Memory Fragment 10

The forgetherealone site was updated with memory fragments 8, 9, and 10 at the same time.

Problem

For fragment 10 there was the following formula:

(1)
\begin{align} \sum_{i=0}^{63} F_i * x^i \end{align}

And a small field named "x" with a button labeled "=" that, when pressed, gave a result. For example entering 63 resulted in:

F(63) = 4594756481115229140062713082413927821823140847615442236532722027145486469523332472724064595038667228332129762087338

We needed to figure out the 64 values for $F_i$ ($F_0$ through $F_{63}$).

Solution

First letter

We first re-wrote the formula as:

(2)
\begin{align} (\sum_{i=0}^{62} F_i * x^i) + (F_{63} * x^{63}) \end{align}

To find $F_{63}$ we considered $F(63)$:

(3)
\begin{align} F(63) = (\sum_{i=0}^{62} F_i * 63^i) + (F_{63} * 63^{63}) \end{align}

and realized that the right side of the addition is very large due to the multiplication by $63^{63}$, while the left side is negligibly small in comparison. So we can get the following approximation:

(4)
\begin{align} F(63) \approx F_{63} * 63^{63} \end{align}

Rewrite to isolate isolate $F_{63}$.

(5)
\begin{align} F_{63} \approx \dfrac{F(63)}{63^{63}} \end{align}

And solve using the number we got from the form.

(6)
\begin{align} F_{63} \approx \dfrac{4594756481115229140062713082413927821823140847615442236532722027145486469523332472724064595038667228332129762087338}{63^{63}} = 20.12833... \end{align}

Since we're looking for a message, that means we're looking for letters, so whole numbers only. So that gives us $F_{63} = 20$.

Second letter

Re-write the formula again as:

(7)
\begin{align} (\sum_{i=0}^{61} F_i * x^i) + (F_{62} * x^{62}) + (F_{63} * x^{63}) \end{align}

Again consider $F(63)$:

(8)
\begin{align} F(63) = (\sum_{i=0}^{61} F_i * 63^i) + (F_{62} * 63^{62}) + (F_{63} * 63^{63}) \end{align}

Substitute that we already know $F_{63} = 20$.

(9)
\begin{align} F(63) = (\sum_{i=0}^{61} F_i * 63^i) + (F_{62} * 63^{62}) + (20 * 63^{63}) \end{align}

Rearrange.

(10)
\begin{align} F(63) - (20 * 63^{63}) = (\sum_{i=0}^{61} F_i * 63^i) + (F_{62} * 63^{62}) \end{align}

Apply that the portion of the addition with $63^{62}$ is so much larger that the other portion is negligible.

(11)
\begin{align} F(63) - (20 * 63^{63}) \approx F_{62} * 63^{62} \end{align}

Rewrite to isolate $F_{62}$.

(12)
\begin{align} F_{62} \approx \dfrac{F(63) - (20 * 63^{63})}{63^{62}} \end{align}

Substitute the known value of $F(63)$ and solve:

(13)
\begin{align} F_{62} \approx \dfrac{4594756481115229140062713082413927821823140847615442236532722027145486469523332472724064595038667228332129762087338 - (20 * 63^{63})}{63^{62}} = 8.08519... \end{align}

So that gives us $F_{62} = 8$.

All the Letters

Repeating this process (using a computer program, for ease) gives us all of the letters:

$i$ $F_i$ Letter
63 20 T
62 8 H
61 5 E
60 23 W
59 9 I
58 19 S
57 5 E
56 19 S
55 20 T
54 1 A
53 13 M
52 15 O
51 14 N
50 7 G
49 19 S
48 20 T
47 20 T
46 8 H
45 5 E
44 9 I
43 18 R
42 14 N
41 21 U
40 13 M
39 2 B
38 5 E
37 18 R
36 9 I
35 13 M
34 2 B
33 21 U
32 5 E
31 4 D
30 22 V
29 9 I
28 1 A
27 12 L
26 19 S
25 23 W
24 9 I
23 20 T
22 8 H
21 20 T
20 8 H
19 5 E
18 9 I
17 18 R
16 13 M
15 9 I
14 14 N
13 4 D
12 19 S
11 6 F
10 15 O
9 18 R
8 1 A
7 14 N
6 5 E
5 23 W
4 19 S
3 20 T
2 1 A
1 18 R
0 20 T
THEWISESTAMONGSTTHEIRNUMBERIMBUEDVIALSWITHTHEIRMINDSFORANEWSTART
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