Factorial Congruence, Complete Inverse Pairing and the Modulus-Two Exception

(2, Yes, [[1, 1]], 1, 1, Yes, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (2, Yes, [[1, 1]], 1, 1, Yes, Yes, Wilson_equivalence_instance).

(3, Yes, [[1, 1], [2, 2]], 2, 2, Yes, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (3, Yes, [[1, 1], [2, 2]], 2, 2, Yes, Yes, Wilson_equivalence_instance).

(4, No, [[1, 1], [2, 2], [3, 2]], 2, 3, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (4, No, [[1, 1], [2, 2], [3, 2]], 2, 3, No, Yes, Wilson_equivalence_instance).

(5, Yes, [[1, 1], [2, 2], [3, 1], [4, 4]], 4, 4, Yes, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (5, Yes, [[1, 1], [2, 2], [3, 1], [4, 4]], 4, 4, Yes, Yes, Wilson_equivalence_instance).

(6, No, [[1, 1], [2, 2], [3, 0], [4, 0], [5, 0]], 0, 5, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (6, No, [[1, 1], [2, 2], [3, 0], [4, 0], [5, 0]], 0, 5, No, Yes, Wilson_equivalence_instance).

(7, Yes, [[1, 1], [2, 2], [3, 6], [4, 3], [5, 1], [6, 6]], 6, 6, Yes, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (7, Yes, [[1, 1], [2, 2], [3, 6], [4, 3], [5, 1], [6, 6]], 6, 6, Yes, Yes, Wilson_equivalence_instance).

(8, No, [[1, 1], [2, 2], [3, 6], [4, 0], [5, 0], [6, 0], [7, 0]], 0, 7, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (8, No, [[1, 1], [2, 2], [3, 6], [4, 0], [5, 0], [6, 0], [7, 0]], 0, 7, No, Yes, Wilson_equivalence_instance).

(9, No, [[1, 1], [2, 2], [3, 6], [4, 6], [5, 3], [6, 0], [7, 0], [8, 0]], 0, 8, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (9, No, [[1, 1], [2, 2], [3, 6], [4, 6], [5, 3], [6, 0], [7, 0], [8, 0]], 0, 8, No, Yes, Wilson_equivalence_instance).

(10, No, [[1, 1], [2, 2], [3, 6], [4, 4], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0]], 0, 9, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (10, No, [[1, 1], [2, 2], [3, 6], [4, 4], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0]], 0, 9, No, Yes, Wilson_equivalence_instance).

(11, Yes, [[1, 1], [2, 2], [3, 6], [4, 2], [5, 10], [6, 5], [7, 2], [8, 5], [9, 1], [10, 10]], 10, 10, Yes, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (11, Yes, [[1, 1], [2, 2], [3, 6], [4, 2], [5, 10], [6, 5], [7, 2], [8, 5], [9, 1], [10, 10]], 10, 10, Yes, Yes, Wilson_equivalence_instance).

(12, No, [[1, 1], [2, 2], [3, 6], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0]], 0, 11, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (12, No, [[1, 1], [2, 2], [3, 6], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0]], 0, 11, No, Yes, Wilson_equivalence_instance).

(13, Yes, [[1, 1], [2, 2], [3, 6], [4, 11], [5, 3], [6, 5], [7, 9], [8, 7], [9, 11], [10, 6], [11, 1], [12, 12]], 12, 12, Yes, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (13, Yes, [[1, 1], [2, 2], [3, 6], [4, 11], [5, 3], [6, 5], [7, 9], [8, 7], [9, 11], [10, 6], [11, 1], [12, 12]], 12, 12, Yes, Yes, Wilson_equivalence_instance).

(14, No, [[1, 1], [2, 2], [3, 6], [4, 10], [5, 8], [6, 6], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0]], 0, 13, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (14, No, [[1, 1], [2, 2], [3, 6], [4, 10], [5, 8], [6, 6], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0]], 0, 13, No, Yes, Wilson_equivalence_instance).

(15, No, [[1, 1], [2, 2], [3, 6], [4, 9], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0]], 0, 14, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (15, No, [[1, 1], [2, 2], [3, 6], [4, 9], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0]], 0, 14, No, Yes, Wilson_equivalence_instance).

(16, No, [[1, 1], [2, 2], [3, 6], [4, 8], [5, 8], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0]], 0, 15, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (16, No, [[1, 1], [2, 2], [3, 6], [4, 8], [5, 8], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0]], 0, 15, No, Yes, Wilson_equivalence_instance).

(17, Yes, [[1, 1], [2, 2], [3, 6], [4, 7], [5, 1], [6, 6], [7, 8], [8, 13], [9, 15], [10, 14], [11, 1], [12, 12], [13, 3], [14, 8], [15, 1], [16, 16]], 16, 16, Yes, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (17, Yes, [[1, 1], [2, 2], [3, 6], [4, 7], [5, 1], [6, 6], [7, 8], [8, 13], [9, 15], [10, 14], [11, 1], [12, 12], [13, 3], [14, 8], [15, 1], [16, 16]], 16, 16, Yes, Yes, Wilson_equivalence_instance).

(18, No, [[1, 1], [2, 2], [3, 6], [4, 6], [5, 12], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0]], 0, 17, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (18, No, [[1, 1], [2, 2], [3, 6], [4, 6], [5, 12], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0]], 0, 17, No, Yes, Wilson_equivalence_instance).

(19, Yes, [[1, 1], [2, 2], [3, 6], [4, 5], [5, 6], [6, 17], [7, 5], [8, 2], [9, 18], [10, 9], [11, 4], [12, 10], [13, 16], [14, 15], [15, 16], [16, 9], [17, 1], [18, 18]], 18, 18, Yes, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (19, Yes, [[1, 1], [2, 2], [3, 6], [4, 5], [5, 6], [6, 17], [7, 5], [8, 2], [9, 18], [10, 9], [11, 4], [12, 10], [13, 16], [14, 15], [15, 16], [16, 9], [17, 1], [18, 18]], 18, 18, Yes, Yes, Wilson_equivalence_instance).

