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# Binomial Number

A binomial number is a number of the form , where , and are integers. Binomial numbers can be factored algebraically as

 (1)

for all ,

 (2)

for odd, and

 (3)

for all positive integers . For example,

 (4) (5) (6) (7) (8) (9) (10) (11) (12)

and

 (13) (14) (15) (16) (17) (18) (19) (20) (21)

Rather surprisingly, the number of factors of with and symbolic and a positive integer is given by , where is the number of divisors of and 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 is given by , where is the number of odd divisors of and 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 , then every odd factor of

 (22)

is of the form . (A number of the form is called a Fermat number.)

If and are primes, then

 (23)

is divisible by every prime factor of not dividing .

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

## References

Guy, R. K. "When Does Divide ." §B47 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 102, 1994.Qi, S and Ming-Zhi, Z. "Pairs where Divides for All ." Proc. Amer. Math. Soc. 93, 218-220, 1985.Schinzel, A. "On Primitive Prime Factors of ." 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."

Binomial Number

## Cite this as:

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