Are 1414 and 2121 the only “interesting” numbers?

The numbers 14 and 21 are quite interesting.

The prime factorisation of 14 is 27 and the prime factorisation of 14+1 is 35. Note that 3 is the prime after 2 and 5 is the prime before 7.

Similarly, the prime factorisation of 21 is 73 and the prime factorisation of 21+1 is 112. Again, 11 is the prime after 7 and 2 is the prime before 3.

In other words, they both satisfy the following definition:

Definition: A positive integer n is called interesting if it has a prime factorisation n=pq with pq such that the prime factorisation of n+1 is pq where p is the prime after p and q the prime before q.

Are there other interesting numbers?

Answer

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:
3p+12=N(p)
2p+13=P(p)
However, by a 1952 result of Jitsuro Nagura, for p25 there is always a prime between p and 65p. In particular, if p31 is a prime:
56p<P(p)<p<N(p)<65p But when p31 the following inequalities are also true:
2p+13<56p65p<3p+12
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.

Attribution
Source : Link , Question Author : Simon Parker , Answer Author :
Parcly Taxel

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