(20, No, [[1, 1], [2, 2], [3, 6], [4, 4], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0]], 0, 19, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (20, No, [[1, 1], [2, 2], [3, 6], [4, 4], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0]], 0, 19, No, Yes, Wilson_equivalence_instance).

(21, No, [[1, 1], [2, 2], [3, 6], [4, 3], [5, 15], [6, 6], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0]], 0, 20, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (21, No, [[1, 1], [2, 2], [3, 6], [4, 3], [5, 15], [6, 6], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0]], 0, 20, No, Yes, Wilson_equivalence_instance).

(22, No, [[1, 1], [2, 2], [3, 6], [4, 2], [5, 10], [6, 16], [7, 2], [8, 16], [9, 12], [10, 10], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0]], 0, 21, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (22, No, [[1, 1], [2, 2], [3, 6], [4, 2], [5, 10], [6, 16], [7, 2], [8, 16], [9, 12], [10, 10], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0]], 0, 21, No, Yes, Wilson_equivalence_instance).

(23, Yes, [[1, 1], [2, 2], [3, 6], [4, 1], [5, 5], [6, 7], [7, 3], [8, 1], [9, 9], [10, 21], [11, 1], [12, 12], [13, 18], [14, 22], [15, 8], [16, 13], [17, 14], [18, 22], [19, 4], [20, 11], [21, 1], [22, 22]], 22, 22, Yes, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (23, Yes, [[1, 1], [2, 2], [3, 6], [4, 1], [5, 5], [6, 7], [7, 3], [8, 1], [9, 9], [10, 21], [11, 1], [12, 12], [13, 18], [14, 22], [15, 8], [16, 13], [17, 14], [18, 22], [19, 4], [20, 11], [21, 1], [22, 22]], 22, 22, Yes, Yes, Wilson_equivalence_instance).

(24, No, [[1, 1], [2, 2], [3, 6], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0]], 0, 23, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (24, No, [[1, 1], [2, 2], [3, 6], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0]], 0, 23, No, Yes, Wilson_equivalence_instance).

(25, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 20], [6, 20], [7, 15], [8, 20], [9, 5], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0]], 0, 24, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (25, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 20], [6, 20], [7, 15], [8, 20], [9, 5], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0]], 0, 24, No, Yes, Wilson_equivalence_instance).

(26, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 16], [6, 18], [7, 22], [8, 20], [9, 24], [10, 6], [11, 14], [12, 12], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0]], 0, 25, No, Yes, Wilson_equivalence_instance)

Compute each running factorial residue through (n−1)! and separately classify primality. Compare the final residue with n−1. The equivalence flag can be Yes for a composite n when both primality and congruence are No. A single instance does not prove the universal equivalence. The exact result, retaining the requested order, is (26, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 16], [6, 18], [7, 22], [8, 20], [9, 24], [10, 6], [11, 14], [12, 12], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0]], 0, 25, No, Yes, Wilson_equivalence_instance).

(2, [[1, 1]], [1], [], single_self_inverse_residue_1_equals_minus_1, 1, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (2, [[1, 1]], [1], [], single_self_inverse_residue_1_equals_minus_1, 1, Yes).

(3, [[1, 1], [2, 2]], [1, 2], [], self_inverse_residues_1_and_p_minus_1, 2, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (3, [[1, 1], [2, 2]], [1, 2], [], self_inverse_residues_1_and_p_minus_1, 2, Yes).

(5, [[1, 1], [2, 3], [3, 2], [4, 4]], [1, 4], [[2, 3, 1]], self_inverse_residues_1_and_p_minus_1, 4, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (5, [[1, 1], [2, 3], [3, 2], [4, 4]], [1, 4], [[2, 3, 1]], self_inverse_residues_1_and_p_minus_1, 4, Yes).

(7, [[1, 1], [2, 4], [3, 5], [4, 2], [5, 3], [6, 6]], [1, 6], [[2, 4, 1], [3, 5, 1]], self_inverse_residues_1_and_p_minus_1, 6, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (7, [[1, 1], [2, 4], [3, 5], [4, 2], [5, 3], [6, 6]], [1, 6], [[2, 4, 1], [3, 5, 1]], self_inverse_residues_1_and_p_minus_1, 6, Yes).

(11, [[1, 1], [2, 6], [3, 4], [4, 3], [5, 9], [6, 2], [7, 8], [8, 7], [9, 5], [10, 10]], [1, 10], [[2, 6, 1], [3, 4, 1], [5, 9, 1], [7, 8, 1]], self_inverse_residues_1_and_p_minus_1, 10, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (11, [[1, 1], [2, 6], [3, 4], [4, 3], [5, 9], [6, 2], [7, 8], [8, 7], [9, 5], [10, 10]], [1, 10], [[2, 6, 1], [3, 4, 1], [5, 9, 1], [7, 8, 1]], self_inverse_residues_1_and_p_minus_1, 10, Yes).

(13, [[1, 1], [2, 7], [3, 9], [4, 10], [5, 8], [6, 11], [7, 2], [8, 5], [9, 3], [10, 4], [11, 6], [12, 12]], [1, 12], [[2, 7, 1], [3, 9, 1], [4, 10, 1], [5, 8, 1], [6, 11, 1]], self_inverse_residues_1_and_p_minus_1, 12, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (13, [[1, 1], [2, 7], [3, 9], [4, 10], [5, 8], [6, 11], [7, 2], [8, 5], [9, 3], [10, 4], [11, 6], [12, 12]], [1, 12], [[2, 7, 1], [3, 9, 1], [4, 10, 1], [5, 8, 1], [6, 11, 1]], self_inverse_residues_1_and_p_minus_1, 12, Yes).

