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!
I think that OP question can also be proved by using: Bertrand's postulate - Wikipedia, the free encyclopedia
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...
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
.
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 .