Page 179 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

The term with the highest power of p is

N!
N
p N = p .
N!0!

Exercise: What is 0!? Please review #9 from Chapter 4 and #11 in Chapter 6.

The next lowest power of p is

N!
p N−1 m = Nmp N−1 .
(N − 1)!1!

All the rest of the terms in the Binomial Formula are non-negative, so this means that

N
N
N
q =(p + m) ≥ p + Nmp N−1 .
Now, let’s look at (p/q) ,
N

N N N N
p p p p
= = ≤ .
N
q q N (p + m) N p + Nmp N−1
The additive identity can sneak into this equation disguised as

p N
1= ,
p N

and so

p N p N 1 1 1
= =1 ∗ = .
N
N
p + Nmp N−1 p 1+ Nm/p 1+ Nm/p 1+ Nm/p

We are looking for a giant number N ∈ N such that

p
N
<.
q

Since

p 1
N
≤ ,
q 1+ Nm/p

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