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# Questions tagged [primitive-roots]

For questions about primitive roots in modular arithmetic, index calculus, and applications in cryptography. For questions about primitive roots of unity, use the (roots-of-unity) tag instead.

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### Number of primitive roots mod $p$ that are not primitive roots mod $p^2$

Consider the primitive roots of a prime $p$ in the range $1...p$ which are not primitive roots mod $p^2$. Let $n(p)$ be this number. While looking for an answer to this question, it seems that the ...
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### Intuition behind this strange heuristic for primitive roots modulo $p$?

Let $p$ be an odd prime. Define $S(p)$ as the sum of all primitive roots modulo $p$ taken from $\left[-\frac{p-1}2,\frac{p-1}2\right]$. Now here's the strange thing. If the primitive roots were '...
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### Given a prime $p$ and a positive integer $a \not \equiv 0,1 \pmod p$, show that S = $\sum^{p-1}_{i=1} a^i \equiv 0 \pmod p$

The origin of this question is actually a different question: Show that all primes except 2 and 5 divide infinitely many elements of $B :=\{1,11,111,1111,\cdots\}$. It's relatively straightforward ...