(17, [[1, 1], [2, 9], [3, 6], [4, 13], [5, 7], [6, 3], [7, 5], [8, 15], [9, 2], [10, 12], [11, 14], [12, 10], [13, 4], [14, 11], [15, 8], [16, 16]], [1, 16], [[2, 9, 1], [3, 6, 1], [4, 13, 1], [5, 7, 1], [8, 15, 1], [10, 12, 1], [11, 14, 1]], self_inverse_residues_1_and_p_minus_1, 16, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (17, [[1, 1], [2, 9], [3, 6], [4, 13], [5, 7], [6, 3], [7, 5], [8, 15], [9, 2], [10, 12], [11, 14], [12, 10], [13, 4], [14, 11], [15, 8], [16, 16]], [1, 16], [[2, 9, 1], [3, 6, 1], [4, 13, 1], [5, 7, 1], [8, 15, 1], [10, 12, 1], [11, 14, 1]], self_inverse_residues_1_and_p_minus_1, 16, Yes).

(19, [[1, 1], [2, 10], [3, 13], [4, 5], [5, 4], [6, 16], [7, 11], [8, 12], [9, 17], [10, 2], [11, 7], [12, 8], [13, 3], [14, 15], [15, 14], [16, 6], [17, 9], [18, 18]], [1, 18], [[2, 10, 1], [3, 13, 1], [4, 5, 1], [6, 16, 1], [7, 11, 1], [8, 12, 1], [9, 17, 1], [14, 15, 1]], self_inverse_residues_1_and_p_minus_1, 18, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (19, [[1, 1], [2, 10], [3, 13], [4, 5], [5, 4], [6, 16], [7, 11], [8, 12], [9, 17], [10, 2], [11, 7], [12, 8], [13, 3], [14, 15], [15, 14], [16, 6], [17, 9], [18, 18]], [1, 18], [[2, 10, 1], [3, 13, 1], [4, 5, 1], [6, 16, 1], [7, 11, 1], [8, 12, 1], [9, 17, 1], [14, 15, 1]], self_inverse_residues_1_and_p_minus_1, 18, Yes).

(23, [[1, 1], [2, 12], [3, 8], [4, 6], [5, 14], [6, 4], [7, 10], [8, 3], [9, 18], [10, 7], [11, 21], [12, 2], [13, 16], [14, 5], [15, 20], [16, 13], [17, 19], [18, 9], [19, 17], [20, 15], [21, 11], [22, 22]], [1, 22], [[2, 12, 1], [3, 8, 1], [4, 6, 1], [5, 14, 1], [7, 10, 1], [9, 18, 1], [11, 21, 1], [13, 16, 1], [15, 20, 1], [17, 19, 1]], self_inverse_residues_1_and_p_minus_1, 22, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (23, [[1, 1], [2, 12], [3, 8], [4, 6], [5, 14], [6, 4], [7, 10], [8, 3], [9, 18], [10, 7], [11, 21], [12, 2], [13, 16], [14, 5], [15, 20], [16, 13], [17, 19], [18, 9], [19, 17], [20, 15], [21, 11], [22, 22]], [1, 22], [[2, 12, 1], [3, 8, 1], [4, 6, 1], [5, 14, 1], [7, 10, 1], [9, 18, 1], [11, 21, 1], [13, 16, 1], [15, 20, 1], [17, 19, 1]], self_inverse_residues_1_and_p_minus_1, 22, Yes).

(29, [[1, 1], [2, 15], [3, 10], [4, 22], [5, 6], [6, 5], [7, 25], [8, 11], [9, 13], [10, 3], [11, 8], [12, 17], [13, 9], [14, 27], [15, 2], [16, 20], [17, 12], [18, 21], [19, 26], [20, 16], [21, 18], [22, 4], [23, 24], [24, 23], [25, 7], [26, 19], [27, 14], [28, 28]], [1, 28], [[2, 15, 1], [3, 10, 1], [4, 22, 1], [5, 6, 1], [7, 25, 1], [8, 11, 1], [9, 13, 1], [12, 17, 1], [14, 27, 1], [16, 20, 1], [18, 21, 1], [19, 26, 1], [23, 24, 1]], self_inverse_residues_1_and_p_minus_1, 28, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (29, [[1, 1], [2, 15], [3, 10], [4, 22], [5, 6], [6, 5], [7, 25], [8, 11], [9, 13], [10, 3], [11, 8], [12, 17], [13, 9], [14, 27], [15, 2], [16, 20], [17, 12], [18, 21], [19, 26], [20, 16], [21, 18], [22, 4], [23, 24], [24, 23], [25, 7], [26, 19], [27, 14], [28, 28]], [1, 28], [[2, 15, 1], [3, 10, 1], [4, 22, 1], [5, 6, 1], [7, 25, 1], [8, 11, 1], [9, 13, 1], [12, 17, 1], [14, 27, 1], [16, 20, 1], [18, 21, 1], [19, 26, 1], [23, 24, 1]], self_inverse_residues_1_and_p_minus_1, 28, Yes).

(31, [[1, 1], [2, 16], [3, 21], [4, 8], [5, 25], [6, 26], [7, 9], [8, 4], [9, 7], [10, 28], [11, 17], [12, 13], [13, 12], [14, 20], [15, 29], [16, 2], [17, 11], [18, 19], [19, 18], [20, 14], [21, 3], [22, 24], [23, 27], [24, 22], [25, 5], [26, 6], [27, 23], [28, 10], [29, 15], [30, 30]], [1, 30], [[2, 16, 1], [3, 21, 1], [4, 8, 1], [5, 25, 1], [6, 26, 1], [7, 9, 1], [10, 28, 1], [11, 17, 1], [12, 13, 1], [14, 20, 1], [15, 29, 1], [18, 19, 1], [22, 24, 1], [23, 27, 1]], self_inverse_residues_1_and_p_minus_1, 30, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (31, [[1, 1], [2, 16], [3, 21], [4, 8], [5, 25], [6, 26], [7, 9], [8, 4], [9, 7], [10, 28], [11, 17], [12, 13], [13, 12], [14, 20], [15, 29], [16, 2], [17, 11], [18, 19], [19, 18], [20, 14], [21, 3], [22, 24], [23, 27], [24, 22], [25, 5], [26, 6], [27, 23], [28, 10], [29, 15], [30, 30]], [1, 30], [[2, 16, 1], [3, 21, 1], [4, 8, 1], [5, 25, 1], [6, 26, 1], [7, 9, 1], [10, 28, 1], [11, 17, 1], [12, 13, 1], [14, 20, 1], [15, 29, 1], [18, 19, 1], [22, 24, 1], [23, 27, 1]], self_inverse_residues_1_and_p_minus_1, 30, Yes).

