Find the least integer $k_n$ s.t. $1^2,2^2,...,n^2$ are all incongruent modulo $k_n$ for $n\geq4$.
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4$\begingroup$ Do your own homework. $\endgroup$– Noah SchweberCommented Jun 6, 2019 at 5:07
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2$\begingroup$ This doesn't really look like homework. But generally it would be better to avoid the imperative mode when posting questions. $\endgroup$– Todd TrimbleCommented Jun 6, 2019 at 5:48
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1$\begingroup$ Thanks Noah Schweber. I'm not good at English. $\endgroup$– quangdoCommented Jun 6, 2019 at 6:35
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$\begingroup$ @quangdo Your guess that Bertrand's postulate applies was completely right! $\endgroup$– Pietro MajerCommented Jun 6, 2019 at 7:37
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1 Answer
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The answer appeared in the following paper:
L. K. Arnold, S. J. Benkoski and B. J. McCabe, The discriminator (a simple application of Bertrand’s postulate), Amer. Math. Monthly 92 (1985), 275–277.
For each $n=5,6,\ldots$, the number $k_n$ is the smallest integer $m \ge 2n$ such that $m$ is $p$ or $2p$ with $p$ an odd prime.
For some further developments, you may look at my talk On primes in arithmetic progressions.
answered Jun 6, 2019 at 5:29
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$\begingroup$ Thanks you very much, Zhi-Wei Sun. $\endgroup$– quangdoCommented Jun 6, 2019 at 6:38