number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

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

positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

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 3-

almost prime In number theory, a natural number is called ''k''-almost prime if it has ''k'' prime factors. More formally, a number ''n'' is ''k''-almost prime if and only if Ω(''n'') = ''k'', where Ω(''n'') is the total number of primes in the prime fa ...

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,

102 102 may refer to: *102 (number), the number * AD 102, a year in the 2nd century AD *102 BC, a year in the 2nd century BC * 102 (ambulance service), an emergency medical transport service in Uttar Pradesh, India * 102 (Clyde) Field Squadron, Royal En ...

, 105, 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 ...

130 130 may refer to: *130 (number) *AD 130 *130 BC Thirteen or 13 may refer to: * 13 (number), the natural number following 12 and preceding 14 * One of the years 13 BC, AD 13, 1913, 2013 Music * 13AD (band), an Indian classic and hard rock band ...

138 138 may refer to: *138 (number) *138 BC *AD 138 Year 138 ( CXXXVIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Niger and Camer ...

154 Year 154 ( CLIV) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelius and Lateranus (or, less frequently, year 907 ''Ab urbe cond ...

, 165, ... the largest known sphenic number is :(2^82,589,933 − 1) × (2^77,232,917 − 1) × (2^74,207,281 − 1). It is the product of 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 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 most of ...

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) isn't a sphenic number. This is easily provable by the multiplication process adding another prime factor, or squaring an existing factor.

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