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Binomial Number

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A binomial number is a number of the form a^n+/-b^n, where a,b, and n are integers. Binomial numbers can be factored algebraically as

a^n-b^n=(a-b)(a^(n-1)+a^(n-2)b+...+ab^(n-2)+b^(n-1))
(1)

for all n,

a^n+b^n=(a+b)(a^(n-1)-a^(n-2)b+...-ab^(n-2)+b^(n-1))
(2)

for n odd, and

a^(nm)-b^(nm)=(a^m-b^m)[a^(m(n-1))+a^(m(n-2))b^m+...+b^(m(n-1))].
(3)

for all positive integers m,n. For example,

a^2-b^2=(a-b)(a+b)
(4)
a^3-b^3=(a-b)(a^2+ab+b^2)
(5)
a^4-b^4=(a-b)(a+b)(a^2+b^2)
(6)
a^5-b^5=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)
(7)
a^6-b^6=(a-b)(a+b)(a^2-ab+b^2)(a^2+ab+b^2)
(8)
a^7-b^7=(a-b)(a^6+a^5b+a^4b^2+a^3b^3+a^2b^4+ab^5+b^6)
(9)
a^8-b^8=(a-b)(a+b)(a^2+b^2)(a^4+b^4)
(10)
a^9-b^9=(a-b)(a^2+ab+b^2)(a^6+a^3b^3+b^6)
(11)
a^(10)-b^(10)=(a-b)(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)×(a^4+a^3b+a^2b^2+ab^3+b^4)
(12)

and

a^2+b^2=a^2+b^2
(13)
a^3+b^3=(a+b)(a^2-ab+b^2)
(14)
a^4+b^4=a^4+b^4
(15)
a^5+b^5=(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)
(16)
a^6+b^6=(a^2+b^2)(a^4-a^2b^2+b^4)
(17)
a^7+b^7=(a+b)(a^6-a^5b+a^4b^2-a^3b^3+a^2b^4-ab^5+b^6)
(18)
a^8+b^8=a^8+b^8
(19)
a^9+b^9=(a+b)(a^2-ab+b^2)(a^6-a^3b^3+b^6)
(20)
a^(10)+b^(10)=(a^2+b^2)(a^8-a^6b^2+a^4b^4-a^2b^6+b^8).
(21)

Rather surprisingly, the number of factors of a^n-b^n with a and b symbolic and n a positive integer is given by d(n), where d(n)=sigma_0(n) is the number of divisors of n and sigma_k(n) is the divisor function. The first few terms are therefore 1, 2, 2, 3, 2, 4, 2, ... (OEIS A000005).

Similarly, the number of factors of a^n+b^n is given by d^((o))(n), where d^((o))(n)=sigma_0^((o))(n) is the number of odd divisors of n and sigma_k^((o))(n) is the odd divisor function. The first few terms are therefore 1, 1, 2, 1, 2, 2, 2, 1,... (OEIS A001227).

In 1770, Euler proved that if (a,b)=1, then every odd factor of

a^(2^n)+b^(2^n)
(22)

is of the form 2^(n+1)K+1. (A number of the form 2^(2^n)+1 is called a Fermat number.)

If p and q are primes, then

((a^(pq)-1)(a-1))/((a^p-1)(a^q-1))-1
(23)

is divisible by every prime factor of a^(p-1) not dividing a^q-1.

See also

Binomial, Cunningham Number, Fermat Number, Mersenne Number, Perfect Cubic Polynomial, Riesel Number, Sierpiński Number of the Second Kind, Waring Formula

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References

Guy, R. K. "When Does 2^a-2^b Divide n^a-n^b." §B47 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 102, 1994.Qi, S and Ming-Zhi, Z. "Pairs where 2^a-2^b Divides n^a-n^b for All n." Proc. Amer. Math. Soc. 93, 218-220, 1985.Schinzel, A. "On Primitive Prime Factors of a^n-b^n." Proc. Cambridge Phil. Soc. 58, 555-562, 1962.Sloane, N. J. A. Sequences A000005/M0246 and A001227 in "The On-Line Encyclopedia of Integer Sequences."

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Binomial Number

Cite this as:

Weisstein, Eric W. "Binomial Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BinomialNumber.html

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