Binomial Number
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,
and
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
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 FormulaExplore with Wolfram|Alpha
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."Referenced on Wolfram|Alpha
Binomial NumberCite this as:
Weisstein, Eric W. "Binomial Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BinomialNumber.html