第 23361 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-paired-roots；資料={"p":37,"a":10}。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-paired-roots；資料={"p":37,"a":10}。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-paired-roots;original snapshot and mathematics retained

explicit ordered source output fields=[p, normalized_class, roots, opposite_first_root, opposite_present, distinct_roots, certificate]

Quadratic ORIGINAL modulus:ORIGINAL integer=37;complete possible trial divisors=[2, 3, 4, 5, 6]

ALL [divisor,remainder] rows=[[2, 1], [3, 1], [4, 1], [5, 2], [6, 1]];prime=是;a composite>1 has a factor at most its square root

Quadratic ORIGINAL nonzero root1 square := mul(1, 1)

Quadratic ORIGINAL nonzero root1 square = 1

ORIGINAL nonzero root x=1 in1..36;actual square=1

Euclidean square=0*37+1;nonzero square class=1;complete root-index order retained

Quadratic ORIGINAL nonzero root2 square := mul(2, 2)

Quadratic ORIGINAL nonzero root2 square = 4

ORIGINAL nonzero root x=2 in1..36;actual square=4

Euclidean square=0*37+4;nonzero square class=4;complete root-index order retained

Quadratic ORIGINAL nonzero root3 square := mul(3, 3)

Quadratic ORIGINAL nonzero root3 square = 9

ORIGINAL nonzero root x=3 in1..36;actual square=9

Euclidean square=0*37+9;nonzero square class=9;complete root-index order retained

Quadratic ORIGINAL nonzero root4 square := mul(4, 4)

Quadratic ORIGINAL nonzero root4 square = 16

ORIGINAL nonzero root x=4 in1..36;actual square=16

Euclidean square=0*37+16;nonzero square class=16;complete root-index order retained

Quadratic ORIGINAL nonzero root5 square := mul(5, 5)

Quadratic ORIGINAL nonzero root5 square = 25

ORIGINAL nonzero root x=5 in1..36;actual square=25

Euclidean square=0*37+25;nonzero square class=25;complete root-index order retained

Quadratic ORIGINAL nonzero root6 square := mul(6, 6)

Quadratic ORIGINAL nonzero root6 square = 36

ORIGINAL nonzero root x=6 in1..36;actual square=36

Euclidean square=0*37+36;nonzero square class=36;complete root-index order retained

Quadratic ORIGINAL nonzero root7 square := mul(7, 7)

Quadratic ORIGINAL nonzero root7 square = 49

ORIGINAL nonzero root x=7 in1..36;actual square=49

Euclidean square=1*37+12;nonzero square class=12;complete root-index order retained

Quadratic ORIGINAL nonzero root8 square := mul(8, 8)

Quadratic ORIGINAL nonzero root8 square = 64

ORIGINAL nonzero root x=8 in1..36;actual square=64

Euclidean square=1*37+27;nonzero square class=27;complete root-index order retained

Quadratic ORIGINAL nonzero root9 square := mul(9, 9)

Quadratic ORIGINAL nonzero root9 square = 81

ORIGINAL nonzero root x=9 in1..36;actual square=81

Euclidean square=2*37+7;nonzero square class=7;complete root-index order retained

Quadratic ORIGINAL nonzero root10 square := mul(10, 10)

Quadratic ORIGINAL nonzero root10 square = 100

ORIGINAL nonzero root x=10 in1..36;actual square=100

Euclidean square=2*37+26;nonzero square class=26;complete root-index order retained

Quadratic ORIGINAL nonzero root11 square := mul(11, 11)

Quadratic ORIGINAL nonzero root11 square = 121

ORIGINAL nonzero root x=11 in1..36;actual square=121

Euclidean square=3*37+10;nonzero square class=10;complete root-index order retained

Quadratic ORIGINAL nonzero root12 square := mul(12, 12)

Quadratic ORIGINAL nonzero root12 square = 144

ORIGINAL nonzero root x=12 in1..36;actual square=144

Euclidean square=3*37+33;nonzero square class=33;complete root-index order retained

Quadratic ORIGINAL nonzero root13 square := mul(13, 13)

