Practical number

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Demonstration of the practicality of the number 12

In 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 (sequence A005153 in the OEIS) begins

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150....

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.

The name "practical number" is due to Srinivasan (1948). He noted that "the subdivisions of money, weights, and measures involve numbers like 4, 12, 16, 20 and 28 which are usually supposed to be so inconvenient as to deserve replacement by powers of 10." He rediscovered the number theoretical property of such numbers and was the first to attempt a classification of these numbers that was completed by Stewart (1954) and Sierpiński (1955). This characterization makes it possible to determine whether a number is practical by examining its prime factorization. Every even perfect number and every power of two is also a practical number.

Practical numbers have also been shown to be analogous with prime numbers in many of their properties.

Characterization of practical numbers

The original characterisation by Srinivasan (1948) stated that a practical number cannot be a deficient number, that is one of which the sum of all divisors (including 1 and itself) is less than twice the number unless the deficiency is one. If the ordered set of all divisors of the practical number ${\displaystyle n}$ is ${\displaystyle {d_{1},d_{2},...,d_{j}}}$ with ${\displaystyle d_{1}=1}$ and ${\displaystyle d_{j}=n}$, then Srinivasan's statement can be expressed by the inequality

${\displaystyle 2n\leq 1+\sum _{i=1}^{j}d_{i}}$.

In other words, the ordered sequence of all divisors ${\displaystyle {d_{1}$ of a practical number has to be a complete sub-sequence.

This partial characterization was extended and completed by Stewart (1954) and Sierpiński (1955) who showed that it is straightforward to determine whether a number is practical from its prime factorization. A positive integer greater than one with prime factorization ${\displaystyle n=p_{1}^{\alpha _{1}}...p_{k}^{\alpha _{k}}}$ (with the primes in sorted order ${\displaystyle p_{1}$) is practical if and only if each of its prime factors ${\displaystyle p_{i}}$ is small enough for ${\displaystyle p_{i}-1}$ to have a representation as a sum of smaller divisors. For this to be true, the first prime ${\displaystyle p_{1}}$ must equal 2 and, for every i from 2 to k, each successive prime ${\displaystyle p_{i}}$ must obey the inequality

${\displaystyle p_{i}\leq 1+\sigma (p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\dots p_{i-1}^{\alpha _{i-1}})=1+\prod _{j=1}^{i-1}{\frac {p_{j}^{\alpha _{j}+1}-1}{p_{j}-1}},}$

where ${\displaystyle \sigma (x)}$ denotes the sum of the divisors of x. For example, 2 × 3² × 29 × 823 = 429606 is practical, because the inequality above holds for each of its prime factors: 3 ≤ σ(2) + 1 = 4, 29 ≤ σ(2 × 3²) + 1 = 40, and 823 ≤ σ(2 × 3² × 29) + 1 = 1171.

The condition stated above is necessary and sufficient for a number to be practical. In one direction, this condition is necessary in order to be able to represent ${\displaystyle p_{i}-1}$ as a sum of divisors of n, because if the inequality failed to be true then even adding together all the smaller divisors would give a sum too small to reach ${\displaystyle p_{i}-1}$. In the other direction, the condition is sufficient, as can be shown by induction. More strongly, if the factorization of n satisfies the condition above, then any ${\displaystyle m\leq \sigma (n)}$ can be represented as a sum of divisors of n, by the following sequence of steps:

• Let ${\displaystyle q=\min\{\lfloor m/p_{k}^{\alpha _{k}}\rfloor ,\sigma (n/p_{k}^{\alpha _{k}})\}}$, and let ${\displaystyle r=m-qp_{k}^{\sigma _{k}}}$.
• Since ${\displaystyle q\leq \sigma (n/p_{k}^{\alpha _{k}})}$ and ${\displaystyle n/p_{k}^{\alpha _{k}}}$ can be shown by induction to be practical, we can find a representation of q as a sum of divisors of ${\displaystyle n/p_{k}^{\alpha _{k}}}$.
• Since ${\displaystyle r\leq \sigma (n)-p_{k}^{\alpha _{k}}\sigma (n/p_{k}^{\alpha _{k}})=\sigma (n/p_{k})}$, and since ${\displaystyle n/p_{k}}$ can be shown by induction to be practical, we can find a representation of r as a sum of divisors of ${\displaystyle n/p_{k}}$.
• The divisors representing r, together with ${\displaystyle p_{k}^{\alpha _{k}}}$ times each of the divisors representing q, together form a representation of m as a sum of divisors of n.

