quadrecip.sagews

Investigating the Legendre symbols $(p|q)$, for primes $p$ and $q$

and discovering the Law of Quadratic Reciprocity

The legendre symbols $(p|q)$, blue for $+1$, red for $-1$

N=200
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=3)
    else:
        G+=point(x, color='red', size=3)

G.show(aspect_ratio=1)

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

N=200
radius=1
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:
    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=0.5, fill=False) # an unfilled circle
    elif l1==-l2:
        G+=circle(x, radius, edgecolor='black', facecolor='black', thickness=0.5, fill=True) # a filled circle

G.show(aspect_ratio=1)

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