16 32 (16/15)] 2;25 38 08 [(½ =) 1;12 49 04] – 2;35 20 40 32 [(½ =) 1;17 40 20 16]

[2;24 (81/80)] 2;25 48 [(½ =) 1;12 54] - 2;27 37 21 [(½ =) 1;13 48 40 30]

[2;33 36 (80/81)] 2;31 42 13 20 [(½ =) 1;15 51 06 40]

[2;36 15 (80/81)] 2;34 19 15 33 20 [(½ =) 1;17 09 37 46 40]

[2;36 15 (81/80)] 2;38 12 11 15 [(½ =) 1;19 06 05 37 30]

[2;52 48 (24/25)] 2;45 53 16 48 [(½ =) 1;22 56 38 24]

[2;48 45 (25/24)] 2;55 46 52 30 [(½ =) 1;27 53 26 15]

[3;00 (80/81)] 2;57 46 40 [(½ =) 1;28 53 20]

References

Britton, John P., Christine Proust, and Steve Shnider. 2011. Plimpton 322: A review and a

different perspective. Archive for History of Exact Sciences 65, no. 5 (September): 519-566.

http://link.springer.com/article/10.1007/s00407-011-0083-4 (last accessed Apr. 2, 2015).

de Solla Price, Derek J. 1964. The Babylonian “Pythagorean Triangle” tablet. Centaurus 10:

219-231.

Friberg, Jøran 1981. Plimpton 322, Pythagorean triples, and the triangle parameter equations.

Historia Mathematica 8: 277-318.

——— 2007. A remarkable collection of Babylonian mathematical texts: Manuscripts in the

Schøyen collection; Cuneiform texts I. New York: Springer.

Muroi, Kazuo 2013. How to construct a large table of reciprocals of Babylonian mathematics.

arXiv:1401.0065 [Math.HO]. http://arxiv.org/abs/1401.0065 (last accessed Apr. 2, 2015).

Neugebauer, Otto 1973. Mathematische Keilschrift-Texte. Berlin: Springer-Verlag.

(Orig. pub. 1935).

——— and Alexander Sachs 1986. Mathematical cuneiform texts, American Oriental Series,

Vol. 29. New Haven: American Oriental Society. (Orig. pub. 1945).

Proust, Christine 2011. On the nature of the table Plimpton 322. Oberwolfach Report 12/2011:

662-665. https://www.mfo.de/document/1109b/OWR_2011_12.pdf.

(last accessed Apr. 2, 2015).

Robson, Eleanor 2001. Neither Sherlock Holmes nor Babylon: A reassessment of Plimpton 322.

Historia Mathematica 28: 167-206.

Roegel, Denis 2011. A reconstruction of Gingerich’s table of regular sexagesimals and a

cuneiform version of the table (1965). http://locomat.loria.fr./gingerich1965/

gingerich1965doc.pdf (last accessed Apr. 2, 2015).

Thureau-Dangin, François 2014. Tablettes d’Uruk à l’usage des prêtres du Temple d’Anu au

temps des Séleucides. Toronto: University of Toronto. (Orig. pub.1922. Paris: Guethner).