Properties

• The only odd practical number is 1, because if n > 2 is an odd number, then 2 cannot be expressed as the sum of distinct divisors of n. More strongly, Srinivasan (1948) observes that other than 1 and 2, every practical number is divisible by 4 or 6 (or both).
• The product of two practical numbers is also a practical number. More strongly the least common multiple of any two practical numbers is also a practical number. Equivalently, the set of all practical numbers is closed under multiplication.
• From the above characterization by Stewart and Sierpiński it can be seen that if n is a practical number and d is one of its divisors then n*d must also be a practical number.
• In the set of all practical numbers there is a primitive set of practical numbers. A primitive practical number is either practical and squarefree or practical and when divided by any of its prime factors whose factorization exponent is greater than 1 is no longer practical. The sequence of primitive practical numbers (sequence A267124 in the OEIS) begins

1, 2, 6, 20, 28, 30, 42, 66, 78, 88, 104, 140, 204, 210, 220, 228, 260, 272, 276, 304, 306, 308, 330, 340, 342, 348, 364, 368, 380, 390, 414, 460 ...

Relation to other classes of numbers

Several other notable sets of integers consist only of practical numbers:

• From the above properties with n a practical number and d one of its divisors (that is, d | n) then n*d must also be a practical number therefore six times every power of 3 must be a practical number as well as six times every power of 2.
• Every power of two is a practical number. Powers of two trivially satisfy the characterization of practical numbers in terms of their prime factorizations: the only prime in their factorizations, p₁, equals two as required.
• Every even perfect number is also a practical number. This follows from Leonhard Euler's result that an even perfect number must have the form 2^n − 1(2ⁿ − 1). The odd part of this factorization equals the sum of the divisors of the even part, so every odd prime factor of such a number must be at most the sum of the divisors of the even part of the number. Therefore, this number must satisfy the characterization of practical numbers.
• Every primorial (the product of the first i primes, for some i) is practical. For the first two primorials, two and six, this is clear. Each successive primorial is formed by multiplying a prime number p_i by a smaller primorial that is divisible by both two and the next smaller prime, p_i − 1. By Bertrand's postulate, p_i < 2p_i − 1, so each successive prime factor in the primorial is less than one of the divisors of the previous primorial. By induction, it follows that every primorial satisfies the characterization of practical numbers. Because a primorial is, by definition, squarefree it is also a primitive practical number.
• Generalizing the primorials, any number that is the product of nonzero powers of the first k primes must also be practical. This includes Ramanujan's highly composite numbers (numbers with more divisors than any smaller positive integer) as well as the factorial numbers.

Practical numbers and Egyptian fractions

If n is practical, then any rational number of the form m/n with 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,

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

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.

Vose (1985) showed that every number x/y has an Egyptian fraction representation with ${\displaystyle \scriptstyle O({\sqrt {\log y}})}$ terms. The proof involves finding a sequence of practical numbers n_i with the property that every number less than n_i may be written as a sum of ${\displaystyle \scriptstyle O({\sqrt {\log n_{i-1}}})}$ distinct divisors of n_i. Then, i is chosen so that n_i − 1 < y ≤ n_i, and xn_i is divided by y giving quotient q and remainder r. It follows from these choices that ${\displaystyle \scriptstyle {\frac {x}{y}}={\frac {q}{n_{i}}}+{\frac {r}{yn_{i}}}}$. Expanding both numerators on the right hand side of this formula into sums of divisors of n_i results in the desired Egyptian fraction representation. Tenenbaum & Yokota (1990) use a similar technique involving a different sequence of practical numbers to show that every number x/y has an Egyptian fraction representation in which the largest denominator is ${\displaystyle \scriptstyle O({\frac {y\log ^{2}y}{\log \log y}})}$.
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. There is a proof for the conjecture on David Eppstein's blog.

