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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, 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 n ...

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 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 because the only ways ...

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''. A ...

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 fact ...

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 ...

, 105,

110 110 may refer to: *110 (number), natural number *AD 110, a year *110 BC, a year *110 film, a cartridge-based film format used in still photography *110 (MBTA bus), Massachusetts Bay Transportation Authority bus route *110 (song), 2019 song by Capi ...

, 114, 130, 138, 154, 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 prime The largest known prime number () is , a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. A prime number is a posi ...

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 In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-fr ...

, 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 oft ...

of any sphenic number is −1. The

cyclotomic polynomials In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primit ...

\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