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I am asked to find all odd primes $p$ such that $ \left( \frac{p}{21} \right) = 1 $ is the Jacobi Symbol.

I have shown that $\left( \frac{p}{21} \right) = 1\Leftrightarrow \left( \frac{p}{3} \right)\left( \frac{p}{7} \right) = 1 \Leftrightarrow \left( \frac{3}{p} \right)\left( \frac{7}{p} \right) = 1$ and that:

$$p \equiv 1 \pmod 3 \\p \in \lbrace1,2,4\rbrace \pmod 7$$

My suspicion is that only finitely many primes are quadratic residues $\pmod {21} $ so I have been trying to arrive at a contradiction somehow, but I'm unsure how I can go about that. Any advice would be greatly appreciated, thank you!

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    $\begingroup$ There are infinitely many primes congruent to $1$ modulo $21$. $\endgroup$ – Lord Shark the Unknown Nov 5 '17 at 16:51
  • $\begingroup$ @LordSharktheUnknown I see so then there are in fact infinitely many primes that are squares mod 21. But how I can determine what they are? $\endgroup$ – user366818 Nov 5 '17 at 16:54
  • $\begingroup$ Since 21 is not a prime, being a square mod 21 is not the same as having the Jacobi symbol 1. $\endgroup$ – user8268 Nov 5 '17 at 17:09
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You missed half of them. $(-1)(-1) = 1.$

Sun Nov  5 10:44:33 PST 2017
   p      mod 3     mod 7   mod 21 
    5        2        5        5
   17        2        3       17
   37        1        2       16
   41        2        6       20
   43        1        1        1
   47        2        5        5
   59        2        3       17
   67        1        4        4
   79        1        2       16
   83        2        6       20
   89        2        5        5
  101        2        3       17
  109        1        4        4
  127        1        1        1
  131        2        5        5
  151        1        4        4
  163        1        2       16
  167        2        6       20
  173        2        5        5
  193        1        4        4
  211        1        1        1
  227        2        3       17
  251        2        6       20
  257        2        5        5
  269        2        3       17
  277        1        4        4
  293        2        6       20
  311        2        3       17
  331        1        2       16
  337        1        1        1
  353        2        3       17
  373        1        2       16
  379        1        1        1
  383        2        5        5
  419        2        6       20
  421        1        1        1
  457        1        2       16
  461        2        6       20
  463        1        1        1
  467        2        5        5
  479        2        3       17
  487        1        4        4
  499        1        2       16
  503        2        6       20
  509        2        5        5
  521        2        3       17
  541        1        2       16
  547        1        1        1
  563        2        3       17
  571        1        4        4
  587        2        6       20
  593        2        5        5
  613        1        4        4
  631        1        1        1
  647        2        3       17
  673        1        1        1
  677        2        5        5
  709        1        2       16
  719        2        5        5
  739        1        4        4
  751        1        2       16
  757        1        1        1
  761        2        5        5
  773        2        3       17
  797        2        6       20
  823        1        4        4
  839        2        6       20
  857        2        3       17
  877        1        2       16
  881        2        6       20
  883        1        1        1
  887        2        5        5
  907        1        4        4
  919        1        2       16
  929        2        5        5
  941        2        3       17
  967        1        1        1
  971        2        5        5
  983        2        3       17
  991        1        4        4
   p      mod 3     mod 7   mod 21 
Sun Nov  5 10:44:33 PST 2017
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