2 [n] b and 3 [n] b for (large) integer n, b

[dyitto]

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)