EnormousInt112201

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

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1 LECTURE NOTES ON ENORMOUS INTEGERS by Harvey M. Friedman Department of Mathematics Ohio State University [email protected] 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].
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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 ).
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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.
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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

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