(37, [[1, 1], [2, 19], [3, 25], [4, 28], [5, 15], [6, 31], [7, 16], [8, 14], [9, 33], [10, 26], [11, 27], [12, 34], [13, 20], [14, 8], [15, 5], [16, 7], [17, 24], [18, 35], [19, 2], [20, 13], [21, 30], [22, 32], [23, 29], [24, 17], [25, 3], [26, 10], [27, 11], [28, 4], [29, 23], [30, 21], [31, 6], [32, 22], [33, 9], [34, 12], [35, 18], [36, 36]], [1, 36], [[2, 19, 1], [3, 25, 1], [4, 28, 1], [5, 15, 1], [6, 31, 1], [7, 16, 1], [8, 14, 1], [9, 33, 1], [10, 26, 1], [11, 27, 1], [12, 34, 1], [13, 20, 1], [17, 24, 1], [18, 35, 1], [21, 30, 1], [22, 32, 1], [23, 29, 1]], self_inverse_residues_1_and_p_minus_1, 36, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (37, [[1, 1], [2, 19], [3, 25], [4, 28], [5, 15], [6, 31], [7, 16], [8, 14], [9, 33], [10, 26], [11, 27], [12, 34], [13, 20], [14, 8], [15, 5], [16, 7], [17, 24], [18, 35], [19, 2], [20, 13], [21, 30], [22, 32], [23, 29], [24, 17], [25, 3], [26, 10], [27, 11], [28, 4], [29, 23], [30, 21], [31, 6], [32, 22], [33, 9], [34, 12], [35, 18], [36, 36]], [1, 36], [[2, 19, 1], [3, 25, 1], [4, 28, 1], [5, 15, 1], [6, 31, 1], [7, 16, 1], [8, 14, 1], [9, 33, 1], [10, 26, 1], [11, 27, 1], [12, 34, 1], [13, 20, 1], [17, 24, 1], [18, 35, 1], [21, 30, 1], [22, 32, 1], [23, 29, 1]], self_inverse_residues_1_and_p_minus_1, 36, Yes).

(41, [[1, 1], [2, 21], [3, 14], [4, 31], [5, 33], [6, 7], [7, 6], [8, 36], [9, 32], [10, 37], [11, 15], [12, 24], [13, 19], [14, 3], [15, 11], [16, 18], [17, 29], [18, 16], [19, 13], [20, 39], [21, 2], [22, 28], [23, 25], [24, 12], [25, 23], [26, 30], [27, 38], [28, 22], [29, 17], [30, 26], [31, 4], [32, 9], [33, 5], [34, 35], [35, 34], [36, 8], [37, 10], [38, 27], [39, 20], [40, 40]], [1, 40], [[2, 21, 1], [3, 14, 1], [4, 31, 1], [5, 33, 1], [6, 7, 1], [8, 36, 1], [9, 32, 1], [10, 37, 1], [11, 15, 1], [12, 24, 1], [13, 19, 1], [16, 18, 1], [17, 29, 1], [20, 39, 1], [22, 28, 1], [23, 25, 1], [26, 30, 1], [27, 38, 1], [34, 35, 1]], self_inverse_residues_1_and_p_minus_1, 40, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (41, [[1, 1], [2, 21], [3, 14], [4, 31], [5, 33], [6, 7], [7, 6], [8, 36], [9, 32], [10, 37], [11, 15], [12, 24], [13, 19], [14, 3], [15, 11], [16, 18], [17, 29], [18, 16], [19, 13], [20, 39], [21, 2], [22, 28], [23, 25], [24, 12], [25, 23], [26, 30], [27, 38], [28, 22], [29, 17], [30, 26], [31, 4], [32, 9], [33, 5], [34, 35], [35, 34], [36, 8], [37, 10], [38, 27], [39, 20], [40, 40]], [1, 40], [[2, 21, 1], [3, 14, 1], [4, 31, 1], [5, 33, 1], [6, 7, 1], [8, 36, 1], [9, 32, 1], [10, 37, 1], [11, 15, 1], [12, 24, 1], [13, 19, 1], [16, 18, 1], [17, 29, 1], [20, 39, 1], [22, 28, 1], [23, 25, 1], [26, 30, 1], [27, 38, 1], [34, 35, 1]], self_inverse_residues_1_and_p_minus_1, 40, Yes).

