Memory Fragment 14

Fragments 12, 13, 14, and 15 dropped at the same time, with no text for Fragment 14.
Fragment 12 linked to F??.png which led us to finding F12131415.png

F12131415.png

The puzzle for Fragment 14 is in the bottom-left quadrant.

The file https://forgethereal.one/assets/F14.dat has the following content:

1657229172964193430185557834892191493
7005108079809315450125272750715986004
4135979842850244987992560307534266588
8513442452810246038831281829119667785
3704822042388146399722224887080304947
1577621808088618200193041116212287770
1341847538613096790475547928801832113
2214812596101118749268951542049500403
4368012437168325370443317448244486693
3247135998232821530957875893260090251
5704957169301965550852041194667872149
8225591981956623918494633529005586304
1689679802517261934506287302895341431
3941840405023829727086564280559455038
3279850296724992478281835782181985658
6900939909563551003865084610162169472
4968812107206867156853792846880246677
3694916050250601455783929554846811247
8787923997389792716527200156422353721
0923890511055086549719459202631043483
8264496035386198082566405265361789450
7497051766167577333296204740268103612
6877329180283667717035945127073759406
7000197485098287813826007978541809509
6721537657692491303467443832248567701
5029852398930370134150842180583447294
9067840417727956149081698698638177793
4335902401542970527626435140613948647
4471145516898520227673746513362900153
5311659613533992018528657751182129159
6673907176299643677389428863209718488
3595995847452393117209273718046295216
8769339116967450093645967995040345311
1548781226590988782825473473669883574
3361264723358562687009430931906588667
8024122822481552292967679341435105138
88391505833795485871649858750=2^?*...

Solution

The problem is laid out in a 37x37 grid. 37 is a prime number. This together with

=2^?*…

indicates that the large number is a multiplication of prime numbers.

Since prime numbers are only divisible by themselves, if you multiply primes together, you can always tell which primes got multiplied.
For example, start with the prime number 5 and multiply it by the prime number 7 to get 35. No matter which other prime numbers we multiply 35 by, the result will always be divisible by 5 and 7. And since none of those other primes are divisible by 5 or 7, we know that multiplying by 5 and by 7 was involved to get the final number.

Similarly, if you multiply a by a prime multiple times, you can still tell how many times that prime was multiplied by.
For example, start with 5 * 5 (= 5^2) and multiply it by 7 * 7 * 7 (= 7^3) to get 8575. No matter which other prime numbers we multiply 8575 by, the result will always be divisible by 5 two times and by 7 three times.

So with =2^?* the puzzle is asking "How many times is the number 2 (aka the first prime number) multiplied into this big number? And then, how many times the next prime? Etc."

n n'th prime times divisible by the n'th prime letter
1 2 1 A
2 3 14 N
3 5 4 D
4 7 9 I
5 11 19 S
6 13 20 T
7 17 21 U
8 19 13 M
9 23 2 B
10 29 12 L
11 31 5 E
12 37 4 D
13 41 1 A
14 43 3 C
15 47 18 R
16 53 15 O
17 59 19 S
18 61 19 S
19 67 1 A
20 71 6 F
21 73 12 L
22 79 1 A
23 83 19 S
24 89 11 K
25 97 9 I
26 101 14 N
27 103 1 A
28 107 14 N
29 109 3 C
30 113 9 I
31 127 5 E
32 131 14 N
33 137 20 T
34 139 18 R
35 149 21 U
36 151 9 I
37 157 14 N
38 163 19 S
39 167 23 W
40 173 8 H
41 179 5 E
42 181 18 R
43 191 5 E
44 193 20 T
45 197 23 W
46 199 15 O
47 211 13 M
48 223 9 I
49 227 14 N
50 229 4 D
51 233 19 S
52 239 2 B
53 241 5 E
54 251 3 C
55 257 1 A
56 263 13 M
57 269 5 E
58 271 1 A
59 277 19 S
60 281 9 I
61 283 14 N
62 293 7 G
63 307 12 L
64 311 5 E

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