An odd prime is x squared plus y squared precisely when it is one modulo four. The example is deliberately concrete; it is a test of the statement, not a substitute for it.
The data
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.
I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.
Derivation
A worked instance is useful here because it exposes every index that the compressed statement hides.
Invariant content
The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.
Scope
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.
The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.