(43, [[1, 1], [2, 22], [3, 29], [4, 11], [5, 26], [6, 36], [7, 37], [8, 27], [9, 24], [10, 13], [11, 4], [12, 18], [13, 10], [14, 40], [15, 23], [16, 35], [17, 38], [18, 12], [19, 34], [20, 28], [21, 41], [22, 2], [23, 15], [24, 9], [25, 31], [26, 5], [27, 8], [28, 20], [29, 3], [30, 33], [31, 25], [32, 39], [33, 30], [34, 19], [35, 16], [36, 6], [37, 7], [38, 17], [39, 32], [40, 14], [41, 21], [42, 42]], [1, 42], [[2, 22, 1], [3, 29, 1], [4, 11, 1], [5, 26, 1], [6, 36, 1], [7, 37, 1], [8, 27, 1], [9, 24, 1], [10, 13, 1], [12, 18, 1], [14, 40, 1], [15, 23, 1], [16, 35, 1], [17, 38, 1], [19, 34, 1], [20, 28, 1], [21, 41, 1], [25, 31, 1], [30, 33, 1], [32, 39, 1]], self_inverse_residues_1_and_p_minus_1, 42, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (43, [[1, 1], [2, 22], [3, 29], [4, 11], [5, 26], [6, 36], [7, 37], [8, 27], [9, 24], [10, 13], [11, 4], [12, 18], [13, 10], [14, 40], [15, 23], [16, 35], [17, 38], [18, 12], [19, 34], [20, 28], [21, 41], [22, 2], [23, 15], [24, 9], [25, 31], [26, 5], [27, 8], [28, 20], [29, 3], [30, 33], [31, 25], [32, 39], [33, 30], [34, 19], [35, 16], [36, 6], [37, 7], [38, 17], [39, 32], [40, 14], [41, 21], [42, 42]], [1, 42], [[2, 22, 1], [3, 29, 1], [4, 11, 1], [5, 26, 1], [6, 36, 1], [7, 37, 1], [8, 27, 1], [9, 24, 1], [10, 13, 1], [12, 18, 1], [14, 40, 1], [15, 23, 1], [16, 35, 1], [17, 38, 1], [19, 34, 1], [20, 28, 1], [21, 41, 1], [25, 31, 1], [30, 33, 1], [32, 39, 1]], self_inverse_residues_1_and_p_minus_1, 42, Yes).

(47, [[1, 1], [2, 24], [3, 16], [4, 12], [5, 19], [6, 8], [7, 27], [8, 6], [9, 21], [10, 33], [11, 30], [12, 4], [13, 29], [14, 37], [15, 22], [16, 3], [17, 36], [18, 34], [19, 5], [20, 40], [21, 9], [22, 15], [23, 45], [24, 2], [25, 32], [26, 38], [27, 7], [28, 42], [29, 13], [30, 11], [31, 44], [32, 25], [33, 10], [34, 18], [35, 43], [36, 17], [37, 14], [38, 26], [39, 41], [40, 20], [41, 39], [42, 28], [43, 35], [44, 31], [45, 23], [46, 46]], [1, 46], [[2, 24, 1], [3, 16, 1], [4, 12, 1], [5, 19, 1], [6, 8, 1], [7, 27, 1], [9, 21, 1], [10, 33, 1], [11, 30, 1], [13, 29, 1], [14, 37, 1], [15, 22, 1], [17, 36, 1], [18, 34, 1], [20, 40, 1], [23, 45, 1], [25, 32, 1], [26, 38, 1], [28, 42, 1], [31, 44, 1], [35, 43, 1], [39, 41, 1]], self_inverse_residues_1_and_p_minus_1, 46, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (47, [[1, 1], [2, 24], [3, 16], [4, 12], [5, 19], [6, 8], [7, 27], [8, 6], [9, 21], [10, 33], [11, 30], [12, 4], [13, 29], [14, 37], [15, 22], [16, 3], [17, 36], [18, 34], [19, 5], [20, 40], [21, 9], [22, 15], [23, 45], [24, 2], [25, 32], [26, 38], [27, 7], [28, 42], [29, 13], [30, 11], [31, 44], [32, 25], [33, 10], [34, 18], [35, 43], [36, 17], [37, 14], [38, 26], [39, 41], [40, 20], [41, 39], [42, 28], [43, 35], [44, 31], [45, 23], [46, 46]], [1, 46], [[2, 24, 1], [3, 16, 1], [4, 12, 1], [5, 19, 1], [6, 8, 1], [7, 27, 1], [9, 21, 1], [10, 33, 1], [11, 30, 1], [13, 29, 1], [14, 37, 1], [15, 22, 1], [17, 36, 1], [18, 34, 1], [20, 40, 1], [23, 45, 1], [25, 32, 1], [26, 38, 1], [28, 42, 1], [31, 44, 1], [35, 43, 1], [39, 41, 1]], self_inverse_residues_1_and_p_minus_1, 46, Yes).

(53, [[1, 1], [2, 27], [3, 18], [4, 40], [5, 32], [6, 9], [7, 38], [8, 20], [9, 6], [10, 16], [11, 29], [12, 31], [13, 49], [14, 19], [15, 46], [16, 10], [17, 25], [18, 3], [19, 14], [20, 8], [21, 48], [22, 41], [23, 30], [24, 42], [25, 17], [26, 51], [27, 2], [28, 36], [29, 11], [30, 23], [31, 12], [32, 5], [33, 45], [34, 39], [35, 50], [36, 28], [37, 43], [38, 7], [39, 34], [40, 4], [41, 22], [42, 24], [43, 37], [44, 47], [45, 33], [46, 15], [47, 44], [48, 21], [49, 13], [50, 35], [51, 26], [52, 52]], [1, 52], [[2, 27, 1], [3, 18, 1], [4, 40, 1], [5, 32, 1], [6, 9, 1], [7, 38, 1], [8, 20, 1], [10, 16, 1], [11, 29, 1], [12, 31, 1], [13, 49, 1], [14, 19, 1], [15, 46, 1], [17, 25, 1], [21, 48, 1], [22, 41, 1], [23, 30, 1], [24, 42, 1], [26, 51, 1], [28, 36, 1], [33, 45, 1], [34, 39, 1], [35, 50, 1], [37, 43, 1], [44, 47, 1]], self_inverse_residues_1_and_p_minus_1, 52, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (53, [[1, 1], [2, 27], [3, 18], [4, 40], [5, 32], [6, 9], [7, 38], [8, 20], [9, 6], [10, 16], [11, 29], [12, 31], [13, 49], [14, 19], [15, 46], [16, 10], [17, 25], [18, 3], [19, 14], [20, 8], [21, 48], [22, 41], [23, 30], [24, 42], [25, 17], [26, 51], [27, 2], [28, 36], [29, 11], [30, 23], [31, 12], [32, 5], [33, 45], [34, 39], [35, 50], [36, 28], [37, 43], [38, 7], [39, 34], [40, 4], [41, 22], [42, 24], [43, 37], [44, 47], [45, 33], [46, 15], [47, 44], [48, 21], [49, 13], [50, 35], [51, 26], [52, 52]], [1, 52], [[2, 27, 1], [3, 18, 1], [4, 40, 1], [5, 32, 1], [6, 9, 1], [7, 38, 1], [8, 20, 1], [10, 16, 1], [11, 29, 1], [12, 31, 1], [13, 49, 1], [14, 19, 1], [15, 46, 1], [17, 25, 1], [21, 48, 1], [22, 41, 1], [23, 30, 1], [24, 42, 1], [26, 51, 1], [28, 36, 1], [33, 45, 1], [34, 39, 1], [35, 50, 1], [37, 43, 1], [44, 47, 1]], self_inverse_residues_1_and_p_minus_1, 52, Yes).