Quadratic ORIGINAL nonzero root13 square = 169

ORIGINAL nonzero root x=13 in1..36;actual square=169

Euclidean square=4*37+21;nonzero square class=21;complete root-index order retained

Quadratic ORIGINAL nonzero root14 square := mul(14, 14)

Quadratic ORIGINAL nonzero root14 square = 196

ORIGINAL nonzero root x=14 in1..36;actual square=196

Euclidean square=5*37+11;nonzero square class=11;complete root-index order retained

Quadratic ORIGINAL nonzero root15 square := mul(15, 15)

Quadratic ORIGINAL nonzero root15 square = 225

ORIGINAL nonzero root x=15 in1..36;actual square=225

Euclidean square=6*37+3;nonzero square class=3;complete root-index order retained

Quadratic ORIGINAL nonzero root16 square := mul(16, 16)

Quadratic ORIGINAL nonzero root16 square = 256

ORIGINAL nonzero root x=16 in1..36;actual square=256

Euclidean square=6*37+34;nonzero square class=34;complete root-index order retained

Quadratic ORIGINAL nonzero root17 square := mul(17, 17)

Quadratic ORIGINAL nonzero root17 square = 289

ORIGINAL nonzero root x=17 in1..36;actual square=289

Euclidean square=7*37+30;nonzero square class=30;complete root-index order retained

Quadratic ORIGINAL nonzero root18 square := mul(18, 18)

Quadratic ORIGINAL nonzero root18 square = 324

ORIGINAL nonzero root x=18 in1..36;actual square=324

Euclidean square=8*37+28;nonzero square class=28;complete root-index order retained

Quadratic ORIGINAL nonzero root19 square := mul(19, 19)

Quadratic ORIGINAL nonzero root19 square = 361

ORIGINAL nonzero root x=19 in1..36;actual square=361

Euclidean square=9*37+28;nonzero square class=28;complete root-index order retained

Quadratic ORIGINAL nonzero root20 square := mul(20, 20)

Quadratic ORIGINAL nonzero root20 square = 400

ORIGINAL nonzero root x=20 in1..36;actual square=400

Euclidean square=10*37+30;nonzero square class=30;complete root-index order retained

Quadratic ORIGINAL nonzero root21 square := mul(21, 21)

Quadratic ORIGINAL nonzero root21 square = 441

ORIGINAL nonzero root x=21 in1..36;actual square=441

Euclidean square=11*37+34;nonzero square class=34;complete root-index order retained

Quadratic ORIGINAL nonzero root22 square := mul(22, 22)

Quadratic ORIGINAL nonzero root22 square = 484

ORIGINAL nonzero root x=22 in1..36;actual square=484

Euclidean square=13*37+3;nonzero square class=3;complete root-index order retained

Quadratic ORIGINAL nonzero root23 square := mul(23, 23)

Quadratic ORIGINAL nonzero root23 square = 529

ORIGINAL nonzero root x=23 in1..36;actual square=529

Euclidean square=14*37+11;nonzero square class=11;complete root-index order retained

Quadratic ORIGINAL nonzero root24 square := mul(24, 24)

Quadratic ORIGINAL nonzero root24 square = 576

ORIGINAL nonzero root x=24 in1..36;actual square=576

Euclidean square=15*37+21;nonzero square class=21;complete root-index order retained

Quadratic ORIGINAL nonzero root25 square := mul(25, 25)

Quadratic ORIGINAL nonzero root25 square = 625

ORIGINAL nonzero root x=25 in1..36;actual square=625

Euclidean square=16*37+33;nonzero square class=33;complete root-index order retained

Quadratic ORIGINAL nonzero root26 square := mul(26, 26)

Quadratic ORIGINAL nonzero root26 square = 676

ORIGINAL nonzero root x=26 in1..36;actual square=676

Euclidean square=18*37+10;nonzero square class=10;complete root-index order retained

Quadratic ORIGINAL nonzero root27 square := mul(27, 27)

Quadratic ORIGINAL nonzero root27 square = 729

ORIGINAL nonzero root x=27 in1..36;actual square=729

Euclidean square=19*37+26;nonzero square class=26;complete root-index order retained

