ancient sources in an attempt to discover the methods the Egyptians used in calculating with Egyptian fractions. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the form in the Rhind papyrus. Although these expansions can generally be described as algebraic identities, the methods used by the Egyptians may not correspond directly to these identities. Additionally, the expansions in the table do not match any single identity; rather, different identities match the expansions for prime and for composite denominators, and more than one identity fits the numbers of each type: For small odd prime denominators , the expansion was used. For larger prime denominators, an expansion of the form was used, where is a number with many divisors (such as a practical number) between and . The remaining term was expanded by representing the number as a sum of divisors of and forming a fraction for each such divisor in this sum. As an example, Ahmes' expansion fits this pattern with and , as and . There may be many different expansions of this type for a given ; however, as K. S. Brown observed, the expansion chosen by the Egyptians was often the one that caused the largest denominator to be as small as possible, among all expansions fitting this pattern. For some composite denominators, factored as , the expansion for has the form of an expansion for with each denominator multiplied by . This method appears to have been used for many of the composite numbers in the Rhind papyrus, but there are exceptions, notably , , and . One can also expand For instance, Ahmes expands . Later scribes used a more general form of this expansion, which works when is a multiple of . The final (prime) expansion in the Rhind papyrus, , does not fit any of these forms, but instead uses an expansion that may be applied regardless of the value of . That is, . A related expansion was also used in the Egyptian Mathematical Leather Roll for several cases. Later usage Egyptian fraction notation continued to be used in Greek times and into the Middle Ages, despite complaints as early as Ptolemy's Almagest about the clumsiness of the notation compared to alternatives such as the Babylonian base-60 notation. Related problems of decomposition into unit fractions were also studied in 9th-century India by Jain mathematician