number theory – Least prime of the form $38^n+31$

The Question :

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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.

The Question Comments :
• If $n$ is odd, then $3 \vert (38^n+31)$. If $n$ is of the form $4k+2$, then $5 \vert (38^n+31)$. The only case to be proved is $38^{4k}+31$ is not a prime.
• @lhf Yes, for multiple of $4$, interestingly the prime factors get larger if $n \neq 12k+4$. If $n=12k+4$, then $7$ divides $38^n+31$. Hence, the only $n$’s that needs to be checked are $n=12k$ and $n=12k+8$.
• Are you trying to crack some secret key? What got you interested in these numbers?
• Obviously, since nothing is lesser than $-\infty$, and since $39^{-\infty}+31=0+31=31$ is prime, this is then “the least n such that $39^n+31$ is prime”. 🙂
• If my idle-time computer calculation is correct, the expression is not prime for $n<185000$.

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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 $(a,b)$.

14 people think this answer is useful

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 $ab^n+c$.

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.