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. For example, if p(x) is the enumerating function of practical numbers, i.e., the number of practical numbers not exceeding x, Saias (1997) proved that for suitable constants c1 and c2:
a formula which resembles the prime number theorem. This result largely resolved a conjecture of Margenstern (1991) that p(x) is asymptotic to cx/log x for some constant c, and it strengthens an earlier claim of Erdős & Loxton (1979) that the practical numbers have density zero in the integers.
Theorems analogous to Goldbach's conjecture and the twin prime conjecture are also known for practical numbers: every positive even integer is the sum of two practical numbers, and there exist infinitely many triples of practical numbers x − 2, x, x + 2. Melfi also showed that there are infinitely many practical Fibonacci numbers (sequence A124105 in OEIS); the analogous question of the existence of infinitely many Fibonacci primes is open. Hausman & Shapiro (1984) showed that there always exists a practical number in the interval for any positive real x, a result analogous to Legendre's conjecture for primes.
Read more about this topic: Practical Number
Famous quotes containing the words analogies, prime and/or numbers:
“Truth on our level is a different thing from truth for the jellyfish, and there must certainly be analogies for truth and error in jellyfish life.”
—T.S. (Thomas Stearns)
“Vanessa wanted to be a ballerina. Dad had such hopes for her.... Corin was the academically brilliant one, and a fencer of Olympic standard. Everything was expected of them, and they fulfilled all expectations. But I was the one of whom nothing was expected. I remember a game the three of us played. Vanessa was the President of the United States, Corin was the British Prime Minister—and I was the royal dog.”
—Lynn Redgrave (b. 1943)
“The barriers of conventionality have been raised so high, and so strangely cemented by long existence, that the only hope of overthrowing them exists in the union of numbers linked together by common opinion and effort ... the united watchword of thousands would strike at the foundation of the false system and annihilate it.”
—Mme. Ellen Louise Demorest 1824–1898, U.S. women’s magazine editor and woman’s club movement pioneer. Demorest’s Illustrated Monthly and Mirror of Fashions, p. 203 (January 1870)