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The conjecture is named after Paul Erdős and Ernst G. Straus, who formulated it in 1948, but it is connected to much more ancient mathematics; sums of unit fractions, like the one in this problem, are known as Egyptian fractions, because of their use in ancient Egyptian mathematics.
An Egyptian fraction is a finite sum of distinct unit fractions, such as + +. That is, each ... The Erdős–Straus conjecture [17] ...
The Erdős–Graham conjecture in combinatorial number theory on monochromatic Egyptian fraction representations of unity, proved by Ernie Croot in 2000. [12] The Erdős–Stewart conjecture on the Diophantine equation n! + 1 = p k a p k+1 b, solved by Florian Luca in 2001. [13]
An Egyptian fraction is a representation of an irreducible fraction as a sum of distinct unit fractions, such as 5 / 6 = 1 / 2 + 1 / 3 . As the name indicates, these representations have been used as long ago as ancient Egypt , but the first published systematic method for constructing such expansions was described in ...
The paragraph beginning The greedy algorithm for Egyptian fractions, first described in 1202... appears to be explaining two separate things: that the number of unit fractions needed to express 2/n and 3/n is well known, so 4/n is the smallest number like this that is unknown; and, it is known that every 4/n needs at most four terms. Consider ...
The topic of Egyptian fractions has also seen interest in modern number theory; for instance, the Erdős–Graham problem [9] and the Erdős–Straus conjecture [10] concern sums of unit fractions, as does the definition of Ore's harmonic numbers.
An Egyptian fraction is the sum of a finite number of reciprocals of positive integers. According to the proof of the Erdős–Graham problem, if the set of integers greater than one is partitioned into finitely many subsets, then one of the subsets can be used to form an Egyptian fraction representation of 1.
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