Timeline for Infinitely many $N$ such that $\langle p\rangle=\langle q\rangle$ mod $N$
Current License: CC BY-SA 3.0
20 events
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| Nov 5, 2013 at 17:47 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Oct 31, 2013 at 13:54 | vote | accept | Huichi Huang | ||
| Oct 31, 2013 at 13:30 | comment | added | Huichi Huang | @GH:Thanks.Nice proof. One more comment: the procedure to produce infinitely many $N$ seems to go as follows: suppose we have $N_1,\,...\,N_i$ as required, then let $M=N_1\timesN_2...\timesN_i$, then we get $N_{i+1}$ satisfying with the required properties but distinct from $N_1,...,N_i$. | |
| Oct 31, 2013 at 12:33 | comment | added | GH from MO | @fedja: I love your version, it is great! Can you also give infinitely many pairwise coprime $N$'s without Dirichlet? | |
| Oct 31, 2013 at 12:29 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Oct 31, 2013 at 10:54 | comment | added | fedja | Lovely. So, all we need is $k$, and a large $N|p^k-q$ with $k$ relatively prime to $\varphi(N)$. That $N$ or $k$ is prime is of no importance really. Let's do it avoiding Dirichlet. The primality of $k$ and the condition $p<q$ are ingenious, so we keep them. Take a huge prime $k>q-p$. Then $p^k-q\equiv p-q\mod k$. Now take the prime factorization of $p^k-q$ and remove all prime factors that are $1$ modulo $k$. Let $N$ be what is left. Then $k\not\mid\varphi(N)$ ($k$ is neither a prime in $N$, nor a divisor of a prime in $N$ minus $1$) and we still have $N\equiv p-q\mod k$, so $N\ge k+p-q$. | |
| S Oct 31, 2013 at 10:17 | history | suggested | Michael Zieve | CC BY-SA 3.0 |
Minor edits for clarity.
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| Oct 31, 2013 at 9:42 | review | Suggested edits | |||
| S Oct 31, 2013 at 10:17 | |||||
| Oct 31, 2013 at 8:31 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Oct 31, 2013 at 8:24 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Oct 31, 2013 at 6:08 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Oct 31, 2013 at 5:57 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Oct 31, 2013 at 5:42 | history | undeleted | GH from MO | ||
| Oct 31, 2013 at 5:42 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Oct 30, 2013 at 18:18 | history | deleted | GH from MO | via Vote | |
| Oct 30, 2013 at 16:32 | comment | added | Will Sawin | How does this produce infinitely many $N$? What if they all divide $p-q$? | |
| Oct 30, 2013 at 16:17 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Oct 30, 2013 at 15:37 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Oct 30, 2013 at 15:25 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Oct 30, 2013 at 15:20 | history | answered | GH from MO | CC BY-SA 3.0 |