number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, a sphenic number (from , 'wedge') is a

positive integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

that is the product of three distinct

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers.

Definition

A sphenic number is a product ''pqr'' where ''p'', ''q'', and ''r'' are three distinct prime numbers. In other words, the sphenic numbers are the

square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. ...

3- almost primes.

Examples

The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are : 30, 42, 66, 70, 78, 102,

105 105 may refer to: *105 (number), the number * AD 105, a year in the 2nd century AD * 105 BC, a year in the 2nd century BC * 105 (telephone number), the emergency telephone number in Mongolia * 105 (MBTA bus), a Massachusetts Bay Transport Authority ...

, 110,

114 114 may refer to: *114 (number) *AD 114 *114 BC *114 (1st London) Army Engineer Regiment, Royal Engineers, an English military unit *114 (Antrim Artillery) Field Squadron, Royal Engineers, a Northern Irish military unit *114 (MBTA bus) *114 (New Je ...

, 130, 138, 154, 165, ... The largest known sphenic number at any time can be obtained by multiplying together the three largest known primes.

Divisors

All sphenic numbers have exactly eight divisors. If we express the sphenic number as

n = p \cdot q \cdot r

, where ''p'', ''q'', and ''r'' are distinct primes, then the set of divisors of ''n'' will be: :

\left\.

The converse does not hold. For example, 24 is not a sphenic number, but it has exactly eight divisors.

Properties

All sphenic numbers are by definition squarefree, because the prime factors must be distinct. The

Möbius function The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and m ...

of any sphenic number is −1. The cyclotomic polynomials

\Phi_n(x)

, taken over all sphenic numbers ''n'', may contain arbitrarily large coefficientsEmma Lehmer, "On the magnitude of the coefficients of the cyclotomic polynomial", ''Bulletin of the American Mathematical Society'' 42 (1936), no. 6, pp. 389–39

(for ''n'' a product of two primes the coefficients are

\pm 1

or 0). Any multiple of a sphenic number (except by 1) is not sphenic. This is easily provable by the multiplication process at a minimum adding another prime factor, or raising an existing factor to a higher power.

Consecutive sphenic numbers

The first case of two consecutive sphenic integers is 230 = 2×5×23 and 231 = 3×7×11. The first case of three is 1309 = 7×11×17, 1310 = 2×5×131, and 1311 = 3×19×23. There is no case of more than three, because every fourth consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree. The numbers 2013 (3×11×61), 2014 (2×19×53), and 2015 (5×13×31) are all sphenic. The next three consecutive sphenic years will be 2665 (5×13×41), 2666 (2×31×43) and 2667 (3×7×127) .

References