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

(3)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)Rewrite to isolate isolate $F_{63}$.

(5)And solve using the number we got from the form.

(6)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)Again consider $F(63)$:

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

(9)Rearrange.

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

(11)Rewrite to isolate $F_{62}$.

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

(13)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`