Lahun Mathematical Papyri

The mathematical texts most commented on are usually named:

• Lahun IV.2 (or Kahun IV.2) (UC 32159[2]): This fragment contains a table of Egyptian fraction representations of numbers of the form 2/n. A more complete version of this table of fractions is given in the Rhind Mathematical Papyrus.[3]
• Lahun IV.3 (or Kahun IV.3) (UC 32160[4]) contains numbers in arithmetical progression and a problem very much like problem 40 of the Rhind Mathematical Papyrus.[3][5][6] Another problem on this fragment computes the volume of a cylindrical granary.[7] In this problem the scribe uses a formula which takes measurements in cubits and computes the volume and expresses it in terms of the unit khar. Given the diameter (d) and height (h) of the cylindrical granary:

${\displaystyle V=((1+1/3)d)^{2}\,((2/3)h)}$.

In modern mathematical notation this is equal to

${\displaystyle V={\frac {32}{27}}d^{2}h={\frac {128}{27}}r^{2}h}$ (measured in khar).

This problem resembles problem 42 of the Rhind Mathematical Papyrus. The formula is equivalent to ${\displaystyle V={\frac {256}{81}}r^{2}h}$ measured in cubic-cubits as used in the other problems.[8]

• Lahun XLV.1 (or Kahun XLV.1) (UC 32161[9]) contains a group of very large numbers (hundreds of thousands).[3][10]
• Lahun LV.3 (or Kahun LV.3) (UC 32134A[11] and UC 32134B[12]) contains a so-called aha problem which asks one to solve for a certain quantity. The problem resembles ones from the Rhind Mathematical Papyrus (problems 24–29).[3][13]
• Lahun LV.4 (or Kahun LV.4) (UC 32162[14]) contains what seems to be an area computation and a problem concerning the value of ducks, geese and cranes.[3][15] The problem concerning fowl is a baku problem and most closely resembles problem 69 in the Rhind Mathematical Papyrus and problems 11 and 21 in the Moscow Mathematical Papyrus.[14]
• Unnamed fragment (UC 32118B[16]). This is a fragmentary piece.[17]

The Lahun papyrus IV.2 reports a 2/n table for odd n, n = 1, ..., 21. The Rhind Mathematical Papyrus reports an odd n table up to 101.[18] These fraction tables were related to multiplication problems and the use of unit fractions, namely n/p scaled by LCM m to mn/mp. With the exception of 2/3, all fractions were represented as sums of unit fractions (i.e. of the form 1/n), first in red numbers. Multiplication algorithms and scaling factors involved repeated doubling of numbers, and other operations. Doubling a unit fraction with an even denominator was simple, dividing the denominator by 2. Doubling a fraction with an odd denominator however results in a fraction of the form 2/n. The RMP 2/n table and RMP 36 rules allowed scribes to find decompositions of 2/n into unit fractions for specific needs, most often to solve otherwise un-scalable rational numbers (i.e. 28/97 in RMP 31, and 30/53 n RMP 36 by substituting 26/97 + 2/97 and 28/53 + 2/53) and generally n/p by (n − 2)/p + 2/p. Decompositions were unique. Red auxiliary numbers selected divisors of denominators mp that best summed to numerator mn.

• List of ancient Egyptian papyri

• John Legon: A Kahun Mathematical Fragment Archived 3 September 2012 at archive.today
• History of Egyptian fractions
• Medical Papyrus, UCL website
• The Kahun Gynaecological Papyrus
• "Kahun papyrus and Arithmetic Progressions". PlanetMath.
• "Egyptian fraction". PlanetMath.