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 x − 2, x, x + 2. Melfi also showed that there are infinitely many practical Fibonacci numbers (sequence A124105 in the 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 [x²,(x + 1)²] for any positive real x, a result analogous to Legendre's conjecture for primes.

Let p(x) count how many practical numbers are at most x. Margenstern (1991) conjectured that p(x) is asymptotic to cx/log x for some constant c, a formula which resembles the prime number theorem, strengthening the earlier claim of Erdős & Loxton (1979) that the practical numbers have density zero in the integers. Saias (1997) proved that for suitable constants c₁ and c₂:

${\displaystyle c_{1}{\frac {x}{\log x}}$

Finally Weingartner (2015) proved Margenstern's conjecture showing that

${\displaystyle p(x)={\frac {cx}{\log x}}\left(1+O\!\left({\frac {\log \log x}{\log x}}\right)\right),}$

for ${\displaystyle x\geq 3}$ and some constant ${\displaystyle c>0}$.

Goldbach's conjecture

From Wikipedia, the free encyclopedia

 

The even integers from 4 to 28 as sums of two primes: Even integers correspond to horizontal lines. For each prime, there are two oblique lines, one red and one blue. The sums of two primes are the intersections of one red and one blue line, marked by a circle. Thus the circles on a given horizontal line give all partitions of the corresponding even integer into the sum of two primes.

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states:

Every even integer greater than 2 can be expressed as the sum of two primes.

The conjecture has been shown to hold for all integers less than 4 × 10¹⁸, but remains unproven despite considerable effort.

Goldbach number

The number of ways an even number can be represented as the sum of two primes.

A Goldbach number is a positive even integer that can be expressed as the sum of two odd primes. Since four is the only even number greater than two that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers.

The expression of a given even number as a sum of two primes is called a Goldbach partition of that number. The following are examples of Goldbach partitions for some even numbers:

6 = 3 + 3

8 = 3 + 5

10 = 3 + 7 = 5 + 5

12 = 7 + 5

...

100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53

...

The number of ways in which 2n can be written as the sum of two primes (for n starting at 1) is:

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 3, 5, 3, 4, 6, 3, 5, 6, 2, 5, 6, 5, 5, 7, 4, 5, 8, 5, 4, 9, 4, 5, 7, 3, 6, 8, 5, 6, 8, 6, 7, 10, 6, 6, 12, 4, 5, 10, 3, ... (sequence A045917 in the OEIS).

Origins

Letter from Goldbach to Euler dated on 7. June 1742 (Latin-German).

 

On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) in which he proposed the following conjecture:

Every integer which can be written as the sum of two primes, can also be written as the sum of as many primes as one wishes, until all terms are units.

He then proposed a second conjecture in the margin of his letter:

Every integer greater than 2 can be written as the sum of three primes.

He considered 1 to be a prime number, a convention subsequently abandoned. The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time. A modern version of Goldbach's marginal conjecture is:

Every integer greater than 5 can be written as the sum of three primes.

Euler replied in a letter dated 30 June 1742, and reminded Goldbach of an earlier conversation they had ("…so Ew vormals mit mir communicirt haben…"), in which Goldbach remarked his original (and not marginal) conjecture followed from the following statement

Every even integer greater than 2 can be written as the sum of two primes,

which is, thus, also a conjecture of Goldbach. In the letter dated 30 June 1742, Euler stated:

"Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann." ("That … every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it.")

