mersenneforum.org No Notice- Binomial Coefficients, Pascal's triangle
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 2005-04-08, 17:01 #1 Vijay   Apr 2005 2·19 Posts No Notice- Binomial Coefficients, Pascal's triangle Ok people, You must have heard about pascal's triangle, the Binomial coefficients... Well What I want to say is that, if you glance at it for a while and notice this: 1.) A Mersenne prime is ALWAYS present as the second coefficient on a random line of the triangle. E.g. look at the 3rd row and second coefficent, you find 3, the first Mersenne prime number, yes?... Well again look at the 7th row, second coefficient, the second Mersenne number, ofcourse doing this you easily can find any mersenne number in the binomial coefficents, but what is also true is that each mersenne number will always be as the second coefficient in the triangle and 2.) a perfect number can be located just underneath the M prime number, so first perfect number is on 4th row, 3rd coeffiicient, second is on 8th row, 3rd coefficent, 3.) hence each perfect number is a product of the sum of a mersenne number and another odd number. Each perfect number can be found underneath the M prime number as the 3rd coefficient in the binomial triangle. 4.) Obviously, the bionial triangle is symmetrical and you'll find the perfect numbers and M primes mirrored on the opposite side of the triangle. Convinced? Not quite... Well what this shows is that the binomial triangle can be made to an infinity of numbers, true, therefore there are an infinite number of M primes and perfect numbers. Also it shows that an odd perfect number can't exist. I think my finding does help somewhat, as the binomial triangle also shows some patterns as to how to get the next Mersenne prime and perfect number, I've found the pattern and got the proof, its time, you people also notice the pattern. AND finally to conclude, there is a pattern. I believe in God
2005-04-08, 17:17   #2
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

23·937 Posts

Quote:
 Originally Posted by Vijay Ok people, You must have heard about pascal's triangle, the Binomial coefficients... Well What I want to say is that, if you glance at it for a while and notice this: 1.) A Mersenne prime is ALWAYS present as the second coefficient on a random line of the triangle. E.g. look at the 3rd row and second coefficent, you find 3, the first Mersenne prime number, yes?... Well again look at the 7th row, second coefficient, the second Mersenne number, ofcourse doing this you easily can find any mersenne number in the binomial coefficents, but what is also true is that each mersenne number will always be as the second coefficient in the triangle and 2.) a perfect number can be located just underneath the M prime number, so first perfect number is on 4th row, 3rd coeffiicient, second is on 8th row, 3rd coefficent, 3.) hence each perfect number is a product of the sum of a mersenne number and another odd number. Each perfect number can be found underneath the M prime number as the 3rd coefficient in the binomial triangle. 4.) Obviously, the bionial triangle is symmetrical and you'll find the perfect numbers and M primes mirrored on the opposite side of the triangle. Convinced? Not quite... Well what this shows is that the binomial triangle can be made to an infinity of numbers, true, therefore there are an infinite number of M primes and perfect numbers. Also it shows that an odd perfect number can't exist. I think my finding does help somewhat, as the binomial triangle also shows some patterns as to how to get the next Mersenne prime and perfect number, I've found the pattern and got the proof, its time, you people also notice the pattern. AND finally to conclude, there is a pattern. I believe in God

Just what we need. Another kook.

Your first observation is worse than trivial because EVERY integer appears
as the second integer in a row of the triangle. C(n,1) = n, so n appears
as the second integer of the n'th row. This has ZIPPO to do with
Mersenne primes.

"binomial triangle can be made to an infinity of numbers" is mathematical
gobbledygook.

Go learn some math. Go learn how to discuss it. And stop making ridiculous
claims out of ignorance.

As for your belief in God:

You are entitled to whatever perversions you like. But belief in God has no
rational basis, while discussions of mathematics does. The two have nothing
to do with one another.

George Carlin said it: There should be another commandment. Keep thy
religion to thyself.

 2005-04-08, 17:23 #3 Vijay   Apr 2005 1001102 Posts Ok Glad to see your awake
 2005-04-09, 15:27 #4 Jwb52z     Sep 2002 3·281 Posts I always thought that a mathematician would be above liking a low comic like George Carlin. Oh well.
 2005-04-09, 18:04 #5 Ken_g6     Jan 2005 Caught in a sieve 5×79 Posts As for the perfect numbers appearing below the mersenne primes, that's natural, too. The third column of Pascal's Triangle adds the two numbers above it, effectively summing all those numbers from column 2. As those numbers count from 1 to (row-2), C(n,2)=1+2+...+n-1, which is a hexagonal number. It is very easy to prove that this equals 1/2*(n-1)*n. Now, if the number n-1 just happened to be a Mersenne prime M, then C(M+1,2)=1/2*M*(M+1). Euclid proved two millenia ago that this is always a perfect number. Euler proved that all even perfect numbers are of this form. It is probable, but not proven, that there are no odd perfect numbers. And you won't find any looking though Pascal's Triangle yourself; all numbers less than 10^300 have been tested.
 2005-04-09, 20:36 #6 Vijay   Apr 2005 1001102 Posts I see

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