On A New Smarandache Type Function

Page 1

On A New Smarandache Type Function

J. Sandor Babes-Bolyai University, 3400 Cluj-Napoca, Romania

(kn)

k Let C n

k

denote a binomial coefficient, ie.

n(n-I) ... (n-k + I)

n!

=

C

for 1 ~k ~n.

1*2* ... *k

n

k!(n-k)!

n-I

I

Clearly, n I C

and nlC n

I

= C . Let us define the following arithmetic function:

n

CCn)=rnax{k:1~k<n-1,nIC

n

k

}

(1)

n

Clearly, this function is well-defined and CCn) ~ 1. We have supposed k < n - 1, otherwise on the basis of n-l

C n

1

C = n, clearly we would have CCn) = n-l. n

k

By a well-known result on primes, pIc

for all primes p and 1 ~ k ~ p-1.

p

Thus we get: C(P) = p-2 for primes p ~ 3.

(2)

Obviously, C(2) = 1 and C(1) = 1. We note that the above result on primes is usually used in the inductive proof of Fermat's "little" theorem. This result can be extended as follows: k

Lemma: For (k,n) = 1, one has n I C n

247


Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.
On A New Smarandache Type Function by Don Hass - Issuu