(59, [[1, 1], [2, 30], [3, 20], [4, 15], [5, 12], [6, 10], [7, 17], [8, 37], [9, 46], [10, 6], [11, 43], [12, 5], [13, 50], [14, 38], [15, 4], [16, 48], [17, 7], [18, 23], [19, 28], [20, 3], [21, 45], [22, 51], [23, 18], [24, 32], [25, 26], [26, 25], [27, 35], [28, 19], [29, 57], [30, 2], [31, 40], [32, 24], [33, 34], [34, 33], [35, 27], [36, 41], [37, 8], [38, 14], [39, 56], [40, 31], [41, 36], [42, 52], [43, 11], [44, 55], [45, 21], [46, 9], [47, 54], [48, 16], [49, 53], [50, 13], [51, 22], [52, 42], [53, 49], [54, 47], [55, 44], [56, 39], [57, 29], [58, 58]], [1, 58], [[2, 30, 1], [3, 20, 1], [4, 15, 1], [5, 12, 1], [6, 10, 1], [7, 17, 1], [8, 37, 1], [9, 46, 1], [11, 43, 1], [13, 50, 1], [14, 38, 1], [16, 48, 1], [18, 23, 1], [19, 28, 1], [21, 45, 1], [22, 51, 1], [24, 32, 1], [25, 26, 1], [27, 35, 1], [29, 57, 1], [31, 40, 1], [33, 34, 1], [36, 41, 1], [39, 56, 1], [42, 52, 1], [44, 55, 1], [47, 54, 1], [49, 53, 1]], self_inverse_residues_1_and_p_minus_1, 58, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (59, [[1, 1], [2, 30], [3, 20], [4, 15], [5, 12], [6, 10], [7, 17], [8, 37], [9, 46], [10, 6], [11, 43], [12, 5], [13, 50], [14, 38], [15, 4], [16, 48], [17, 7], [18, 23], [19, 28], [20, 3], [21, 45], [22, 51], [23, 18], [24, 32], [25, 26], [26, 25], [27, 35], [28, 19], [29, 57], [30, 2], [31, 40], [32, 24], [33, 34], [34, 33], [35, 27], [36, 41], [37, 8], [38, 14], [39, 56], [40, 31], [41, 36], [42, 52], [43, 11], [44, 55], [45, 21], [46, 9], [47, 54], [48, 16], [49, 53], [50, 13], [51, 22], [52, 42], [53, 49], [54, 47], [55, 44], [56, 39], [57, 29], [58, 58]], [1, 58], [[2, 30, 1], [3, 20, 1], [4, 15, 1], [5, 12, 1], [6, 10, 1], [7, 17, 1], [8, 37, 1], [9, 46, 1], [11, 43, 1], [13, 50, 1], [14, 38, 1], [16, 48, 1], [18, 23, 1], [19, 28, 1], [21, 45, 1], [22, 51, 1], [24, 32, 1], [25, 26, 1], [27, 35, 1], [29, 57, 1], [31, 40, 1], [33, 34, 1], [36, 41, 1], [39, 56, 1], [42, 52, 1], [44, 55, 1], [47, 54, 1], [49, 53, 1]], self_inverse_residues_1_and_p_minus_1, 58, Yes).

(61, [[1, 1], [2, 31], [3, 41], [4, 46], [5, 49], [6, 51], [7, 35], [8, 23], [9, 34], [10, 55], [11, 50], [12, 56], [13, 47], [14, 48], [15, 57], [16, 42], [17, 18], [18, 17], [19, 45], [20, 58], [21, 32], [22, 25], [23, 8], [24, 28], [25, 22], [26, 54], [27, 52], [28, 24], [29, 40], [30, 59], [31, 2], [32, 21], [33, 37], [34, 9], [35, 7], [36, 39], [37, 33], [38, 53], [39, 36], [40, 29], [41, 3], [42, 16], [43, 44], [44, 43], [45, 19], [46, 4], [47, 13], [48, 14], [49, 5], [50, 11], [51, 6], [52, 27], [53, 38], [54, 26], [55, 10], [56, 12], [57, 15], [58, 20], [59, 30], [60, 60]], [1, 60], [[2, 31, 1], [3, 41, 1], [4, 46, 1], [5, 49, 1], [6, 51, 1], [7, 35, 1], [8, 23, 1], [9, 34, 1], [10, 55, 1], [11, 50, 1], [12, 56, 1], [13, 47, 1], [14, 48, 1], [15, 57, 1], [16, 42, 1], [17, 18, 1], [19, 45, 1], [20, 58, 1], [21, 32, 1], [22, 25, 1], [24, 28, 1], [26, 54, 1], [27, 52, 1], [29, 40, 1], [30, 59, 1], [33, 37, 1], [36, 39, 1], [38, 53, 1], [43, 44, 1]], self_inverse_residues_1_and_p_minus_1, 60, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (61, [[1, 1], [2, 31], [3, 41], [4, 46], [5, 49], [6, 51], [7, 35], [8, 23], [9, 34], [10, 55], [11, 50], [12, 56], [13, 47], [14, 48], [15, 57], [16, 42], [17, 18], [18, 17], [19, 45], [20, 58], [21, 32], [22, 25], [23, 8], [24, 28], [25, 22], [26, 54], [27, 52], [28, 24], [29, 40], [30, 59], [31, 2], [32, 21], [33, 37], [34, 9], [35, 7], [36, 39], [37, 33], [38, 53], [39, 36], [40, 29], [41, 3], [42, 16], [43, 44], [44, 43], [45, 19], [46, 4], [47, 13], [48, 14], [49, 5], [50, 11], [51, 6], [52, 27], [53, 38], [54, 26], [55, 10], [56, 12], [57, 15], [58, 20], [59, 30], [60, 60]], [1, 60], [[2, 31, 1], [3, 41, 1], [4, 46, 1], [5, 49, 1], [6, 51, 1], [7, 35, 1], [8, 23, 1], [9, 34, 1], [10, 55, 1], [11, 50, 1], [12, 56, 1], [13, 47, 1], [14, 48, 1], [15, 57, 1], [16, 42, 1], [17, 18, 1], [19, 45, 1], [20, 58, 1], [21, 32, 1], [22, 25, 1], [24, 28, 1], [26, 54, 1], [27, 52, 1], [29, 40, 1], [30, 59, 1], [33, 37, 1], [36, 39, 1], [38, 53, 1], [43, 44, 1]], self_inverse_residues_1_and_p_minus_1, 60, Yes).

