a^n + b prime?

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hcf(a,b)=1, else divisible by hcf(a,b)
a+b odd, else divisible by 2
a=2^k, k=2^j only, Fermat [b=1]
a<>4k+3, else divisible by 2k+1 [b=2k]
a<>6k+1, else divisible by 3 [b=2]
a<>6, else divisible by 5 [b=19]
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2^n + 78557 prime? [a=2, b=78557, n=1]
NO - 78557 Sierpinski Number [proof at http://www.teamprimerib.com/sob/78557.php]
and 2^36=1 [mod 3,5,7,13,19,37,73] => b.2^n+1 == 2^n+b [mod 3,5,7,13,19,37,73] [mod 36] - Robert Israel

3^n + b prime? [a=3, b=2, n=1]
tested up to
b=130532, n=4400
b=130532, n=17411 prob.prime [Anthony C]
NO - b=1581339422187635434 [Haugland, Carmody]

9^n + b prime? [a=9, b=2, n=1]
tested up to
b=1396, n=1500

5^n + b prime? [a=5, b=2, n=1]
tested up to
b=788, n=1900

7^n + 2 prime? [a=7, b=2, n=1]
NO - all divisible by 3

11^n + 4 prime? [a=11, b=4, n=1]
NO - all divisible by 5

13^n + 2 prime [a=13, b=2, n=1]
NO - all divisible by 3

17^n + b prime? [a=17, b=2, n=1]
tested up to
b=32, n=1200