# How many odd primes are Quadratic Residues $\pmod {21}$

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!

• There are infinitely many primes congruent to $1$ modulo $21$. – Lord Shark the Unknown Nov 5 '17 at 16:51
• @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? – user366818 Nov 5 '17 at 16:54
• Since 21 is not a prime, being a square mod 21 is not the same as having the Jacobi symbol 1. – user8268 Nov 5 '17 at 17:09

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