Numbers of the form b^n +/- 1 get a lot of attention (Fermat primes, Mersenne primes, Cunningham project). We go off the beaten path and examine b^n +/- 2. These numbers can be prime if b is odd.
We restrict our attention to numbers less than 2^1200. We restrict our attention to numbers with b<=n, i.e., the exponent is equal to or larger than the base, i.e., top-heavy.
Here is some Pari/GP code to find primes of the form b^n+2, also printing out its size in bits.
thresh=1200 ; for(i=3 , +oo , if( i*log(i)/log(2) > thresh , break) ; if(i%2 && (isprimepower(i)<2) , for(j=i , +oo , p=i^j+2 ; z=log(p)/log(2) ; if(z>thresh , break) ; if(ispseudoprime(p) , print(i , " ^ " , j , " + 2 " , z)))))
Below we list all the primes of both forms, sorted by size.
There are 182 primes on the list. 73 are +2. 109 are -2. Is this disparity expected?
The first (and only) instance of base 11 on the list is 11^287+2. There are no primes 11^n-2 for 11 <= n <= 13000. (However, 11^4-2 and 11^6-2 are non-top-heavy primes.) Is 11 especially sparse? If so, why?
3^4+2 and 7^12-2 are the only safe primes on the list. Intriguingly, the multiplicative group order of prime b^n+2 is b^n+1, which is a Cunningham form.
The only cousin primes (primes separated by 4) straddling a perfect power on the list are around 3^4 (79 and 83) and 51^66.
3 ^ 3 + 2
3 ^ 4 - 2
3 ^ 4 + 2
3 ^ 5 - 2
3 ^ 6 - 2
3 ^ 8 + 2
3 ^ 9 - 2
3 ^ 10 + 2
7 ^ 7 - 2
3 ^ 14 + 2
7 ^ 8 - 2
3 ^ 15 + 2
5 ^ 14 - 2
7 ^ 12 - 2
3 ^ 22 - 2
3 ^ 24 + 2
5 ^ 17 + 2
3 ^ 26 + 2
7 ^ 15 - 2
3 ^ 36 + 2
3 ^ 37 - 2
5 ^ 26 - 2
15 ^ 16 + 2
3 ^ 41 - 2
15 ^ 17 - 2
15 ^ 19 - 2
15 ^ 20 + 2
7 ^ 28 - 2
19 ^ 19 - 2
19 ^ 20 - 2
7 ^ 31 - 2
21 ^ 21 - 2
19 ^ 23 - 2
17 ^ 24 - 2
3 ^ 63 + 2
21 ^ 24 + 2
23 ^ 24 - 2
21 ^ 25 - 2
5 ^ 50 - 2
21 ^ 27 - 2
17 ^ 30 - 2
21 ^ 32 - 2
3 ^ 90 - 2
3 ^ 98 + 2
19 ^ 38 - 2
3 ^ 102 - 2
3 ^ 105 - 2
3 ^ 110 + 2
21 ^ 40 + 2
23 ^ 39 + 2
3 ^ 123 + 2
15 ^ 51 - 2
3 ^ 126 + 2
15 ^ 52 + 2
29 ^ 42 - 2
3 ^ 139 + 2
7 ^ 84 - 2
15 ^ 65 - 2
35 ^ 50 - 2
21 ^ 60 - 2
7 ^ 98 - 2
45 ^ 52 - 2
13 ^ 78 - 2
51 ^ 51 - 2
5 ^ 126 - 2
15 ^ 75 + 2
13 ^ 80 - 2
33 ^ 60 + 2
29 ^ 63 + 2
5 ^ 143 + 2
13 ^ 90 - 2
5 ^ 144 - 2
31 ^ 69 - 2
7 ^ 128 - 2
3 ^ 235 + 2
51 ^ 66 - 2
51 ^ 66 + 2
3 ^ 243 + 2
21 ^ 88 - 2
57 ^ 67 - 2
51 ^ 69 - 2
51 ^ 72 + 2
43 ^ 76 - 2
15 ^ 106 + 2
63 ^ 70 + 2
29 ^ 87 + 2
35 ^ 83 + 2
17 ^ 105 + 2
13 ^ 117 - 2
17 ^ 106 - 2
15 ^ 112 + 2
59 ^ 75 + 2
13 ^ 120 - 2
71 ^ 74 - 2
61 ^ 78 - 2
21 ^ 106 - 2
35 ^ 92 - 2
57 ^ 81 + 2
39 ^ 93 + 2
21 ^ 112 + 2
3 ^ 315 + 2
3 ^ 317 - 2
15 ^ 132 + 2
39 ^ 99 + 2
21 ^ 120 - 2
31 ^ 109 - 2
35 ^ 110 - 2
45 ^ 104 + 2
65 ^ 95 + 2
3 ^ 363 + 2
69 ^ 96 - 2
83 ^ 92 - 2
87 ^ 92 + 2
5 ^ 260 - 2
5 ^ 261 + 2
3 ^ 386 + 2
3 ^ 391 + 2
37 ^ 119 - 2
45 ^ 114 + 2
47 ^ 113 + 2
91 ^ 97 - 2
93 ^ 98 - 2
21 ^ 146 - 2
33 ^ 128 - 2
7 ^ 238 - 2
69 ^ 111 - 2
87 ^ 106 - 2
39 ^ 132 + 2
77 ^ 113 + 2
99 ^ 109 - 2
45 ^ 135 + 2
95 ^ 113 + 2
17 ^ 184 - 2
61 ^ 127 - 2
103 ^ 113 - 2
69 ^ 127 - 2
3 ^ 494 + 2
75 ^ 128 - 2
3 ^ 520 - 2
51 ^ 146 - 2
73 ^ 136 - 2
115 ^ 123 - 2
7 ^ 302 - 2
33 ^ 169 - 2
3 ^ 541 - 2
63 ^ 144 + 2
105 ^ 129 + 2
87 ^ 136 + 2
55 ^ 153 - 2
95 ^ 135 + 2
3 ^ 561 - 2
93 ^ 137 - 2
29 ^ 189 + 2
113 ^ 135 + 2
17 ^ 232 - 2
133 ^ 136 - 2
65 ^ 161 + 2
69 ^ 159 - 2
111 ^ 144 + 2
69 ^ 161 - 2
115 ^ 144 - 2
45 ^ 180 - 2
11 ^ 287 + 2
123 ^ 144 + 2
87 ^ 156 + 2
3 ^ 648 - 2
45 ^ 190 - 2
53 ^ 184 - 2
39 ^ 204 - 2
33 ^ 218 + 2
35 ^ 220 - 2
39 ^ 216 - 2
153 ^ 158 + 2
39 ^ 217 + 2
21 ^ 264 - 2
29 ^ 239 + 2
43 ^ 214 - 2
57 ^ 200 - 2
117 ^ 171 - 2
91 ^ 181 - 2
137 ^ 166 - 2
111 ^ 176 + 2
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