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 Hyperoperators [n] basics for large n dyitto Junior Fellow Posts: 13 Threads: 3 Joined: Mar 2011 03/06/2011, 01:20 AM (This post was last modified: 03/12/2011, 10:15 PM by dyitto.) Hyperoperators [n] basics for large n We could inductively define hyperoperators as follows: a, b, n being positive integers: a[1]b := a + b n > 1 -> a[n]1 := a a[n+1](b + 1) := a[n](a[n+1]b) From this some lemmas can be proven: 1. a[2]b = a * b 2. a[3]b = a ^ b 3. 2[n]2 = 4 4. n > 2 -> 1[n]b = 1 5. a > 1 -> a[n](b + 1) > a[n]b 6. (a + 1)[n]b > a[n]b 7. ((a > 2 or b > 2) and a > 1 and b > 1) -> a[n+1]b > a[n]b 8. 1 < a < b c = $^a\log(b)$ rounded up to integer m > 0, k >= 0 Then: a [4] m >= c * (b + k) -> a [4] (m + k + 1) >= b [4] (k + 2) Is this the common definition here? Have proofs been given somewhere for the lemmas? I wrote them down long time ago, and I was about to do it again before I discovered this forum. Some minor corrections made in this post[/edit] « Next Oldest | Next Newest »

 Messages In This Thread Hyperoperators [n] basics for large n - by dyitto - 03/06/2011, 01:20 AM RE: Hyperoperators [n] basics for large n - by bo198214 - 03/06/2011, 08:52 AM RE: Hyperoperators [n] basics for large n - by dyitto - 03/06/2011, 03:19 PM RE: Hyperoperators [n] basics for large n - by dyitto - 03/06/2011, 03:24 PM RE: Hyperoperators [n] basics for large n - by dyitto - 03/06/2011, 03:32 PM RE: Hyperoperators [n] basics for large n - by dyitto - 03/06/2011, 08:56 PM RE: Hyperoperators [n] basics for large n - by dyitto - 03/06/2011, 09:41 PM RE: Hyperoperators [n] basics for large n - by dyitto - 03/07/2011, 11:26 AM RE: Hyperoperators [n] basics for large n - by bo198214 - 03/07/2011, 01:35 PM RE: Hyperoperators [n] basics for large n - by dyitto - 03/12/2011, 10:19 PM

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