The three-valued Legendre symbol from squares

Goal and scope

Modulo 7 the symbol on 0,1,2,3,4,5,6 is respectively 0,1,1,-1,1,-1,-1.

Complete square-table classification at odd prime 7, without Euler criterion or Jacobi extension.

Givens and symbols

• calc

• G1 — p=7; L(a)=(a/7), with a first normalized to 0 through 6.

• L(a) — Legendre value (a/7)
• p — fixed odd prime 7

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• calc · prerequisite — Assume exact integer arithmetic, division with remainder, and definitions of divisibility and gcd; every congruence has an explicit modulus. Assume the complete trial-division primality criterion.
• mko_definition · cited — Normalize a modulo odd prime p first. If p divides a, (a/p)=0; otherwise the value is 1 for a nonzero square class and−1 for another nonzero class. Classification uses a complete nonzero square table or root witness, not a prematurely assumed Euler criterion. Modulus2 and composite moduli are outside this node domain, and a few multiplicativity samples do not prove a general result. · mko-legendre-symbol

Main derivation

p=7

7\text{ odd prime};\quad L(a)=L(a\bmod7)

x=1,2,3,4,5,6

\begin{array}{c|rrrrrr}x&1&2&3&4&5&6\\\hline x^2\bmod7&1&4&2&2&4&1\end{array}

a\equiv0\pmod7

L(a)=0

a\bmod7\in\{1,2,4\}

L(1)=L(2)=L(4)=1

a\bmod7\in\{3,5,6\}

L(3)=L(5)=L(6)=-1

a=0,1,2,3,4,5,6

L(a)=0,1,1,-1,1,-1,-1\text{ respectively}

-4\equiv3,\quad9\equiv2,\quad14\equiv0\pmod7

L(-4)=-1,\quad L(9)=1,\quad L(14)=0

L(2)=1,\quad L(3)=-1,\quad2\cdot3=6

L(6)=-1=L(2)L(3)\text{ in this instance}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S06, S07 — The classification used no half-power congruence, avoiding a circular Euler-based definition.
• S07 · premise_audit · G1, S06 — 9's integer-square appearance is not the classification method; residue 2 has root 3.
• S06 · premise_audit · G1, S03, S04, S05 — The zero,+1,-1 branches are disjoint and classify each class once.
• S05 · premise_audit · G1, S02, S03, S04 — Testing only x=1,2 would not prove absence.
• S04 · premise_audit · G1, S02, S03 — Roots are witnesses; the complete table also certifies absence for other classes.
• S03 · premise_audit · G1, S01 — a=14 is the zero class and has value 0.
• S02 · premise_audit · G1, S01 — The complete table supports both root witnesses and nonsquare absence.
• S01 · premise_audit · G1 — The denominator in the symbol is not division; a may be negative while p is an odd prime.

Error controls and negative controls

Zero first

After normalization test p dividing a before nonzero square/nonsquare classification.

No circularity

The symbol comes from complete squares, not its later Euler criterion.

• (14/7)=1 because zero is a square. → reject — The zero branch specifically returns 0.
• The same table directly defines Legendre (a/8). → reject — 8 is not an odd prime and is outside this symbol's domain.
• One multiplication example proves general multiplicativity. → reject — It checks only the pair 2,3 and not the universal statement.

Result and limitations

Modulo 7 the symbol on 0,1,2,3,4,5,6 is respectively 0,1,1,-1,1,-1,-1.

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: bf6613da5aa41f52a17e16f75b5886f4fb614337c244c624146c4ed59148ad74
• original source SHA-256: 90e7d3c68f6ec4970520bf9ddc0f5ddc1a6905fc6b9958fe26e324b623ca2856
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)