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Contents Next: Bernoulli numbers and Up: Arithmetic Properties of Binomial Previous: Studying binomial coefficients

Sums of Binomial Coefficients

Proof of (1.12): \ Let

be a primitive p th root of unity and recall that

as ideals in Q

. Define

to be the sum on the left side of (1.12) for each j, so that

which belongs to the ideal

, for

. Therefore

, belongs to

. However, since each

is a rational integer, it must be divisible by

where

is the smallest multiple of

, which is

, and (1.12) follows immediately. Proof of (1.13): \ Let d be a quadratic non--residue

and a, b and n any positive integers. Define the sequence

of integers by

so that, from the binomial theorem,

as

, where

is the sum in (1.13).

Now, by Kummer's Theorem,

and so, if p+1 divides n then p divides

, which divides

, by (7.1). So by selecting a=b=1 and letting d run through all quadratic non--residues

, we have

equations in the

unknowns

. Therefore each

must be divisible by p as these equations give rise to a Vandermonde matrix whose determinant is not divisible by p.

On the other hand if (1.13) holds for all odd j then is divisible by p for any admissible choices of a,b and d, by (7.2). Now fix d and select a and b so that is a primitive root modulo p in the field Q. Note that , so that . By (7.1), we see that , and so . But is a primitive root modulo p and so divides , giving that p+1 divides n.

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Contents Next: Bernoulli numbers and Up: Arithmetic Properties of Binomial Previous: Sums of Binomial