Graham Numbers

Let's define $a\uparrow [n]a=a\underset{\text{n arrows}}{\underbrace{\uparrow \uparrow \cdots \uparrow }}a$ where $\uparrow$ represents the Knuth's arrow notation

We define :

• $g_0=4$
• $g_1=3\uparrow [g_0]3$
• $\dots$
• $g_{64}=3\uparrow [g_{63}]3$

$g_{64}$ is the notorious Graham's number.

The last digits of this humongous number are known:

$\dots 03222348723967018485186439059104575627262464195387$

We define $G(3)=g_{64}$ and the generalized Graham's number G(p) where we replace all 3s with p.

What is the smallest prime number p such as G(p) contains the string '1415' in its last 100 digits.

Answer format: p,the last digits starting with '1415'

Example: 3,8485186439059104575627262464195387 // If we ask for the smallest p such as G(p) contains '8485'.

[My timing: 4 sec]