2 [n] b and 3 [n] b for (large) integer n, b - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: 2 [n] b and 3 [n] b for (large) integer n, b (/showthread.php?tid=607) 2 [n] b and 3 [n] b for (large) integer n, b - dyitto - 03/08/2011 There's one thing I can easily proof about this: n > 2 -> 3 [n] 2 > 2 [n] 3 Proof: n = 3: 3 [3] 2 = 3 ^ 2 = 9 > 8 = 2 ^ 3 = 2 [3] 3 Suppose for some n > 2 : 3 [n] 2 > 2 [n] 3 Then we wish to prove that 3 [n+1] 2 > 2 [n+1] 3 3 [n+1] 2 = 3 [n] 3 = 3 [n-1] (3 [n] 2) > 3 [n-1] (2 [n] 3) > 2 [n-1] (2 [n] 3) = 2 [n] 4 = 2 [n] (2 [n+1] 2) = 2 [n+1] 3 But now I also suspect that for each n: 2 [n+1] b > 3 [n] b will be true for sufficiently large b n = 1: b > 3 -> 2 * b > 3 + b n = 2: b > 3 -> 2 ^ b > 3 * b But haven't yet proved it for any n > 2. RE: 2 [n] b and 3 [n] b for (large) integer n, b - dyitto - 03/09/2011 n = 3: b > 3 -> 2 [4] b > 3 [3] b Proof: For b = 4: 2 [4] 4 = 2 ^ (2 ^ (2 ^ 2)) = 2 ^ (2 ^ 16) = 2 ^ 65536 > 81 = 3 ^ 4 = 3 [3] 4 Let's assume that 2 [4] b > 3 [3] b for some b > 3. Then we wish to prove that 2 [4] (b + 1) > 3 [3] (b + 1) 2 [4] (b + 1) = 2 ^ (2 [4] b) > 2 ^ (3 [3] b) > 3 * (3 [3] b) = 3 [3] (b + 1) RE: 2 [n] b and 3 [n] b for (large) integer n, b - dyitto - 03/12/2011 n = 4: b > 3 -> 2 [5] b > 3 [4] b Proof: For b = 4: Apply lemma 8 having a = 2, b = 3, c = 2, m = 4, k = 2: 2 [4] 4 > 2 * (3 + 2) is certainly true -> 2 [4] 7 >= 3 [4] 4 2 [5] 4 = 2 [4] 65536 > 2 [4] 7 >= 3 [4] 4 Let's assume that 2 [5] b > 3 [4] b for some b > 3. Then we wish to prove that 2 [5] (b + 1) > 3 [4] (b + 1) 2 [5] (b + 1) = 2 [4] (2 [5] b) > 2 [4] (3 [4] b) > 3 [3] (3 [4] b) = 3 [4] (b + 1)