# Are 1414 and 2121 the only “interesting” numbers?

The numbers $$1414$$ and $$2121$$ are quite interesting.

The prime factorisation of $$1414$$ is $$2⋅72\cdot 7$$ and the prime factorisation of $$14+114+1$$ is $$3⋅53\cdot 5$$. Note that $$33$$ is the prime after $$22$$ and $$55$$ is the prime before $$77$$.

Similarly, the prime factorisation of $$2121$$ is $$7⋅37\cdot 3$$ and the prime factorisation of $$21+121+1$$ is $$11⋅211\cdot 2$$. Again, $$1111$$ is the prime after $$77$$ and $$22$$ is the prime before $$33$$.

In other words, they both satisfy the following definition:

Definition: A positive integer $$nn$$ is called interesting if it has a prime factorisation $$n=pqn=pq$$ with $$p≠qp\ne q$$ such that the prime factorisation of $$n+1n+1$$ is $$p′q′p'q'$$ where $$p′p'$$ is the prime after $$pp$$ and $$q′q'$$ the prime before $$qq$$.

Are there other interesting numbers?

Note that exactly one of $n$ and $n+1$ is even. It follows that for $n$ to be interesting, either $n=3p$ and $n+1=2N(p)$ or $n=2p$ and $n+1=3P(p)$, where $P(p)$ and $N(p)$ are the previous and next primes to $p$ respectively. Rearranging we get that $p$ must satisfy one of the following two equations:

However, by a 1952 result of Jitsuro Nagura, for $p\ge25$ there is always a prime between $p$ and $\frac65p$. In particular, if $p\ge31$ is a prime:
But when $p\ge31$ the following inequalities are also true:

Therefore, if $p$ is to satisfy $(1)$ or $(2)$ above, it must be less than 31. This leaves a handful of cases to check for $p$, and we find that the only interesting numbers are 14 and 21 as conjectured.

The Nagura paper is a reference in the Wikipedia article on Bertrand's postulate. While those in the comments had saw it, sketching out the approach I use here, I already knew what to do; I did not read those comments in detail until after posting my answer.