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## Main Question or Discussion Point

In pascal triangle, if all the elements of a row(except 1 in both end) are divisible by the row number, then the row number is a prime.

Or if the coefficient of binomial expansion (except 1's) are divisible by the power, then the power is a prime.

It is inefficient to check all the coefficient upto the middle term(since the coefficient repeat, thereafter) with the power of the binomial expansion, for divisibility.

But if we take this way,

Say 15361, i.e

(1+1)

So, if 15361 is a prime, then it will also be divisible, to the sum of coefficient, too. But it is highly possible, composite number will also divide the sum of coefficient.

But if we take summation of coefficient in parts, then it becomes almost zero, that each summation part,which will have different values, will also be divisible by power of composite numbers.

That is, in the above example.

If we divide ,7680 coefficient(upto middle term), of the above examples, into say 12 equal parts, i.e each part will have 640 coefficent and each summation part will be of different values. Then, the possibility become highly, that only a power of prime will only divide all the 12 parts summation.

C(15361,1)+.....+C(15361,640) = will be divisible by the prime power

C(15361,641)+.....+C(15361,1280) = will be divisible by the prime power

C(15361,1281)+.....+C(15361,1920) = will be divisible by the prime power

C(15361,1921)+.....+C(15361,2560) = will be divisible by the prime power

C(15361,2561)+.....+C(15361,3200) = will be divisible by the prime power

C(15361,3201)+.....+C(15361,3840) = will be divisible by the prime power

C(15361,3841)+.....+C(15361,4480) = will be divisible by the prime power

C(15361,4481)+.....+C(15361,5120) = will be divisible by the prime power

C(15361,5121)+.....+C(15361,5760) = will be divisible by the prime power

C(15361,5761)+.....+C(15361,6400) = will be divisible by the prime power

C(15361,6401)+.....+C(15361,7040) = will be divisible by the prime power

C(15361,7041)+.....+C(15361,7680) = will be divisible by the prime power

So, what is the efficieny of such check, for primality.

Thanks.

Or if the coefficient of binomial expansion (except 1's) are divisible by the power, then the power is a prime.

It is inefficient to check all the coefficient upto the middle term(since the coefficient repeat, thereafter) with the power of the binomial expansion, for divisibility.

But if we take this way,

Say 15361, i.e

(1+1)

^{15361}= 1 + C(15361,1) + C(15361,2) + .... + C(15361,7680) + C(15361,7681) + ... + C(15361,15360) + 1So, if 15361 is a prime, then it will also be divisible, to the sum of coefficient, too. But it is highly possible, composite number will also divide the sum of coefficient.

But if we take summation of coefficient in parts, then it becomes almost zero, that each summation part,which will have different values, will also be divisible by power of composite numbers.

That is, in the above example.

If we divide ,7680 coefficient(upto middle term), of the above examples, into say 12 equal parts, i.e each part will have 640 coefficent and each summation part will be of different values. Then, the possibility become highly, that only a power of prime will only divide all the 12 parts summation.

C(15361,1)+.....+C(15361,640) = will be divisible by the prime power

C(15361,641)+.....+C(15361,1280) = will be divisible by the prime power

C(15361,1281)+.....+C(15361,1920) = will be divisible by the prime power

C(15361,1921)+.....+C(15361,2560) = will be divisible by the prime power

C(15361,2561)+.....+C(15361,3200) = will be divisible by the prime power

C(15361,3201)+.....+C(15361,3840) = will be divisible by the prime power

C(15361,3841)+.....+C(15361,4480) = will be divisible by the prime power

C(15361,4481)+.....+C(15361,5120) = will be divisible by the prime power

C(15361,5121)+.....+C(15361,5760) = will be divisible by the prime power

C(15361,5761)+.....+C(15361,6400) = will be divisible by the prime power

C(15361,6401)+.....+C(15361,7040) = will be divisible by the prime power

C(15361,7041)+.....+C(15361,7680) = will be divisible by the prime power

So, what is the efficieny of such check, for primality.

Thanks.