Least prime of the form $38^n+31$ I search the least n such that 
$$38^n+31$$ 
is prime. 
I checked the $n$ upto $3000$ and found none, so the least prime of that form must have more than $4000$ digits. I am content with a probable prime, it need not be a proven prime.
 A: This should be a comment for @Einar Rødland, but it's too long so I'm making it an answer.
Your argument gives a heuristic that there should be infinitely many primes of this form, but I'm not sure I believe it and here's why:
You are only taking into account the first two "bad" arithmetic sequences when you restrict to looking at things divisible by 4.  In fact we have infinitely many "bad" arithmetic sequences we need to throw out.
If we try the same argument for numbers of the form $2^n + 1$ and we throw out $n \equiv 1 \mod 2$ and $n \equiv 2 \mod 4$ as they will all be divisible by 3 or 5, then running your argument would tell us to expect infinitely many primes of this form. However if we throw out all the "bad" arithmetic sequences (which in this case are much easier to classify) then we only are left with $n = 2^k$, taking this into account your heuristic gives finite expectation.
These two problems don't seem that different to me on the surface, and without a better understanding of the higher order "bad" arithmetic sequences, it's not clear to me what the right heuristic should be.
A: This is not a proof, but does not conveniently fit into a comment.
I'll take into account that $n=4k$ is required, otherwise $38^n+31$ will be divisible by $3$ or $5$ as pointed in the comments.
Now, if we treat the primes as "pseudorandom" in the sense that any large number $n$ has a likelihood $1/\ln(n)$ of being prime (which is the prime number density for large $n$), the expected number of primes for $n=4,8,\ldots,4N$ will increase with $N$ as
$$
\sum_{k=1}^N\frac{1}{\ln(38^{4k}+31)}
\approx\frac{\ln N+\gamma}{4\ln 38}
\text{ where }\gamma=0.57721566\ldots
$$
and for the expected number of primes to exceed 1, you'll need $N$ in the order of 1,200,000.
Of course, you could get lucky and find it at much lower $n$, but a priori I don't see any particular reason why it should be...or shouldn't.
Basically, in general for numbers $a^n+b$, the first prime will usually come fairly early, otherwise often very late (or not at all if $a$ and $b$ have a common factor).
Of course, this argument depends on assuming "pseudorandom" behaviour of the primes, and so cannot be turned into a formal proof. However, it might perhaps be possible to say something about the distribution of $n$ values giving the first prime over different pairs $(a,b)$.
A: Primality of numbers of the form $a^n+b$ is a very hard problem in general. For instance, existence of primes of the form $4294967296^n+1=(2^{32})^n+1$ is an old open problem in number theory (wiki), although it is also easy to show that this can be a prime only for $n$ of a special form (powers of $2$). Your problem $2085136^n+31=(38^4)^n+31$ does not seem much easier.
In other words, a theory-based answer to your problem is very unlikely in the near future. For a practice-based answer you will probably need to use some distributed computing project for searching for prime numbers like PrimeGrid, which has found most of the known large primes of the form $ab^n+c$.
A: In general, is there any prime of the form $\frac{ab^n+c}{\gcd(a+c,b-1)}$ with integer $n\ge1$, for fixed integers $a\ge1$, $b\ge2$, $c\ne0$, $\gcd(a,c)=1$, $\gcd(b,c)=1$? It is unknown whether this problem is solvable, e.g. let $a=4$, $b=72$, $c=-1$, the smallest prime of the form $\frac{4\cdot72^n-1}{\gcd(4+1,72-1)}$ with $n\ge1$ is at $n=1119849$
If we want to solve this problem, we must for every such $(a,b,c)$ integer triple, either find a prime of this form or prove that there is no prime of this form, e.g.:
$(a,b,c) = (78557,2,1)$, in which all numbers are divisible by at least one of $3, 5, 7, 13, 19, 37, 73$
$(a,b,c) = (11047,3,1)$, in which all numbers are divisible by at least one of $2, 5, 7, 13, 73$
$(a,b,c) = (47,8,1)$, in which all numbers are divisible by at least one of $3, 5, 13$
$(a,b,c) = (334,10,-1)$, in which all numbers are divisible by at least one of $3, 7, 13, 37$
$(a,b,c) = (9,4,-1)$, in which all numbers factored as difference of squares
$(a,b,c) = (1,8,1)$, in which all numbers factored as sum of cubes
$(a,b,c) = (25,12,-1)$, in which even $n$ factored as difference of squares and odd $n$ is divisible by $13$
$(a,b,c) = (64,936,-1)$, in which even $n$ factored as difference of squares and $n$ divisible by $3$ factored as difference of cubes and other $n$ divisible by either $37$ or $109$, etc.
