$2^n + 1518781$ is never a prime. Why?

Question

$2^n + 1518781$ is never a prime. Why ?

A general rule-of-thumb for "is there a prime of the form $f(n)$?" questions is, unless there exists a set of small divisors $D$, called a covering set, that divide every number of the form $f(n)$, then there will eventually be a prime.

So I assume some kind of covering set can be constructed here.

But I have no idea how to do that.

I know $1518781 = 11 * 138071$.

edit

$$2^{2n+1} + 1518781 = 3 k$$

So we reduce to studying.

$$4^n + 1518781$$

Similarly

$$4^{2n + 1} + 1518781 = 5 m$$

So we reduce to studying

$$16^n + 1518781$$

Answer 1

All the $2^n + 1518781, $ for $n \geq 1,$ are divisible by at least one prime of these six: $3,5,7,13,17,241.$ Which one (when only one) can be found from reducing $n \pmod {24}.$ That's it.

I have a factoring routine that considers only small prime factors, up to some bound I say. For this output I told it to stop at 10000, that is stop and print "big" if the remaining factor is over 10000 and is composite.

1   = 3 7 31  2333
2   = 5 89  3413
3   = 3  506263
4   = 7 17  12763
5   = 3^2 37  4561
6   = 5 31 41  239
7   = 3 7 151  479
8   = 13  116849
9   = 3 107  4733
10   = 5 7 173  251
11   = 3^3 23 31  79
12   = 17 29  3089
13   = 3 7 19 43  89
14   = 5  307033
15   = 3  517183
16   = 7^3 31  149
17   = 3^2  183317
18   = 5^2  71237
19   = 3 7 271  359
20   = 13 17  11617
21   = 3 31 59  659
22   = 5 7 23 47  151
23   = 3^2 313  3517
24   = 89 241  853
25   = 3 7 499  3347
26   = 5 31 41  10799
27   = 3 43  1052221
28   = 7 17 1303  1741
29   = 3^3  19940359
30   = 5 397  541693
31   = 3 7 19 31  173741
32   = 13  cdot mbox{BIG} 
33   = 3 23  cdot mbox{BIG} 
34   = 5 7  cdot mbox{BIG} 
35   = 3^2 89 4517  9497
36   = 17 31 61 563  3797
37   = 3 7^2 151  6191849
38   = 5^2  cdot mbox{BIG} 
39   = 3  cdot mbox{BIG} 
40   = 7 29 887  6106337
41   = 3^2 31 37 43  4953997
42   = 5 4507  cdot mbox{BIG} 
43   = 3 7 409  cdot mbox{BIG} 
44   = 13 17 23  cdot mbox{BIG} 
45   = 3 47  cdot mbox{BIG} 
46   = 5 7 31^2 41 89  573343
47   = 3^4 263 7013  942031
48   = 241  cdot mbox{BIG} 
49   = 3 7 19  cdot mbox{BIG} 
50   = 5 53 79  cdot mbox{BIG} 
51   = 3 31 1579  cdot mbox{BIG} 
52   = 7 17 151 823  cdot mbox{BIG} 
53   = 3^2  cdot mbox{BIG} 
54   = 5 193  cdot mbox{BIG} 
55   = 3 7 23 43 197 613  14365061
56   = 13 31  cdot mbox{BIG} 
57   = 3 89  cdot mbox{BIG} 
58   = 5^2 7^2 389 757  cdot mbox{BIG} 
59   = 3^2  cdot mbox{BIG} 
60   = 17 251  cdot mbox{BIG} 
61   = 3 7 31 191 463  cdot mbox{BIG} 
62   = 5 1381  cdot mbox{BIG} 
63   = 3 67  cdot mbox{BIG} 
64   = 7 83 2633 4129  cdot mbox{BIG} 
65   = 3^3 101  cdot mbox{BIG} 
66   = 5 23 31 41 229 2801  cdot mbox{BIG} 
67   = 3 7 19 151 181  cdot mbox{BIG} 
68   = 13 17 29 47 89 137  cdot mbox{BIG} 
69   = 3 43  cdot mbox{BIG} 
70   = 5 7  cdot mbox{BIG} 
71   = 3^2 31  cdot mbox{BIG} 
72   = 241  cdot mbox{BIG} 
73   = 3 7 1153  cdot mbox{BIG} 
74   = 5 397  cdot mbox{BIG} 
75   = 3 3313  cdot mbox{BIG} 
76   = 7 17 31 2777  cdot mbox{BIG} 
77   = 3^2 23 37 733 839  cdot mbox{BIG} 
78   = 5^2  cdot mbox{BIG} 
79   = 3 7^2 59 89  cdot mbox{BIG} 
80   = 13  cdot mbox{BIG} 
81   = 3 31  cdot mbox{BIG} 
82   = 5 7 151 167  cdot mbox{BIG} 
83   = 3^3 43  cdot mbox{BIG} 
84   = 17 2447  cdot mbox{BIG} 
85   = 3 7 19 139 2851  cdot mbox{BIG} 
86   = 5 31 41  cdot mbox{BIG} 
87   = 3  cdot mbox{BIG} 
88   = 7 23^2 6323  cdot mbox{BIG} 
89   = 3^2 79  cdot mbox{BIG} 
90   = 5 89  cdot mbox{BIG} 
91   = 3 7 31 47 5011  cdot mbox{BIG} 
92   = 13 17  cdot mbox{BIG} 
93   = 3  cdot mbox{BIG} 
94   = 5 7  cdot mbox{BIG} 
95   = 3^2  cdot mbox{BIG} 
96   = 29 31 61 241  cdot mbox{BIG} 
97   = 3 7 43 151  cdot mbox{BIG} 
98   = 5^3  cdot mbox{BIG} 
99   = 3 23  cdot mbox{BIG} 
100   = 7^2 17 1019  cdot mbox{BIG} 
101   = 3^5 31 89  cdot mbox{BIG} 
102   = 5 53 1427 4933  cdot mbox{BIG} 
103   = 3 7 19  cdot mbox{BIG} 
104   = 13^2 8839  cdot mbox{BIG} 
105   = 3  cdot mbox{BIG} 
106   = 5 7 31 41 317  cdot mbox{BIG} 
107   = 3^2 2389  cdot mbox{BIG} 
108   = 17  cdot mbox{BIG} 
109   = 3 7 1439  cdot mbox{BIG} 
110   = 5 23 251  cdot mbox{BIG} 
111   = 3 31 43  cdot mbox{BIG} 
112   = 7 89 151  cdot mbox{BIG} 
113   = 3^2 37  cdot mbox{BIG} 
114   = 5 47 787  cdot mbox{BIG} 
115   = 3 7 107  cdot mbox{BIG} 
116   = 13 17^2 31  cdot mbox{BIG} 
117   = 3  cdot mbox{BIG} 
118   = 5^2 7 397  cdot mbox{BIG} 
119   = 3^3 617  cdot mbox{BIG} 
120   = 241  cdot mbox{BIG}