(67, [[1, 1], [2, 34], [3, 45], [4, 17], [5, 27], [6, 56], [7, 48], [8, 42], [9, 15], [10, 47], [11, 61], [12, 28], [13, 31], [14, 24], [15, 9], [16, 21], [17, 4], [18, 41], [19, 60], [20, 57], [21, 16], [22, 64], [23, 35], [24, 14], [25, 59], [26, 49], [27, 5], [28, 12], [29, 37], [30, 38], [31, 13], [32, 44], [33, 65], [34, 2], [35, 23], [36, 54], [37, 29], [38, 30], [39, 55], [40, 62], [41, 18], [42, 8], [43, 53], [44, 32], [45, 3], [46, 51], [47, 10], [48, 7], [49, 26], [50, 63], [51, 46], [52, 58], [53, 43], [54, 36], [55, 39], [56, 6], [57, 20], [58, 52], [59, 25], [60, 19], [61, 11], [62, 40], [63, 50], [64, 22], [65, 33], [66, 66]], [1, 66], [[2, 34, 1], [3, 45, 1], [4, 17, 1], [5, 27, 1], [6, 56, 1], [7, 48, 1], [8, 42, 1], [9, 15, 1], [10, 47, 1], [11, 61, 1], [12, 28, 1], [13, 31, 1], [14, 24, 1], [16, 21, 1], [18, 41, 1], [19, 60, 1], [20, 57, 1], [22, 64, 1], [23, 35, 1], [25, 59, 1], [26, 49, 1], [29, 37, 1], [30, 38, 1], [32, 44, 1], [33, 65, 1], [36, 54, 1], [39, 55, 1], [40, 62, 1], [43, 53, 1], [46, 51, 1], [50, 63, 1], [52, 58, 1]], self_inverse_residues_1_and_p_minus_1, 66, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (67, [[1, 1], [2, 34], [3, 45], [4, 17], [5, 27], [6, 56], [7, 48], [8, 42], [9, 15], [10, 47], [11, 61], [12, 28], [13, 31], [14, 24], [15, 9], [16, 21], [17, 4], [18, 41], [19, 60], [20, 57], [21, 16], [22, 64], [23, 35], [24, 14], [25, 59], [26, 49], [27, 5], [28, 12], [29, 37], [30, 38], [31, 13], [32, 44], [33, 65], [34, 2], [35, 23], [36, 54], [37, 29], [38, 30], [39, 55], [40, 62], [41, 18], [42, 8], [43, 53], [44, 32], [45, 3], [46, 51], [47, 10], [48, 7], [49, 26], [50, 63], [51, 46], [52, 58], [53, 43], [54, 36], [55, 39], [56, 6], [57, 20], [58, 52], [59, 25], [60, 19], [61, 11], [62, 40], [63, 50], [64, 22], [65, 33], [66, 66]], [1, 66], [[2, 34, 1], [3, 45, 1], [4, 17, 1], [5, 27, 1], [6, 56, 1], [7, 48, 1], [8, 42, 1], [9, 15, 1], [10, 47, 1], [11, 61, 1], [12, 28, 1], [13, 31, 1], [14, 24, 1], [16, 21, 1], [18, 41, 1], [19, 60, 1], [20, 57, 1], [22, 64, 1], [23, 35, 1], [25, 59, 1], [26, 49, 1], [29, 37, 1], [30, 38, 1], [32, 44, 1], [33, 65, 1], [36, 54, 1], [39, 55, 1], [40, 62, 1], [43, 53, 1], [46, 51, 1], [50, 63, 1], [52, 58, 1]], self_inverse_residues_1_and_p_minus_1, 66, Yes).