Goldbach's third version (equivalent to the two other versions) is the form in which the conjecture is usually expressed today. It is also known as the "strong", "even", or "binary" Goldbach conjecture, to distinguish it from a weaker conjecture, known today variously as the Goldbach's weak conjecture, the "odd" Goldbach conjecture, or the "ternary" Goldbach conjecture. This weak conjecture asserts that all odd numbers greater than 7 are the sum of three odd primes, and appears to have been proved in 2013. The weak conjecture is a corollary of the strong conjecture, as, if n – 3 is a sum of two primes, then n is a sum of three primes. The converse implication, and the strong Goldbach conjecture remain unproven.

Verified results

For small values of n, the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, Nils Pipping in 1938 laboriously verified the conjecture up to n ≤ 10⁵. With the advent of computers, many more values of n have been checked; T. Oliveira e Silva is running a distributed computer search that has verified the conjecture for n ≤ 4 × 10¹⁸ (and double-checked up to 4 × 10¹⁷) as of 2013. One record from this search is that 3,325,581,707,333,960,528 is the smallest number that has no Goldbach partition with a prime below 9781.

Heuristic justification

Statistical considerations that focus on the probabilistic distribution of prime numbers present informal evidence in favour of the conjecture (in both the weak and strong forms) for sufficiently large integers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, and the more "likely" it becomes that at least one of these representations consists entirely of primes.

Number of ways to write an even number n as the sum of two primes (4 ≤ n ≤ 1,000), (sequence A002375 in the OEIS)

 

Number of ways to write an even number n as the sum of two primes (4 ≤ n ≤ 1,000,000)

A very crude version of the heuristic probabilistic argument (for the strong form of the Goldbach conjecture) is as follows. The prime number theorem asserts that an integer m selected at random has roughly a ${\displaystyle 1/\ln m}$ chance of being prime. Thus if n is a large even integer and m is a number between 3 and n/2, then one might expect the probability of m and n − m simultaneously being prime to be ${\displaystyle 1{\big /}{\big [}\ln m\,\ln(n-m){\big ]}}$. If one pursues this heuristic, one might expect the total number of ways to write a large even integer n as the sum of two odd primes to be roughly

${\displaystyle \sum _{m=3}^{n/2}{\frac {1}{\ln m}}{1 \over \ln(n-m)}\approx {\frac {n}{2(\ln n)^{2}}}.}$

Since this quantity goes to infinity as n increases, we expect that every large even integer has not just one representation as the sum of two primes, but in fact has very many such representations.

This heuristic argument is actually somewhat inaccurate, because it assumes that the events of m and n − m being prime are statistically independent of each other. For instance, if m is odd then n − m is also odd, and if m is even, then n − m is even, a non-trivial relation because, besides the number 2, only odd numbers can be prime. Similarly, if n is divisible by 3, and m was already a prime distinct from 3, then n − m would also be coprime to 3 and thus be slightly more likely to be prime than a general number. Pursuing this type of analysis more carefully, Hardy and Littlewood in 1923 conjectured (as part of their famous Hardy–Littlewood prime tuple conjecture) that for any fixed c ≥ 2, the number of representations of a large integer n as the sum of c primes ${\displaystyle n=p_{1}+\cdots +p_{c}}$ with ${\displaystyle p_{1}\leq \cdots \leq p_{c}}$ should be asymptotically equal to 

${\displaystyle \left(\prod _{p}{\frac {p\gamma _{c,p}(n)}{(p-1)^{c}}}\right)\int _{2\leq x_{1}\leq \cdots \leq x_{c}:x_{1}+\cdots +x_{c}=n}{\frac {dx_{1}\cdots dx_{c-1}}{\ln x_{1}\cdots \ln x_{c}}}}$

where the product is over all primes p, and ${\displaystyle \gamma _{c,p}(n)}$ is the number of solutions to the equation ${\displaystyle n=q_{1}+\cdots +q_{c}\mod p}$ in modular arithmetic, subject to the constraints ${\displaystyle q_{1},\ldots ,q_{c}\neq 0\mod p}$. This formula has been rigorously proven to be asymptotically valid for c ≥ 3 from the work of Vinogradov, but is still only a conjecture when ${\displaystyle c=2}$. In the latter case, the above formula simplifies to 0 when n is odd, and to

