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Please help me find all $a, b, c \in \mathbb{P}$ such that $a^b+c$ is a prime

Example I just can find: $2^3 + 5 = 13 \in \mathbb{P}$

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there is probably (I'll be as cautious too :-)) an infinity of solutions : consider just $a=b=2$ you'll get $c\in{3,7,13,19,37,43,67,79,\cdots}$. Are there more hypothesis than indicated? –  Raymond Manzoni Nov 25 '12 at 10:18
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short table for $a>2$ of first [a,b,c] solutions : [3, 2, 2] [3, 3, 2] [3, 139, 2] [5, 3, 2] [5, 17, 2] [11, 5, 2] [23, 11, 2] [29, 3, 2] [41, 23, 2] [47, 113, 2] [53, 7, 2] [71, 3, 2] [71, 13, 2] [71, 31, 2] [83, 3, 2] [113, 3, 2] [131, 23, 2] [149, 5, 2] [149, 13, 2] [149, 17, 2] [173, 3, 2] [179, 5, 2] [191, 29, 2] [197, 5, 2] [197, 41, 2] –  Raymond Manzoni Nov 25 '12 at 10:25
    
Thank you to all, teacher's challenge is so ... :P –  Xeing Nov 25 '12 at 10:34

1 Answer 1

up vote 5 down vote accepted

There are almost certainly infinitely many: classifying them is probably very difficult.

For example, take $a=b=2$. It has long been conjectured that there are infinitely many primes of the form $4+c$, where $c$ is prime. Tis problem is related to the famous twin primes problem. There is a great deal of numerical evidence suggesting that there are infinitely many primes $c$ such that $4+c$ is prime. But (so far) there is no proof. Examples are easy: $4+3$, $4+7$, $4+13$, and so on.

It is in fact conjectured that for any even number $e$, there are infinitely many primes of the form $e+c$, where $c$ is prime. That would mean that in particular there are infinitely many primes of the form $2^b+c$, where $c$ is any prime. Again, there are many examples: $2^3+3$, $2^3+5$, $2^3+11$, $2^5+5$, and so on.

For odd $a$, th problem is different, for then the only possibility is $c=2$. But it is not known whether there are infinitely many primes of the form $a^b+2$, where $a$ is an odd prime and $b$ is prime. Examples are not hard to find, such as $3^2+2$, $3^3+2$, $5^3+2$, $11^2+2$.

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