The complete nonzero square table modulo 7

Goal and scope

Modulo 7, the nonzero quadratic residues are {1,2,4}; the class of -5 equals the class of 2 and has root classes 3 and 4, while 3 has no square root modulo 7.

Bounded odd prime 7 with the entire nonzero square table; no Euler criterion or Legendre symbol.

Givens and symbols

• calc

• G1 — p=7, candidates a=-5 and 3; nonzero root representatives are exactly 1,2,3,4,5,6.

• Q_7,N_7 — nonzero square and nonsquare classes
• p — qualified odd prime

Rules and sources

• calc · prerequisite — Assume exact integer arithmetic, division with remainder, and definitions of divisibility and gcd; every congruence has an explicit modulus. Assume complete trial division to the square root decides primality.
• mko_definition · cited — For odd prime p normalize a, then enumerate every nonzero square to decide nonzero quadratic-residue membership. Separate zero, p=2 and composite moduli; no Euler criterion or Legendre symbol is used here. · mko-quadratic-residue

Main derivation

p=7

7>2,\quad2\nmid7,\quad\sqrt7<3\Rightarrow7\text{ odd prime}

a=-5

a\equiv2\pmod7,\quad0<2<7

x=1,2,3

1^2\equiv1,\quad2^2\equiv4,\quad3^2\equiv2\pmod7

x=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}

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

Q_7=\{1,2,4\},\quad N_7=\{3,5,6\}

a\equiv2\pmod7

\operatorname{roots}(2)=\{3,4\},\quad\operatorname{roots}(1)=\{1,6\},\quad\operatorname{roots}(4)=\{2,5\}

a=3,\quad3\notin\{1,2,4\}

x^2\not\equiv3\pmod7\text{ for every }x\in\mathbb Z

p=2\text{ or }m=8

\operatorname{roots}_{2}(1)=\{1\};\quad\operatorname{roots}_{8}(1)=\{1,3,5,7\}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S01, S06 — All four odd classes modulo 8 square to 1; this introduces no Legendre symbol or Euler criterion.
• S07 · premise_audit · G1, S04, S05 — Nonexistence follows from the complete residue table, not a bounded integer search.
• S06 · premise_audit · G1, S02, S05 — The paired roots are distinct: nonzero x modulo odd prime 7 cannot satisfy 2x=0.
• S05 · premise_audit · G1, S04 — Neither Q7 nor N7 contains zero; they partition only the nonzero carrier.
• S04 · premise_audit · G1, S03 — Zero is omitted because this is the nonzero-square classification; zero is a separate case.
• S03 · premise_audit · G1, S01 — 3^2=9 has residue 2, not integer 9.
• S02 · premise_audit · G1, S01 — -5 is valid; normalization changes the representative, not the residue class.
• S01 · premise_audit · G1 — The two-root conclusion cannot be transferred to composite moduli.

Error controls and negative controls

Zero class

Normalize first and handle zero separately from nonzero residue/nonresidue classes.

Complete root table

List every x from 1 to 6 before asserting absence.

• Zero is a nonzero quadratic residue here. → reject — Zero is separate and outside nonzero Q7.
• 3 is a square mod 7 because 3^2=9. → reject — 9 is 2, not 3, modulo 7, and the full table omits 3.
• 1 modulo 8 also has exactly two roots. → reject — 1,3,5,7 are four distinct roots.

Result and limitations

Modulo 7, the nonzero quadratic residues are {1,2,4}; the class of -5 equals the class of 2 and has root classes 3 and 4, while 3 has no square root modulo 7.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 384acd699a63eb402427b8835c9873f84548e160871fa06b63517b0138ea17d3
• original source SHA-256: d000894426cdf201a44c3ff9da8d4a815803ac12c4d484007b9cbefb594772c4
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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