A001466 - Denominator of Egyptian fraction for pi - 3
A001466 - Denominator of Egyptian fraction for pi - 3
8, 61, 5020, 128541455, …
While staying in the Thebes region of Egypt in the 1850s, Scottish lawyer and antiquarian Alexander Henry Rhind purchased an ancient six-foot papyrus. First decoded by Hultsch in 1895, the scroll contains dozens of algebraic and geometric techniques and problems. The scroll includes, for instance, an equation for finding cylindrical volume using π ≈ 256/81 ≈ 3.1605. It also includes a table of fractions 2/n for odd n between 5 and 101. The ancient Egyptians’ shorthand for reciprocals led them to prefer to write fractions with non-unit numerators (with the sole exception of 2/3) instead as the sum of unit fractions. These sums have become known as Egyptian fractions.
The sums are non-trivial. For instance, in the Rhind papyrus’s table, 2/15 is not equivalent to 1/15 + 1/15. Instead, it is 1/10 + 1/30. In 1202, Fibonacci proved that any fraction could be represented in this way as a sum of unit fractions, and even provided an algorithm for calculating them.
Although four terms are necessary for 2/101, in the 1950s, Paul Erdős and E.G. Straus conjectured that only three terms are necessary for any 4/n. Sierpiński then suggested the same was true for any 5/n. Neither of these conjectures have ever been proven, but no counter-examples have been found for n < 1014.
