Practical number

As Stewart (1954) showed, it is straightforward to determine whether a number is practical from its prime factorization. A positive integer $n=p_1^{\alpha_1}...p_k^{\alpha_k}$ with $n > 1$ and $p_1<p_2<\dots<p_k$ primes is practical if and only if $p 1 = 2$ and for $i=2,\dots,k$

$p_i\leq1+\sigma(p_1^{\alpha_1}\dots p_{i-1}^{\alpha_{i-1}})=1+\prod_{j=1}^{i-1}\frac{p_j^{\alpha_j+1}-1}{p_j-1},$

where $σ(x)$ denotes the sum of the divisors of x.

In one direction, this condition is clearly necessary in order to be able to represent $p i - 1$ as a sum of divisors of n. In the other direction, the condition is sufficient, as can be shown by induction. More strongly, one can show that, if the factorization of n satisfies the condition above, then any $m \le \sigma(n)$ can be represented as a sum of divisors of n, by the following sequence of steps:

Let $q = \min\{\lfloor m/p_k^{\alpha_k}\rfloor, \sigma(n/p_k^{\alpha_k})\}$ , and let $r = m - qp_k^{\sigma_k}$ .
Since $q\le\sigma(n/p_k^{\alpha_k})$ and $n/p_k^{\alpha_k}$ can be shown by induction to be practical, we can find a representation of q as a sum of divisors of $n/p_k^{\alpha_k}$ .
Since $r\le \sigma(n) - p_k^{\alpha_k}\sigma(n/p_k^{\alpha_k}) = \sigma(n/p_k)$ , and since $n / p k$ can be shown by induction to be practical, we can find a representation of r as a sum of divisors of $n / p k$ .
The divisors representing r, together with $p_k^{\alpha_k}$ times each of the divisors representing q, together form a representation of m as a sum of divisors of n.

For example, 3 ≤ σ(2)+1 = 4, 29 ≤ σ(2 × 3²)+1 = 40, and 823 ≤ σ(2 × 3² × 29)+1=1171, so 2 × 3² × 29 × 823 = 429606 is practical.

[edit] 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 $c 1$ and $c 2$ :

$c_1\frac x{\log x}<p(x)<c_2\frac x{\log x},$

a formula which resembles the prime number theorem. This result largely resolved a conjecture of Margenstern (1991) that p(x) is asymptotic to $c x / log x$ for some constant c.

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 1996). 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 and Shapiro (1984) showed that there always exists a practical number in the interval $[x 2,(x + 1) 2]$ for any positive real x, a result analogous to Legendre's conjecture for primes.

[edit] Practical numbers and Egyptian fractions

If n is practical, then any rational number of the form m/n may be represented as a sum ∑d_i/n where each d_i is a distinct divisor of n. Each term in this sum simplifies to a unit fraction, so such a sum provides a representation of m/n as an Egyptian fraction. For instance,

$\frac{13}{20}=\frac{10}{20}+\frac{2}{20}+\frac{1}{20}=\frac12+\frac1{10}+\frac1{20}.$

Fibonacci, in his 1202 book Liber Abaci (Sigler 2002) 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.

[edit] References

Hausman, Miriam; Shapiro, Harold N. (1984). "On practical numbers". Communications on Pure and Applied Mathematics 37 (5): 705–713. MR 0752596.

Margenstern, Maurice (1991). "Les nombres pratiques: théorie, observations et conjectures". Journal of Number Theory 37 (1): 1–36. DOI:10.1016/S0022-314X(05)80022-8. MR 1089787.

Melfi, Giuseppe (1996). "On two conjectures about practical numbers". Journal of Number Theory 56 (1): 205–210. DOI:10.1006/jnth.1996.0012. MR 1370203.