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:
“There must be a solemn and terrible aloneness that comes over the child as he takes those first independent steps. All this is lost to memory and we can only reconstruct it through analogies in later life....To the child who takes his first steps and finds himself walking alone, this moment must bring the first sharp sense of the uniqueness and separateness of his body and his person, the discovery of the solitary self.”
—Selma H. Fraiberg (20th century)
“One wants in a Prime Minister a good many things, but not very great things. He should be clever but need not be a genius; he should be conscientious but by no means strait-laced; he should be cautious but never timid, bold but never venturesome; he should have a good digestion, genial manners, and, above all, a thick skin.”
—Anthony Trollope (1815–1882)
“What culture lacks is the taste for anonymous, innumerable germination. Culture is smitten with counting and measuring; it feels out of place and uncomfortable with the innumerable; its efforts tend, on the contrary, to limit the numbers in all domains; it tries to count on its fingers.”
—Jean Dubuffet (1901–1985)