In her paper Periodic Decimal Fractions, Bennet uses the phrase "the exponent to which $10$ belongs modulo $p$" to mean "the least positive integer $k$ such that $10^k\equiv 1\pmod p$."
After looking around, it seems that this language is somewhat commonplace, and it can be found on documents by Wolfram and other papers. As a native English speaker, I have never encountered this language and it took me a while to figure out the author's intent.
- Does anybody know where this language came from?
- How can we explain the use of "belongs to" here?
It comes from Gauss's Disquisitiones Arithmeticae, and his development of primitive roots.
In the section on primitive roots he proves, amongst others, two results:
If $a^k\equiv 1 \pmod p$, then $k|(p-1)$
The number of elements that (Gauss's word) 'belong' to $k$ is $\phi(k)$, with $\phi(k)$ being the totient function.
For example, using $p=19$ and primitive root $3$ we get the following table.
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline 1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\ \hline 3&9&8&5&15&7&2&6&18&16&10&11&14&4&12&17&13&1\\ \hline \end{array}
The divisors of $p-1=18$ are $1,2,3,6,9,18$, with totient values $1,1,2,2,6,6$.
The order of an element is the lowest value of $k$ such that $a^k\equiv 1\pmod p$, which gives the following table:
\begin{array}{|c|c|c|} \hline k&\phi(k)&elements\\ \hline 1&1&18\\\hline 2&1&9\\\hline 3&2&7,11\\\hline 6&2&8,12\\\hline 9&6&4,5,6,9,16,17\\\hline 18&6&2,3,10,13,14,15\\ \hline \end{array}
At this point, Gauss says that, for example, '$16$ belongs to the exponent $9$, or '$8$ belongs to the exponent $6$'.
To clear up, $7$ belongs to exponent $3$ because $7^3\equiv 1\pmod{19}$. If we look up $7$ in the first table, we can see that $3^6\equiv 7\pmod{19}$, and then that $(3^6)^3=3^{18}\equiv 1\pmod{19}$.