${\displaystyle 2\Pi _{2}\left(\prod _{p\mid n;p\geq 3}{\frac {p-1}{p-2}}\right)\int _{2}^{n}{\frac {dx}{(\ln x)^{2}}}\approx 2\Pi _{2}\left(\prod _{p\mid n;p\geq 3}{\frac {p-1}{p-2}}\right){\frac {n}{(\ln n)^{2}}}}$

when n is even, where ${\displaystyle \Pi _{2}}$ is Hardy–Littlewood's twin prime constant

${\displaystyle \Pi _{2}:=\prod _{p\geq 3}\left(1-{\frac {1}{(p-1)^{2}}}\right)=0.6601618158\ldots .}$

This is sometimes known as the extended Goldbach conjecture. The strong Goldbach conjecture is in fact very similar to the twin prime conjecture, and the two conjectures are believed to be of roughly comparable difficulty.

The Goldbach partition functions shown here can be displayed as histograms which informatively illustrate the above equations. See Goldbach's comet.

Rigorous results

The strong Goldbach conjecture is much more difficult than the weak Goldbach conjecture. Using Vinogradov's method, Chudakov, Van der Corput, and Estermann showed that almost all even numbers can be written as the sum of two primes (in the sense that the fraction of even numbers which can be so written tends towards 1). In 1930, Lev Schnirelmann proved that any natural number greater than 1 can be written as the sum of not more than C prime numbers, where C is an effectively computable constant, see Schnirelmann density. Schnirelmann's constant is the lowest number C with this property. Schnirelmann himself obtained C < 800,000. This result was subsequently enhanced by many authors, such as Olivier Ramaré, who in 1995 showed that every even number n  ≥ 4 is in fact the sum of at most six primes. The best known result currently stems from the proof of the weak Goldbach conjecture by Harald Helfgott, which directly implies that every even number n  ≥ 4 is the sum of at most four primes.

In 1924 Hardy and Littlewood showed under the assumption of the GRH that amount of even numbers up to X violating Goldbach conjecture is much less than ${\displaystyle X^{0.5+c}}$ for small c.

Chen Jingrun showed in 1973 using the methods of sieve theory that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes).

In 1975, Hugh Montgomery and Robert Charles Vaughan showed that "most" even numbers are expressible as the sum of two primes. More precisely, they showed that there exist positive constants c and C such that for all sufficiently large numbers N, every even number less than N is the sum of two primes, with at most ${\displaystyle CN^{1-c}}$ exceptions. In particular, the set of even integers which are not the sum of two primes has density zero.

Linnik proved in 1951 the existence of a constant K such that every sufficiently large even number is the sum of two primes and at most K powers of 2. Roger Heath-Brown and Jan-Christoph Schlage-Puchta in 2002 found that K = 13 works.

As with many famous conjectures in mathematics, there are a number of purported proofs of the Goldbach conjecture, none of which are accepted by the mathematical community.

Related problems

Although Goldbach's conjecture implies that every positive integer greater than one can be written as a sum of at most three primes, it is not always possible to find such a sum using a greedy algorithm that uses the largest possible prime at each step. The Pillai sequence tracks the numbers requiring the largest number of primes in their greedy representations.

One can consider similar problems in which primes are replaced by other particular sets of numbers, such as the squares.

• It was proven by Lagrange that every positive integer is the sum of four squares.
• Hardy and Littlewood listed as their Conjecture I: "Every large odd number (n > 5) is the sum of a prime and the double of a prime." (Mathematics Magazine, 66.1 (1993): 45–47.) This conjecture is known as Lemoine's conjecture (also called Levy's conjecture).
• The Goldbach conjecture for practical numbers, a prime-like sequence of integers, was stated by Margenstern in 1984, and proved by Melfi in 1996: every even number is a sum of two practical numbers.