Lagmental Vicfred

Quadratic Reciprocity Turns One Odd Prime around Another by Vicfred

tags: mathematics, quadratic residues, quadratic reciprocity, sums of squares

For distinct odd primes, the two Legendre symbols differ only when both primes are three modulo four. This is a compact note, but the quantifiers and hypotheses stay on the page.

Objects and notation

For an odd prime \(p\), the Legendre symbol \(\left(\frac ap\right)\) records whether \(a\) is a nonzero square modulo \(p\). Reciprocity exchanges numerator and denominator up to a sign.

$$ p,q\ \text{distinct odd primes} $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ \left(\frac pq\right)\left(\frac qp\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}} $$

Push the symbols

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ \left(\frac{19}{43}\right)=-\left(\frac{43}{19}\right)=-\left(\frac5{19}\right)=-1 $$

Structural reading

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad p,q\ \text{distinct odd primes},\\[5pt] \mathsf{C}\;&:\quad \left(\frac pq\right)\left(\frac qp\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}. \end{aligned} $$

A hypothesis worth keeping

The symbol is defined modulo an odd prime and is not ordinary division. Composite odd denominators require the Jacobi symbol, which can equal one without certifying a square.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \left(\frac pq\right)\left(\frac qp\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}} \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Thu 06 February 2014. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.