Hello!

I have a question! Is it true that the central binomial coefficient is divisible by a prime number if , where means fractional part?

Thanks!

Printable View

- Jul 10th 2010, 01:34 PMBen92Divisibility of central binomial coefficient
Hello!

I have a question! Is it true that the central binomial coefficient is divisible by a prime number if , where means fractional part?

Thanks! - Jul 10th 2010, 02:03 PMchiph588@
- Jul 10th 2010, 02:11 PMchiph588@
- Jul 10th 2010, 02:58 PMBen92
Thanks! :)

- Jul 10th 2010, 03:03 PMAlso sprach Zarathustra
I think that OP question can also be proved by using: Bertrand's postulate - Wikipedia, the free encyclopedia

- Jul 10th 2010, 04:09 PMmelese
Here is what I've tried. , with (becuase is the fractional part). Then and (since ). So, . (Remember this).

In general, , so . Therefore, , since when we have a strict inequality.

Now, due to Legendre, the exponent of the largest power of that divides is , and this expresstion is positive. This implies that divides .

I hope this helps... - Jul 10th 2010, 04:22 PMchiph588@
- Jul 10th 2010, 05:00 PMmeleseReply to comments.
- Jul 10th 2010, 05:28 PMchiph588@
- Jul 10th 2010, 06:39 PMsimplependulum

Then

or so the value of the power of the prime still be contained in the central coefficient is :

.

We need to ensure that there exists such that or equivalently . I believe is not necessarily to be . For example , we have

and but we still have - Jul 10th 2010, 06:56 PMmelese
- Jul 10th 2010, 07:01 PMchiph588@
- Jul 10th 2010, 07:13 PMsimplependulum
Yes , so the if-then statement is correct , by the way i am not sure if this can help in this situation :

The max. power of a given prime denoted by such that can be obtained by this formula ( without a formal proof , could be wrong ) : where is the sum of the digits of in the representation of base .

For example , we have so so which is true because

. - Jul 11th 2010, 04:31 AMBen92
My proof:

Let a prime number, so the highest power of which divides is , so the highest power of which divides the central binomial coefficient is . And that is completely trivial will be 1 for all real number if , and we know that there are only two values possible: 0 or 1, so the summation is nonnegative. Thus, it is anough if happens at least once in the summation, and the initial condition shows that, because . - Jul 11th 2010, 01:54 PMmelese