Quadratic ORIGINAL nonzero root28 square := mul(28, 28)

Quadratic ORIGINAL nonzero root28 square = 784

ORIGINAL nonzero root x=28 in1..36;actual square=784

Euclidean square=21*37+7;nonzero square class=7;complete root-index order retained

Quadratic ORIGINAL nonzero root29 square := mul(29, 29)

Quadratic ORIGINAL nonzero root29 square = 841

ORIGINAL nonzero root x=29 in1..36;actual square=841

Euclidean square=22*37+27;nonzero square class=27;complete root-index order retained

Quadratic ORIGINAL nonzero root30 square := mul(30, 30)

Quadratic ORIGINAL nonzero root30 square = 900

ORIGINAL nonzero root x=30 in1..36;actual square=900

Euclidean square=24*37+12;nonzero square class=12;complete root-index order retained

Quadratic ORIGINAL nonzero root31 square := mul(31, 31)

Quadratic ORIGINAL nonzero root31 square = 961

ORIGINAL nonzero root x=31 in1..36;actual square=961

Euclidean square=25*37+36;nonzero square class=36;complete root-index order retained

Quadratic ORIGINAL nonzero root32 square := mul(32, 32)

Quadratic ORIGINAL nonzero root32 square = 1024

ORIGINAL nonzero root x=32 in1..36;actual square=1024

Euclidean square=27*37+25;nonzero square class=25;complete root-index order retained

Quadratic ORIGINAL nonzero root33 square := mul(33, 33)

Quadratic ORIGINAL nonzero root33 square = 1089

ORIGINAL nonzero root x=33 in1..36;actual square=1089

Euclidean square=29*37+16;nonzero square class=16;complete root-index order retained

Quadratic ORIGINAL nonzero root34 square := mul(34, 34)

Quadratic ORIGINAL nonzero root34 square = 1156

ORIGINAL nonzero root x=34 in1..36;actual square=1156

Euclidean square=31*37+9;nonzero square class=9;complete root-index order retained

Quadratic ORIGINAL nonzero root35 square := mul(35, 35)

Quadratic ORIGINAL nonzero root35 square = 1225

ORIGINAL nonzero root x=35 in1..36;actual square=1225

Euclidean square=33*37+4;nonzero square class=4;complete root-index order retained

Quadratic ORIGINAL nonzero root36 square := mul(36, 36)

Quadratic ORIGINAL nonzero root36 square = 1296

ORIGINAL nonzero root x=36 in1..36;actual square=1296

Euclidean square=35*37+1;nonzero square class=1;complete root-index order retained

Quadratic distinct nonzero residue count := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

Quadratic distinct nonzero residue count = 18

Quadratic ORIGINAL modulus minus1 := sub(37, 1)

Quadratic ORIGINAL modulus minus1 = 36

Quadratic half of all nonzero roots := div(36, 2)

Quadratic half of all nonzero roots = 18

ORIGINAL ODD PRIME p=37;complete nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 12], [8, 27], [9, 7], [10, 26], [11, 10], [12, 33], [13, 21], [14, 11], [15, 3], [16, 34], [17, 30], [18, 28], [19, 28], [20, 30], [21, 34], [22, 3], [23, 11], [24, 21], [25, 33], [26, 10], [27, 26], [28, 7], [29, 27], [30, 12], [31, 36], [32, 25], [33, 16], [34, 9], [35, 4], [36, 1]];distinct classes=[1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26, 27, 28, 30, 33, 34, 36]

x^2=y^2 implies p divides(x-y)(x+y),so x=y or x=-y modp;odd p and nonzero x make opposite roots DISTINCT;each image has2 roots,so residue count=18=(37-1)/2=18

Quadratic ORIGINAL candidate class Euclidean quotient times modulus := mul(0, 37)

Quadratic ORIGINAL candidate class Euclidean quotient times modulus = 0

Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple := sub(10, 0)

Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple = 10

Quadratic ORIGINAL candidate class:ORIGINAL signed value=10;positive modulus=37;actual quotient=0

canonical residue=10;value=quotient*modulus+residue and0<=residue<modulus

Quadratic negative of FIRST source root Euclidean quotient times modulus := mul(-1, 37)

Quadratic negative of FIRST source root Euclidean quotient times modulus = -37

Quadratic negative of FIRST source root ORIGINAL signed value minus multiple := sub(-11, -37)

Quadratic negative of FIRST source root ORIGINAL signed value minus multiple = 26

Quadratic negative of FIRST source root:ORIGINAL signed value=-11;positive modulus=37;actual quotient=-1

canonical residue=26;value=quotient*modulus+residue and0<=residue<modulus

逐步反向核對

• S084 · 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。
• S083 · premise_and_result_audit — 以精確有理數重算 Quadratic negative of FIRST source root ORIGINAL signed value minus multiple，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 Quadratic negative of FIRST source root Euclidean quotient times modulus，不以小數近似替代。
• S081 · 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。
• S080 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple，不以小數近似替代。
• S079 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL candidate class Euclidean quotient times modulus，不以小數近似替代。
• S078 · 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。
• S077 · premise_and_result_audit — 以精確有理數重算 Quadratic half of all nonzero roots，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL modulus minus1，不以小數近似替代。
• S075 · premise_and_result_audit — 以精確有理數重算 Quadratic distinct nonzero residue count，不以小數近似替代。
• S074 · 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。
• S073 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root36 square，不以小數近似替代。
• S072 · 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。
• S071 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root35 square，不以小數近似替代。
• S070 · 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。
• S069 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root34 square，不以小數近似替代。
• S068 · 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。
• S067 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root33 square，不以小數近似替代。
• S066 · 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。
• S065 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root32 square，不以小數近似替代。
• S064 · 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。
• S063 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root31 square，不以小數近似替代。
• S062 · 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。
• S061 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root30 square，不以小數近似替代。
• S060 · 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。
• S059 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root29 square，不以小數近似替代。
• S058 · 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。
• S057 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root28 square，不以小數近似替代。
• S056 · 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。
• S055 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root27 square，不以小數近似替代。
• S054 · 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。
• S053 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root26 square，不以小數近似替代。
• S052 · 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。
• S051 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root25 square，不以小數近似替代。
• S050 · 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。
• S049 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root24 square，不以小數近似替代。
• S048 · 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。
• S047 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root23 square，不以小數近似替代。
• S046 · 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。
• S045 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root22 square，不以小數近似替代。
• S044 · 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。
• S043 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root21 square，不以小數近似替代。
• S042 · 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。
• S041 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root20 square，不以小數近似替代。
• S040 · 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。
• S039 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root19 square，不以小數近似替代。
• S038 · 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。
• S037 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root18 square，不以小數近似替代。
• S036 · 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。
• S035 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root17 square，不以小數近似替代。
• S034 · 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。
• S033 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root16 square，不以小數近似替代。
• S032 · 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。
• S031 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root15 square，不以小數近似替代。
• S030 · 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。
• S029 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root14 square，不以小數近似替代。
• S028 · 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。
• S027 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root13 square，不以小數近似替代。
• S026 · 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。
• S025 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root12 square，不以小數近似替代。
• S024 · 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。
• S023 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root11 square，不以小數近似替代。
• S022 · 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。
• S021 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root10 square，不以小數近似替代。
• S020 · 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。
• S019 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root9 square，不以小數近似替代。
• S018 · 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。
• S017 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root8 square，不以小數近似替代。
• S016 · 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。
• S015 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root7 square，不以小數近似替代。
• S014 · 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。
• S013 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root6 square，不以小數近似替代。
• S012 · 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。
• S011 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root5 square，不以小數近似替代。
• S010 · 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。
• S009 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root4 square，不以小數近似替代。
• S008 · 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。
• S007 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root3 square，不以小數近似替代。
• S006 · 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。
• S005 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root2 square，不以小數近似替代。
• 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 nonzero root1 square，不以小數近似替代。
• S002 · 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。
• 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

(37, 10, [11, 26], 26, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)

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。