{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

EnormousInt112201

# EnormousInt112201 - 1 LECTURE NOTES ON ENORMOUS INTEGERS by...

This preview shows pages 1–4. Sign up to view the full content.

1 LECTURE NOTES ON ENORMOUS INTEGERS by Harvey M. Friedman Department of Mathematics Ohio State University http://www.math.ohio-state.edu/~friedman/ November 22, 2001 Abstract. We discuss enormous integers and rates of growth after [PH77]. This breakthrough was based on a variant of the classical finite Ramsey theorem. Since then, examples have been given of greater relevance to a number of standard mathematical and computer science contexts, often involving even more enormous integers and rates of growth. 1.F(x 1 ,...,x k ) = F(x 2 ,...,x k+1 ) N = the nonnegative integers. THEOREM 1.Let F:N k {1,. ..,r}. There exists x 1 < . .. < x k+1 such that F(x 1 ,...,x k ) = F(x 2 ,...,x k+1 ). This is an immediate consequence of a more general combinatorial theorem called Ramsey’s theorem, but it is much simpler to state. We call this adjacent Ramsey theory. There are inherent finite estimates here. THEOREM 1.2. For all k,r there exists t such that the following holds. Let F:N k {1,. ..,r}. There exists x 1 < . .. < x k+1 £ t such that F(x 1 ,...,x k ) = F(x 2 ,...,x k+1 ). QUESTION: What is the least such t = Adj(k,r)? THEOREM 1.3. Adj(k,1) = k. Adj(k,2) = 2k. THEOREM 1.4. Let k 5. Adj(k,3) is greater than an exponential stack of k-2 1.5’s topped off with k-1. E.g., Adj(6,3) > 10 173 , Adj(7,3) > 10^10 172 . THEOREM 1.5. Adj(k,r) is at most an exponential stack of k-1 2’s topped off with a reasonable function of k and r. Our adjacent Ramsey theory from the 80’s is lurking in the background in [DLR95].

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 2. THE ACKERMANN HIERARCHY There is a good notation for really big numbers - up to a point. We use a streamlined version of the Ackerman hierarchy. Let f:Z + Z + be strictly increasing. We define the critical function f’:Z + Z + of f by: f’(n) = the result of applying f n times at 1. Define f 1 :Z + Z + to be the doubling function, and f n+1 :Z + Z + be f n ’. Thus f 1 is doubling, f 2 is exponentiation, f 3 is iterated exponentiation; i.e., f 3 (n) = E*(n) = an exponential stack of n 2’s. f 4 is confusing. We can equivalently present this by the recursion equations f 1 (n) = 2n, f k+1 (1) = f k (1), f k+1 (n+1) = f k (f k+1 (n)), where k,n = 1. We define A(k,n) = f k (n). Note that A(k,1) = 2, A(k,2) = 4. For k 3, A(k,3) > A(k- 2,k-2), and as a function of k, eventually strictly dominates each f n , n 1. A(3,4) = 65,536. A(4,3) = 65,536. A(4,4) = E*(65,536). And A(4,5) is E*(E*(65,536)). It seems safe to assert, e.g., that A(5,5) is incomprehensib- ly large. We propose this number as a sort of benchmark. 3. VECTOR REDUCTION Let k 1 and x N k . We perform the "reduction" on x = (x 1 ,...,x k ) as follows. Find the greatest i < k such that x i > 0, and replace x i ,x i+1 by x i -1,x 1 +...+x k . THEOREM 3.1. For all k 1 and x N k , this reduction can be performed only finitely many times. The number of times this reduction can be performed at x N k is very large. E.g., THEOREM 3.2. The number of times this reduction can be performed at (2,0,0,0,0) is greater than E*(2 1,000,000 ).
3 THEOREM 3.3. For all k 3 and n 2, the number of times this reduction can be applied to (n,0,. ..,0) N k is greater than A(k-1,n) and less than A(k,n+c), where c is a universal constant.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 11

EnormousInt112201 - 1 LECTURE NOTES ON ENORMOUS INTEGERS by...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online