this requirement can be relaxed to allow repeated denominators. However, this relaxed form of Egyptian fractions does not allow for any number to be represented using fewer fractions, as any expansion with repeated fractions can be converted to an Egyptian fraction of equal or smaller length by repeated application of the replacement if k is odd, or simply by replacing by if k is even. This result was first proven by . Graham and Jewett proved that it is similarly possible to convert expansions with repeated denominators to (longer) Egyptian fractions, via the replacement This method can lead to long expansions with large denominators, such as had originally used this replacement technique to show that any rational number has Egyptian fraction representations with arbitrarily large minimum denominators. Any fraction has an Egyptian fraction representation in which the maximum denominator is bounded by and a representation with at most terms. The number of terms must sometimes be at least proportional to ; for instance this is true for the fractions in the sequence , , , , , ... whose denominators form Sylvester's sequence. It has been conjectured that terms are always enough. It is also possible to find representations in which both the maximum denominator and the number of terms are small. characterized the numbers that can be represented by Egyptian fractions in which all denominators are nth powers. In particular, a rational number q can be represented as an Egyptian fraction with square denominators if and only if q lies in one of the two half-open intervals showed that any rational number has very dense expansions, using a constant fraction of the denominators up to N for any sufficiently large N. Engel expansion, sometimes called an Egyptian product, is a form of Egyptian fraction expansion in which each denominator is a multiple of the previous one: In addition, the sequence of multipliers ai is required to be nondecreasing. Every rational number has a finite Engel expansion, while irrational numbers have an infinite Engel expansion. study numbers that have multiple distinct Egyptian fraction representations with the same number of terms and the same product of denominators; for instance, one of the examples they supply is Unlike the ancient Egyptians, they allow denominators to be repeated in these expansions. They apply their results for this problem to the characterization of free products of Abelian groups by a small number of numerical