I'm writing a paper right now, in which Sophie Germain's theorem is included. Can anybody explain auxiliary prime θ to me?


Sophie Germain proved that the product $xyz$ must be divisible by $p^2$ if an auxiliary prime θ can be found such that two conditions are satisfied:

No two $p$th powers differ by one modulo θ; and $p$ is itself not a $p$th power modulo θ.


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    $\begingroup$ I think you need to add way more context. $\endgroup$ – pjs36 Dec 16 '15 at 18:36
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    $\begingroup$ @pjs3 thanks mate :) $\endgroup$ – Luzzi Dec 16 '15 at 18:40
  • $\begingroup$ It would improve the question a lot, if you told the readers that you are discussing the cases of FLT studied by Sophie Germain. For example $xyz$ does not mean much at all unless it is stated that they form a putative counterexample to FLT. $\endgroup$ – Jyrki Lahtonen Dec 19 '15 at 7:59

In this case, auxiliary prime simply means a prime related to the inputs and conditions. Given that $x,y,z,p$ satisfy

$$x^p + y^p = z^p$$

then $p^2$ divides $xyz$ if we can find a prime (the "auxiliary prime" $\theta$) such that

  1. $p^n - p^m \ne 1 \text{ mod } \theta$ for all $m,n$, and
  2. $p$ is itself not a $p$th power modulo $\theta$.

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