Practical number

In mathematics, and in particular number theory, a practical number or panarithmic number is a positive integer "n" such that all smaller positive integers can be represented as sums of distinct divisors of "n". For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5=3+2, 7=6+1, 8=6+2, 9=6+3, 10=6+3+1, and 11=6+3+2.

The sequence of practical numbers OEIS | id = A005153 begins:1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, ...

Practical numbers were used by Fibonacci in his Liber Abaci (1202) in connection with the problem of representing rational numbers as Egyptian fractions. Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators harv|Sigler|2002. In the modern mathematical literature, beginning with harvtxt|Srinivasan|1948, practical numbers have been studied for their similarities with prime numbers. A characterization by Stewart makes it possible to determine whether a number is practical by examining its prime factorization. Any even perfect number and any power of two is also a practical number.

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 harv|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.

Characterization of practical numbers

As harvtxt|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

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 $q\; =\; min\{lfloor\; m/p\_k^\{alpha\_k\}\; floor,\; sigma(n/p\_k^\{alpha\_k\})\}$, and let $r\; =\; m\; -\; qp\_k^\{sigma\_k\}$.
* Since $qlesigma(n/p\_k^\{alpha\_k\})$ 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 $rle\; sigma(n)\; -\; p\_k^\{alpha\_k\}sigma(n/p\_k^\{alpha\_k\})\; =\; sigma(n/p\_k)$, 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".

Any power of two is a practical number, because it trivially satisfies this characterization: the only prime in its factorization, "p"₁, equals two as required. Any even perfect number is also a practical number: due to Euler's result that these numbers must have the form 2^{"n" − 1}(2^"n" − 1), every odd prime factor of an even perfect number must be at most the sum of the divisors of the even part of the number, and therefore the number must satisfy Stewart's characterization.

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", harvtxt|Saias|1997 proved that for suitable constants "c"₁ and "c"₂:

:

a formula which resembles the prime number theorem. This result largely resolved a conjecture of harvtxt|Margenstern|1991 that "p"("x") is asymptotic to "cx"/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 harv|Melfi|1996. Melfi also showed that there are infinitely many practical Fibonacci numbers OEIS|id=A124105; the analogous question of the existence of infinitely many Fibonacci primes is open. harvtxt|Hausman|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.

References

*citation
last1 = Hausman | first1 = Miriam | last2 = Shapiro | first2 = Harold N.
title = On practical numbers
journal = Communications on Pure and Applied Mathematics
volume = 37
year = 1984
issue = 5
pages = 705–713
id = MathSciNet | id = 0752596
doi = 10.1002/cpa.3160370507.

*citation
last = Margenstern | first = Maurice
title = Les nombres pratiques: théorie, observations et conjectures
journal = Journal of Number Theory
volume = 37
year = 1991
issue = 1
pages = 1–36
id = MathSciNet | id = 1089787
doi = 10.1016/S0022-314X(05)80022-8.

*citation
authorlink = Giuseppe Melfi | last = Melfi | first = Giuseppe
title = On two conjectures about practical numbers
journal = Journal of Number Theory
volume = 56
year = 1996
issue = 1
pages = 205–210
id = MathSciNet | id = 1370203
doi = 10.1006/jnth.1996.0012.

*citation
last = Saias | first = Eric
title = Entiers à diviseurs denses, I
journal = Journal of Number Theory
volume = 62
issue = 1
year = 1997
pages = 163–191
id = MathSciNet | id = 1430008
doi = 10.1006/jnth.1997.2057.

*citation
title = Fibonacci's Liber Abaci
last = Sigler | first = Laurence E. (trans.)
publisher = Springer-Verlag
year = 2002
isbn = 0-387-95419-8
pages = 119–121.

*citation
last = Srinivasan | first = A. K.
title = Practical numbers
journal = Current Science
volume = 17
year = 1948
pages = 179–180
id = MathSciNet | id = 0027799.

*citation
last = Stewart | first = B. M.
title = Sums of distinct divisors
journal = American Journal of Mathematics
volume = 76
year = 1954
pages = 779–785
id = MathSciNet | id = 0064800
doi = 10.2307/2372651.

External links

* [http://www.dm.unipi.it/gauss-pages/melfi/public_html/pratica.html Tables of practical numbers] compiled by Giuseppe Melfi
*PlanetMath | urlname = PracticalNumber | title = Practical Number
*Mathworld | urlname = PracticalNumber | title = Practical Number