Linear support for the prime number sequence and the first and second Hardy-Littlewood conjectures

Abstract

Servais and Grün used results about linear support for the prime number sequence to obtain upper bounds on the smallest prime in odd perfect numbers. This was extended by Cohen and Hendy who proved that for every $n \in \mathbb {N}$ there exists an integer $b_n$ such that $p_i \ge p_1 + ni - b_n$ for any sequence $(p_i)$ of odd primes, and found minimal values of $b_n$ for $3 \le n \le 5$, and conjectured minimal values for $6 \le n \le 10$. We give a new proof of the existence of $b_n$ and the values for $3 \le n \le 5$. We also show that if we assume that the second Hardy-Littlewood conjecture, $\pi (a+b) \le \pi (a) + \pi (b)$ for $a, b \ge 2$, is true, then we can in a finite number of steps determine numbers, $T_n$, that give quite close bounds for the values of $b_n$, namely $T_n \le b_n \le T_n + n - 2$, and determine the values of $T_n$ for $6 \le n \le 20$.

We also consider the question of whether the values of $b_n$ can be replaced by smaller numbers if we assume that $p_1 > 3$. We will show that if we assume that the first Hardy-Littlewood conjecture is true, then we can determine the minimum such values, $a_n$, for $3 \le n \le 5$. We also determine some lower bounds for $a_n$ for $n \ge 6$. It is well-known that the two Hardy-Littlewood conjectures are mutually exclusive, but we never use both conjectures simultaneously.

These results give us the upper bound $p_1<\frac {n}{2^n-1}t+b_n$ on the smallest prime in odd perfect numbers, where $t$ is the number of distinct primes dividing the odd perfect number. This bound was also found by Cohen and Hendy, but we improve the constants $b_n$.