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# Value k

When discussing Mersenne numbers, all factors can be expressed in the following form:

$\displaystyle{ 2kp+1 }$ where $\displaystyle{ p }$ is the exponent in $\displaystyle{ 2^p-1 }$.

An example:

\displaystyle{ \begin{align}2^{23}-1 &= 8388607\\&= 47 * 178481\\ \\ 47 &= 2 * 1 * 23 + 1\\ 178481 &= 2 * 3880 * 23 + 1 \end{align} }

So, we see that for the factor 47, $\displaystyle{ k=1 }$ and for 178481, $\displaystyle{ k=3880 }$.

When trial factoring finds a factor, it can either be written out and stored long hand, or just the k-value may be recorded.