第 23305 題：二次剩餘

原題與已知條件

odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={"mode":"even_prime","modulus":2,"a":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={"mode":"even_prime","modulus":2,"a":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-quadratic-residue

逐步推導

source operation=quadratic-domain-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[mode, modulus, normalized_class, declared_model_qualified, boundary]

Quadratic ORIGINAL boundary class Euclidean quotient times modulus := mul(2, 2)

Quadratic ORIGINAL boundary class Euclidean quotient times modulus = 4

Quadratic ORIGINAL boundary class ORIGINAL signed value minus multiple := sub(5, 4)

Quadratic ORIGINAL boundary class ORIGINAL signed value minus multiple = 1

Quadratic ORIGINAL boundary class:ORIGINAL signed value=5;positive modulus=2;actual quotient=2

canonical residue=1;value=quotient*modulus+residue and0<=residue<modulus

ORIGINAL mode=even_prime,modulus=2,a=5;normalized class=1

declared_model_qualified=否;zero,p2 and composite conventions handled separately;NOT a claim this class has no square root under another convention

逐步反向核對

• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL boundary class ORIGINAL signed value minus multiple，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL boundary class Euclidean quotient times modulus，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(even_prime, 2, 1, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。