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Prove that there are infinitely many primes $q$ such that $q \equiv 1 \pmod{n}$, when $n$ is prime.

Use the hint: Consider the order of $a + kN$ in the multiplicative group of $\mathbb{Z}/N\mathbb{Z}$, where $N=a^n-1$ and $k \in \mathbb{Z}$

This is copied from this question, the answer I found doesn't quite make sense to me, more specifically why is it that the $q_i \not\equiv 1 \pmod n$ ? Would anyone be able to explain it, here is the answer provided

Let $q$ be a prime divisor of $$ A = \frac{a^n-1}{a-1} = a^{n-1}+a^{n-2} + \dots + a + 1, $$ then $$ a^n-1 = A (a-1) \equiv 0 \pmod{q},$$ so by Fermat's little theorem either $q \vert a-1$ or $n \vert q-1$ (or both). Now if $q$ is also a divisor of $a-1$, then $$ 0 \equiv A \equiv 1^{n-1} + 1^{n-2} + \dots + 1 \equiv n \pmod{q} $$ and so $q$ is also a divisor of $n$, so if we chose $a$ equal to a multiple of $n$ then we are sure that a prime divisor $q$ of $A$ is not a divisor of $a-1$.

Now we can use Euclid's argument in the following way: suppose that there is only a finite number of primes $\equiv 1 \pmod{n}$, say $q_1,q_2,\dots,q_k$, set $a = nq_1q_2\dots q_k$ or $a = n$ if $k=0$. Let $q$ a divisor of $a^{n-1}+a^{n-2}+\dots+1$, then by the previous discussion $q$ is not a divisor of $a-1$, but $a^n \equiv 1\pmod{q}$ so by Fermat's little theorme $n \vert q-1$, and we found a prime $q \equiv 1 \pmod{n}$, which by construction is not possibly any of the $q_i$'s a contradiction.

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  • $\begingroup$ Look at what $q$ divides. Can any $q_i$ divide the same thing ? $\endgroup$ – Maxime Ramzi Oct 27 '18 at 22:24
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The fact you need is that $q$ divides $a^n-1$, and $q_i$ cannot, since it divides $a^n$.

This is a special case of Dirichlet's theorem on primes in arithmetic progression. The full theorem is harder to prove.

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  • $\begingroup$ Where does it state that q divides $a^n$? $\endgroup$ – J. Masterson Oct 28 '18 at 18:23
  • $\begingroup$ Also is it necessary to state $a=n$ if $k=0$? $\endgroup$ – J. Masterson Oct 28 '18 at 18:23
  • $\begingroup$ That just depends on whether you've established a convention that the empty product is equal to 1. $\endgroup$ – C Monsour Oct 28 '18 at 21:46
  • $\begingroup$ It doesn't say that $q$ divides $a^n$. But you set $a=nq_1\cdot\cdot\cdot q_k$, so $q_i$ divided $a$ and thus also $a^n$, for $1\le i\le k$. $\endgroup$ – C Monsour Oct 28 '18 at 21:48

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