Saturday, 23 February 2013


Today is the $54^{th}$ day of the year.

$54 = 2 \times 3^3$

There are some prime numbers that are one greater than a square number. $2,917$ is one such number, see A002496. $2,917 = 2,916 + 1 = 54^2 + 1$, thus $54$ is one of the sequence of numbers $n$ such that $n^2 + 1$ is prime, see A005574. $n$ could be thought of as a Near Root Prime.

It is interesting to speculate on whether there is a number that is in this sequence, let's call it $p$, that when one is added to its square is also a member of this sequence i.e is there a number $p$ such that $p^2 + 1$ is a Near Root Prime. (cf Cunningham chains).

It is fairly easy to show that no such number exists. Assume that $p$ is a Near Root Prime then we know by definition that $p^2 + 1$ is prime. All primes greater than 2 are odd, therefore $p^2 + 1$ is odd. This means that $p^2$ is even which imples that $p$ is even assuming $p$ is greater than $1$. Inspection of A005574 bears this out.
If all the members of the Near Root Prime sequence are even , except the first member, then $p^2 + 1$ cannot be a Near Root Prime because it is odd.

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