and Egyptian fractions If is practical, then any rational number of the form with may be represented as a sum where each is a distinct divisor of . Each term in this sum simplifies to a unit fraction, so such a sum provides a representation of as an Egyptian fraction. For instance, Fibonacci, in his 1202 book Liber Abaci lists several methods for finding Egyptian fraction representations of a rational number. Of these, the first is to test whether the number is itself already a unit fraction, but the second is to search for a representation of the numerator as a sum of divisors of the denominator, as described above. This method is only guaranteed to succeed for denominators that are practical. Fibonacci provides tables of these representations for fractions having as denominators the practical numbers 6, 8, 12, 20, 24, 60, and 100. showed that every rational number has an Egyptian fraction representation with terms. The proof involves finding a sequence of practical numbers with the property that every number less than may be written as a sum of distinct divisors of . Then, is chosen so that , and is divided by giving quotient and remainder . It follows from these choices that . Expanding both numerators on the right hand side of this formula into sums of divisors of results in the desired Egyptian fraction representation. use a similar technique involving a different sequence of practical numbers to show that every rational number has an Egyptian fraction representation in which the largest denominator is . According to a September 2015 conjecture by Zhi-Wei Sun, every positive rational number has an Egyptian fraction representation in which every denominator is a practical number. The conjecture was proved by . Analogies with prime numbers One reason for interest in practical numbers is that many of their properties are similar to properties of the prime numbers. Indeed, theorems analogous to Goldbach's conjecture and the twin prime conjecture are known for practical numbers: every positive even integer is the sum of two practical numbers, and there exist infinitely many triples of practical numbers . Melfi also showed that there are infinitely many practical Fibonacci numbers ; the analogous question of the existence of infinitely many Fibonacci primes is open. showed that there always exists a practical number in the interval for any positive real , a result analogous to Legendre's conjecture for