32 Jean-Marie De Koninck
98
the smallest solution of γ(n + 1) γ(n) = 19: the
only38
solutions n
109
of this equation are 98, 135 and 11 375; in fact, below is the table of all the
solutions n
109
of equation γ(n + 1) γ(n) = k for 1 k 100 (observe
that this equation has no solution n
109
for k = 5, 9, 25, 33, 35, 37, 51, 57,
61, 63, 65, 77, 81, 87 and 95):
k n 109 such that
γ(n + 1) γ(n) = k
3 4, 49
7 9, 12
11 20, 27, 288, 675, 71199
13 18, 152, 3024
15 16, 28
17 1681, 59535, 139239, 505925
19 98, 135, 11375
21 25, 2299, 18490
23 75, 1215, 1647, 2624
27 52, 39325
29 171, 847, 1616, 4374
31 32, 36, 40, 45, 60, 1375
39 76, 775
41 50, 63000
43 56, 84
45 22747, 182182
47 92, 1444, 250624
49 54, 584, 21375, 23762, 71874, 177182720
53 147, 315, 9152, 52479
55 512, 9408, 12167, 129311
59 324, 4239
67 72, 88, 132, 5576255
69 82075, 656914
71 140, 3509, 114375
73 872, 1274, 3249
75 148, 105412, 843637
79 81, 104, 117, 156, 343, 375, 7100, 47375, 76895
83 164, 275, 5967, 33124, 89375, 7870625,38850559
85 126, 1016, 16128, 471968, 10028976
89 531, 11736
91 96, 100, 1050624
93 832, 201019, 1608574
97 3807, 4067, 12716, 73304
99 112, 1975, 8575
38It
seems plausible that, for each odd number k 1, the number of solutions of equation
γ(n + 1) γ(n) = k is finite. In 2003, J.M. De Koninck & F. Luca [53] proved that if the abc
Conjecture is true, then, for each odd integer k = ±1, equation γ(n + 1) γ(n) = k has only a finite
number of solutions.
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