Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2006-10-04 03:43:37

ganesh
Moderator
Registered: 2005-06-28
Posts: 12,919

Breaking the record of Graham's number

Let n=6->6->6->6->6
Let n1 equal to the smallest number apart from zero and one which is a perfect square, cube, fourth, fifth, sixth and nth power.
n1+1 is a prime number.
Proof:- n1-1 is divisible to almost any prime number, because of the fact that (a^n-b^n) is divisible by (a-b).
This is a better proof than Graham's Number for Ramsay theory, he he big_smile big_smile
Such numbers are enormously large.
For example, the smallest number apart from zero and one which is a perfect square, cube, fourth, fifht and sixth power is 1152921504606846976.
(That is how many bytes make an exabyte).
And extrapolating this to tenths power, the result is 3.940842x10^758, a number containing 759 digits.
If this is extrapolated to 20th power, the result is 2.125219 x 10^70077543 approximately, containg 70077544 digits.
And when this extrapolated to 1000th power, the result is 10^10^433, much much larger than a googolplex smile


Character is who you are when no one is looking.

Offline

#2 2006-10-04 03:56:28

Devantè
Real Member
Registered: 2006-07-14
Posts: 6,400

Re: Breaking the record of Graham's number

Cool. The modern age just keeps coming up with new things.

Offline

#3 2006-10-04 04:07:47

ganesh
Moderator
Registered: 2005-06-28
Posts: 12,919

Re: Breaking the record of Graham's number

Prime numbers are never-ending. It is very very difficult to prove that n1+1 is a composite number, well impossible, because of this property i mentioned. However, Graham's Number is a shame to mathematics and to Ramsay theory. The proof isn't sound at all, as said by some other mathematicians. Hence, I was interested in giving a more difficult aspect to prove or disprove by counter-proof.


Character is who you are when no one is looking.

Offline

#4 2006-10-04 04:12:47

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Breaking the record of Graham's number

Let n=6->6->6->6->6
Let n1 equal to the smallest number apart from zero and one which is a perfect square, cube, fourth, fifth, sixth and nth power.
n1+1 is a prime number.
Proof:- n1-1 is divisible to almost any prime number, because of the fact that (a^n-b^n) is divisible by (a-b).

I don't understand your 6->6->6->6 notation.

I also don't see how stating n1-1 is divible by almost any prime number proves n1+1 is prime.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Offline

#5 2006-10-04 16:43:25

ganesh
Moderator
Registered: 2005-06-28
Posts: 12,919

Re: Breaking the record of Graham's number

Ricky,
the arrow in the notation denotes John Conway's chained arrow notation which is much much larger than knuth's up-arrow notation.
If n-1 is divisible by almost all prime numbers, n+1 should be prime.
Thats a hypothetic conclusion.


Character is who you are when no one is looking.

Offline

#6 2006-10-04 17:47:52

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Breaking the record of Graham's number

If n-1 is divisible by almost all prime numbers, n+1 should be prime.
Thats a hypothetic conclusion.

By hypothetic, do you mean one that remains to be unproven?

And what is meant by most?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Offline

#7 2006-10-04 20:37:59

ganesh
Moderator
Registered: 2005-06-28
Posts: 12,919

Re: Breaking the record of Graham's number

Yes, hypothetic is because, it is next to impossible to prove.
There already exists a proof that (a^n-b^n) is divisible by (a-b).
Therefore, I used the word 'most'.
Like for example, 63 is divisible by 3, 7.
4095 has many prime factors.
These are numbers of the kind I have mentioned in my post #1.


Character is who you are when no one is looking.

Offline

#8 2007-05-21 05:53:56

Laterally Speaking
Real Member
Registered: 2007-05-21
Posts: 356

Re: Breaking the record of Graham's number

I'm pretty sure the reason why people have disclaimed Graham's number is because it is the upper bound to a (rather hairy) problem in Ramsey Theory, the lower bound having been identified as 6 by the same people that "discovered" the upper bound.

As one of these people said, "there's room for improvement".


"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

Offline

#9 2007-05-23 00:33:57

Laterally Speaking
Real Member
Registered: 2007-05-21
Posts: 356

Re: Breaking the record of Graham's number

ganesh wrote:

Ricky,
the arrow in the notation denotes John Conway's chained arrow notation which is much much larger than knuth's up-arrow notation.

In fact, The chained-arrow notation here is pretty huge, although I'm not sure I really got the concept of the notation in full.


"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

Offline

Board footer

Powered by FluxBB