Prime Sequences
Jackson Capper
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Given a prime P(m):
\(
\begin{align}
\\
a &= \left\lfloor\frac{P(m)}{2}\right\rfloor
\\
{A} &= \left\{ n : P(m) < n < \left(P(m)+\left(a – \mod{\frac{P(m)}{a}}\right)\right) \right\}
\\
{B} &= \left\{ n : P(m) + \left((a – n) – \mod{\frac{P(m)}{(a – n)}}\right), 0 < n \leq a \right\}
\\&= \left\{ P(m) + \left((a – 0) – \mod{\frac{P(m)}{(a – 0)}}\right), P(m) + \left((a – 1) – \mod{\frac{P(m)}{(a – 1)}}\right), \dots \right\}
\\
P(m + 1) &= \min{ \left({B} – {A}\right) }
\end{align}
\)
Example: For P(m) = 11,
\(
\begin{align}
a &= \left\lfloor\frac{11}{2}\right\rfloor
\\ &= 5
\\
{A} &= \left\{ 12, 13, 14, 15 \right\}
\\
{B} &= \left\{ 11+\left(5 – \mod{\frac{11}{(5)}}\right), 11+\left(4 – \mod{\frac{11}{(4)}}\right), \dots \right\}
\\ &= \left\{ 12, 14, 15 \right\}
\\
P(m + 1) &= \min{ (\left\{ 12, 13, 14, 15 \right\} – (\left\{ 12, 14, 15 \right\}) }
\\ &= \min{ \left\{ 13 \right\} }
\\ &= 13
\end{align}
\)