mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, specifically in

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 binomial number is an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

which can be obtained by evaluating a

homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables ...

containing two terms. It is a generalization of a Cunningham number.

Definition

A binomial number is an integer obtained by evaluating a

containing two terms, also called a

binomial Binomial may refer to: In mathematics *Binomial (polynomial), a polynomial with two terms *Binomial coefficient, numbers appearing in the expansions of powers of binomials *Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition * ...

. The form of this binomial is

x^n \!\pm y^n

, with

x > y

and

n > 1

. However, since

x^n \!- y^n

is always divisible by

x - y

, when studying the numbers generated from the version with the negative sign, they are usually divided by

x - y

first. Binomial numbers formed this way form

Lucas sequence In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation : x_n = P \cdot x_ - Q \cdot x_ where P and Q are fixed integers. Any sequence satisfying this rec ...

s. Specifically: :

U_n(a+b,ab) = \frac,

and

V_n(a+b,ab) = a^n \!+ b^n

Binomial numbers are a generalization of a Cunningham numbers, and it will be seen that the Cunningham numbers are binomial numbers where

y = 1

. Other subsets of the binomial numbers are the Mersenne numbers and the

repunit In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his book ''Recr ...

Factorization

The main reason for studying these numbers is to obtain their factorizations. Aside from algebraic factors, which are obtained by factoring the underlying

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

(binomial) that was used to define the number, such as

difference of two squares In elementary algebra, a difference of two squares is one squared number (the number multiplied by itself) subtracted from another squared number. Every difference of squares may be factored as the product of the sum of the two numbers and the ...

and

sum of two cubes In mathematics, the sum of two cubes is a cubed number added to another cubed number. Factorization Every sum of cubes may be factored according to the identity a^3 + b^3 = (a + b)(a^2 - ab + b^2) in elementary algebra. Binomial numbers g ...

, there are other

prime 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 ...

factors (called primitive prime factors, because for a given

x^n \!\pm y^n

they do not factorize

x^m \!\pm y^m

with

m < n

, except for a small number of exceptions as stated in Zsigmondy's theorem) which occur seemingly at random, and it is these which the number theorist is looking for. Some binomial numbers' underlying binomials have Aurifeuillian factorizations, which can assist in finding prime factors.

Cyclotomic polynomial 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 prim ...

s are also helpful in finding factorizations. The amount of work required in searching for a factor is considerably reduced by applying Legendre's theorem. This theorem states that all factors of a binomial number are of the form

kn + 1

n

is even or

2kn + 1

if it is odd.

Observation

Some people write "binomial number" when they mean

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

, but this usage is not standard and is deprecated.

Notes

References

* {{cite book , last=Riesel , first=Hans , title=Prime numbers and computer methods for factorization , publisher=Birkhauser , location=Boston, MA , edition = 2nd , year=1994 , isbn=0-8176-3743-5 , zbl=0821.11001 , series=Progress in Mathematics , volume=126

External links