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discovering the law of quadratic reciprocity

Project: KOB1
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Investigating the Legendre symbols (p∣q)(p|q), for primes pp and qq and discovering the Law of Quadratic Reciprocity

Plotting the legendre symbols (p∣q)(p|q), blue for +1+1, red for −1-1

N=200
pointsize = 16
G=Graphics()

points =[(p,q) for p in primes(3,N) for q in primes(3,N) 
         if not p == q]

for x in points:
    if legendre_symbol(x[0],x[1])==1:
        G+=point(x, color='blue', size=pointsize)
    elif legendre_symbol(x[0],x[1])==-1:
        G+=point(x, color='red', size=pointsize)

G.show(figsize=10, aspect_ratio=1, axes_labels=['$p$','$q$'])

Comparing (p∣q)(p|q) with (q∣p)(q|p), unfilled circle for equal, filled circle for unequal

N=200
radius=1
G=Graphics()

points =[(p,q) for q in primes(3,N) for p in primes(q,N) 
         if not p == q]

for x in points:
    l1 = legendre_symbol(x[0],x[1]) # (p|q)
    l2 = legendre_symbol(x[1],x[0]) # (q|p)
    
    if l1 == l2:
        G+=circle(x, radius, edgecolor='black', facecolor='black', 
                  thickness=1, fill=False) 
    elif l1 == -l2:
        G+=circle(x, radius, edgecolor='black', facecolor='black', 
                  thickness=1, fill=True)



G.show(figsize = 10, aspect_ratio=1, axes_labels=['$p$','$q$'])

This plot reveals the Law of Quadratic Reciprocity (p∣q)={−(q∣p), if p≡q≡3(mod4)(q∣p), otherwise (p|q) = \left \{ \begin{array}{rl} -(q|p), & \text{ if } p \equiv q \equiv 3 \pmod{4}\\ (q|p), &\text{ otherwise} \end{array}\right.