Definition:Balanced Prime

Definition

Definition 1

Let $\left({p_{n - 1}, p_n, p_{n + 1} }\right)$ be a triplet of consecutive prime numbers.

$p_n$ is a balanced prime if and only if:

$p_n = \dfrac {p_{n - 1} + p_{n + 1} } 2$

Definition 2

Let $\paren {p_{n - 1}, p_n, p_{n + 1} }$ be a triplet of consecutive prime numbers.

$p_n$ is a balanced prime if and only if:

\(\ds p_{n - 1} + d\)

\(=\)

\(\ds p_n\)

\(\ds p_{n - 1} + 2 d\)

\(=\)

\(\ds p_{n + 1}\)

for some $d \in \Z$.

Definition 3

Let $p$ be a prime number.

$p$ is a balanced prime if and only if the prime gaps either side of it are equal.

Sequence

The sequence of balanced primes begins:

$5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, \ldots$

Examples

\(\ds 2 \times 5\)

\(=\)

\(\ds 3 + 7\)

\(\ds 2 \times 53\)

\(=\)

\(\ds 47 + 59\)

\(\ds 2 \times 157\)

\(=\)

\(\ds 151 + 163\)

\(\ds 2 \times 173\)

\(=\)

\(\ds 167 + 179\)

\(\ds 2 \times 211\)

\(=\)

\(\ds 199 + 223\)

\(\ds 2 \times 257\)

\(=\)

\(\ds 251 + 263\)

\(\ds 2 \times 263\)

\(=\)

\(\ds 257 + 269\)

\(\ds 2 \times 373\)

\(=\)

\(\ds 367 + 379\)

\(\ds 2 \times 563\)

\(=\)

\(\ds 557 + 569\)

\(\ds 2 \times 593\)

\(=\)

\(\ds 587 + 599\)

\(\ds 2 \times 607\)

\(=\)

\(\ds 601 + 613\)

Also see

• Equivalence of Definitions of Balanced Prime

• Results about balanced primes can be found here.