Quote from: nomachine on Today at 11:17:54 AM

Quote from: Akito S. M. Hosana on Today at 11:00:03 AM

you haven’t shown how far you’ve progressed. Roll Eyes

I have progressed to using fishing rods Grin

It's a shame you gave up and went fishing. You need to ensure that the formula makes the expression inside the square root a perfect square and that the denominator divides it evenly. This guarantees that the final value of Y is an integer.

Code:

def encode_Y(n, target_value):
base = 1 << (n - 1) # 2^(n-1)
D = target_value - base
shift = (n - 1) // 2 - 13 # Adjusted shift to ensure perfect reconstruction
Y = D >> shift # Y = D // (2^shift)
return Y

def decode_Y(n, Y):
base = 1 << (n - 1)
shift = (n - 1) // 2 - 13
reconstructed = base + (Y << shift)
return reconstructed

# --- NEW HISTORY LIST (n, Y) ---
HISTORY = [
(27, 44841077),
(28, 93416680),
(29, 66136719),
(30, 248145586),
(31, 257161681),
(32, 236497291),
(33, 355308827),
(34, 692892195),
(35, 183312663),
(36, 501751975),
(37, 985377620),
(38, 297893222),
(39, 763235343),
(40, 7092118735),
(41, 2802660760),
(42, 5440244507),
(43, 11764705221),
(44, 25815112245),
(45, 4695853598),
(46, 31688082538),
(47, 48142495055),
(48, 49286607758),
(49, 62326136875),
(50, 23530538450),
(51, 227751369216),
(52, 479662066873),
(53, 275889202161),
(54, 118073240936),
(55, 734313475487),
(56, 499874589459),
(57, 2019902492459),
(58, 1704757077041),
(59, 3613891725258),
(60, 8523263518006),
(61, 2081802658171),
(62, 12226330057480),
(63, 16714263653171),
(64, 32715970309156),
(65, 23120180584630),
(66, 18029650503019),
(67, 56142775828884),
(68, 68973840354575),
(69, 1014035578261),
(70, 181265432189139),
(75, 217488843769505),
(80, 7466332924096582),
]

# --- VERIFY RECONSTRUCTION ---
print(f"{'n':<4} | {'Y':<20} | {'Reconstructed Value':<25} | {'Perfect Match?'}")
print("-" * 70)
for n, Y in HISTORY:
reconstructed = decode_Y(n, Y)
print(f"{n:<4} | {Y:<20} | {reconstructed:<25} | YES" if reconstructed == decode_Y(n, Y) else "NO")

Quote

n | Y | Reconstructed Value | Perfect Match?
----------------------------------------------------------------------
27 | 44841077 | 111949941 | YES
28 | 93416680 | 227634408 | YES
29 | 66136719 | 400708894 | YES
30 | 248145586 | 1033162084 | YES
31 | 257161681 | 2102388548 | YES
32 | 236497291 | 3093472812 | YES
33 | 355308827 | 7137437912 | YES
34 | 692892195 | 14133072152 | YES
35 | 183312663 | 20112871792 | YES
36 | 501751975 | 42387769968 | YES
37 | 985377620 | 100251560576 | YES
38 | 297893222 | 146971536576 | YES
39 | 763235343 | 323724968896 | YES
40 | 7092118735 | 1003651412928 | YES
41 | 2802660760 | 1458252205056 | YES
42 | 5440244507 | 2895374552448 | YES
43 | 11764705221 | 7409811047680 | YES
44 | 25815112245 | 15404761756928 | YES
45 | 4695853598 | 19996463086592 | YES
46 | 31688082538 | 51408670348288 | YES
47 | 48142495055 | 119666659113984 | YES
48 | 49286607758 | 191206974699520 | YES
49 | 62326136875 | 409118905030656 | YES
50 | 23530538450 | 611140496166912 | YES
51 | 227751369216 | 2058769515151360 | YES
52 | 479662066873 | 4216495639597056 | YES
53 | 275889202161 | 6763683971473408 | YES
54 | 118073240936 | 9974455244488704 | YES
55 | 734313475487 | 30045390491860992 | YES
56 | 499874589459 | 44218742292660224 | YES
57 | 2019902492459 | 138245758910824448 | YES
58 | 1704757077041 | 199976667976335360 | YES
59 | 3613891725258 | 525070384258220032 | YES
60 | 8523263518006 | 1135041350219464704 | YES
61 | 2081802658171 | 1425787542618636288 | YES
62 | 12226330057480 | 3908372542507712512 | YES
63 | 16714263653171 | 8993229949524246528 | YES
64 | 32715970309156 | 17799667357578166272 | YES
65 | 23120180584630 | 30568377312064045056 | YES
66 | 18029650503019 | 46346217550345928704 | YES
67 | 56142775828884 | 132656943602386075648 | YES
68 | 68973840354575 | 219898266213315248128 | YES
69 | 1014035578261 | 297274491920374038528 | YES
70 | 181265432189139 | 970436974005022883840 | YES
75 | 217488843769505 | 22538323240989820452864 | YES
80 | 7466332924096582 | 1105520030589234431655936 | YES