Sylvester's sequence 2, 3, 7, 43, 1807, ... can be viewed as generated by an infinite greedy expansion of this type for the number 1, where at each step we choose the denominator instead of , and sometimes Fibonacci's greedy algorithm is attributed to James Joseph Sylvester. After his description of the greedy algorithm, Fibonacci suggests yet another method, expanding a fraction by searching for a number c having many divisors, with , replacing by , and expanding ac as a sum of divisors of bc, similar to the method proposed by Hultsch and Bruins to explain some of the expansions in the Rhind papyrus. Modern number theory Although Egyptian fractions are no longer used in most practical applications of mathematics, modern number theorists have continued to study many different problems related to them. These include problems of bounding the length or maximum denominator in Egyptian fraction representations, finding expansions of certain special forms or in which the denominators are all of some special type, the termination of various methods for Egyptian fraction expansion, and showing that expansions exist for any sufficiently dense set of sufficiently smooth numbers. One of the earliest publications of Paul Erdős proved that it is not possible for a harmonic progression to form an Egyptian fraction representation of an integer. The reason is that, necessarily, at least one denominator of the progression will be divisible by a prime number that does not divide any other denominator. The latest publication of Erdős, nearly 20 years after his death, proves that every integer has a representation in which all denominators are products of three primes. The Erdős–Graham conjecture in combinatorial number theory states that, if the integers greater than 1 are partitioned into finitely many subsets, then one of the subsets has a finite subset of itself whose reciprocals sum to one. That is, for every , and every r-coloring of the integers greater than one, there is a finite monochromatic subset S of these integers such that The conjecture was proven in 2003 by Ernest S. Croot III. Znám's problem and primary pseudoperfect numbers are closely related to the existence of Egyptian fractions of the form For instance, the primary pseudoperfect number 1806 is the product of the prime numbers 2, 3, 7, and 43, and gives rise to the Egyptian fraction . Egyptian fractions are normally defined as requiring all denominators to be distinct, but