(71, [[1, 1], [2, 36], [3, 24], [4, 18], [5, 57], [6, 12], [7, 61], [8, 9], [9, 8], [10, 64], [11, 13], [12, 6], [13, 11], [14, 66], [15, 19], [16, 40], [17, 46], [18, 4], [19, 15], [20, 32], [21, 44], [22, 42], [23, 34], [24, 3], [25, 54], [26, 41], [27, 50], [28, 33], [29, 49], [30, 45], [31, 55], [32, 20], [33, 28], [34, 23], [35, 69], [36, 2], [37, 48], [38, 43], [39, 51], [40, 16], [41, 26], [42, 22], [43, 38], [44, 21], [45, 30], [46, 17], [47, 68], [48, 37], [49, 29], [50, 27], [51, 39], [52, 56], [53, 67], [54, 25], [55, 31], [56, 52], [57, 5], [58, 60], [59, 65], [60, 58], [61, 7], [62, 63], [63, 62], [64, 10], [65, 59], [66, 14], [67, 53], [68, 47], [69, 35], [70, 70]], [1, 70], [[2, 36, 1], [3, 24, 1], [4, 18, 1], [5, 57, 1], [6, 12, 1], [7, 61, 1], [8, 9, 1], [10, 64, 1], [11, 13, 1], [14, 66, 1], [15, 19, 1], [16, 40, 1], [17, 46, 1], [20, 32, 1], [21, 44, 1], [22, 42, 1], [23, 34, 1], [25, 54, 1], [26, 41, 1], [27, 50, 1], [28, 33, 1], [29, 49, 1], [30, 45, 1], [31, 55, 1], [35, 69, 1], [37, 48, 1], [38, 43, 1], [39, 51, 1], [47, 68, 1], [52, 56, 1], [53, 67, 1], [58, 60, 1], [59, 65, 1], [62, 63, 1]], self_inverse_residues_1_and_p_minus_1, 70, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (71, [[1, 1], [2, 36], [3, 24], [4, 18], [5, 57], [6, 12], [7, 61], [8, 9], [9, 8], [10, 64], [11, 13], [12, 6], [13, 11], [14, 66], [15, 19], [16, 40], [17, 46], [18, 4], [19, 15], [20, 32], [21, 44], [22, 42], [23, 34], [24, 3], [25, 54], [26, 41], [27, 50], [28, 33], [29, 49], [30, 45], [31, 55], [32, 20], [33, 28], [34, 23], [35, 69], [36, 2], [37, 48], [38, 43], [39, 51], [40, 16], [41, 26], [42, 22], [43, 38], [44, 21], [45, 30], [46, 17], [47, 68], [48, 37], [49, 29], [50, 27], [51, 39], [52, 56], [53, 67], [54, 25], [55, 31], [56, 52], [57, 5], [58, 60], [59, 65], [60, 58], [61, 7], [62, 63], [63, 62], [64, 10], [65, 59], [66, 14], [67, 53], [68, 47], [69, 35], [70, 70]], [1, 70], [[2, 36, 1], [3, 24, 1], [4, 18, 1], [5, 57, 1], [6, 12, 1], [7, 61, 1], [8, 9, 1], [10, 64, 1], [11, 13, 1], [14, 66, 1], [15, 19, 1], [16, 40, 1], [17, 46, 1], [20, 32, 1], [21, 44, 1], [22, 42, 1], [23, 34, 1], [25, 54, 1], [26, 41, 1], [27, 50, 1], [28, 33, 1], [29, 49, 1], [30, 45, 1], [31, 55, 1], [35, 69, 1], [37, 48, 1], [38, 43, 1], [39, 51, 1], [47, 68, 1], [52, 56, 1], [53, 67, 1], [58, 60, 1], [59, 65, 1], [62, 63, 1]], self_inverse_residues_1_and_p_minus_1, 70, Yes).

(73, [[1, 1], [2, 37], [3, 49], [4, 55], [5, 44], [6, 61], [7, 21], [8, 64], [9, 65], [10, 22], [11, 20], [12, 67], [13, 45], [14, 47], [15, 39], [16, 32], [17, 43], [18, 69], [19, 50], [20, 11], [21, 7], [22, 10], [23, 54], [24, 70], [25, 38], [26, 59], [27, 46], [28, 60], [29, 68], [30, 56], [31, 33], [32, 16], [33, 31], [34, 58], [35, 48], [36, 71], [37, 2], [38, 25], [39, 15], [40, 42], [41, 57], [42, 40], [43, 17], [44, 5], [45, 13], [46, 27], [47, 14], [48, 35], [49, 3], [50, 19], [51, 63], [52, 66], [53, 62], [54, 23], [55, 4], [56, 30], [57, 41], [58, 34], [59, 26], [60, 28], [61, 6], [62, 53], [63, 51], [64, 8], [65, 9], [66, 52], [67, 12], [68, 29], [69, 18], [70, 24], [71, 36], [72, 72]], [1, 72], [[2, 37, 1], [3, 49, 1], [4, 55, 1], [5, 44, 1], [6, 61, 1], [7, 21, 1], [8, 64, 1], [9, 65, 1], [10, 22, 1], [11, 20, 1], [12, 67, 1], [13, 45, 1], [14, 47, 1], [15, 39, 1], [16, 32, 1], [17, 43, 1], [18, 69, 1], [19, 50, 1], [23, 54, 1], [24, 70, 1], [25, 38, 1], [26, 59, 1], [27, 46, 1], [28, 60, 1], [29, 68, 1], [30, 56, 1], [31, 33, 1], [34, 58, 1], [35, 48, 1], [36, 71, 1], [40, 42, 1], [41, 57, 1], [51, 63, 1], [52, 66, 1], [53, 62, 1]], self_inverse_residues_1_and_p_minus_1, 72, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (73, [[1, 1], [2, 37], [3, 49], [4, 55], [5, 44], [6, 61], [7, 21], [8, 64], [9, 65], [10, 22], [11, 20], [12, 67], [13, 45], [14, 47], [15, 39], [16, 32], [17, 43], [18, 69], [19, 50], [20, 11], [21, 7], [22, 10], [23, 54], [24, 70], [25, 38], [26, 59], [27, 46], [28, 60], [29, 68], [30, 56], [31, 33], [32, 16], [33, 31], [34, 58], [35, 48], [36, 71], [37, 2], [38, 25], [39, 15], [40, 42], [41, 57], [42, 40], [43, 17], [44, 5], [45, 13], [46, 27], [47, 14], [48, 35], [49, 3], [50, 19], [51, 63], [52, 66], [53, 62], [54, 23], [55, 4], [56, 30], [57, 41], [58, 34], [59, 26], [60, 28], [61, 6], [62, 53], [63, 51], [64, 8], [65, 9], [66, 52], [67, 12], [68, 29], [69, 18], [70, 24], [71, 36], [72, 72]], [1, 72], [[2, 37, 1], [3, 49, 1], [4, 55, 1], [5, 44, 1], [6, 61, 1], [7, 21, 1], [8, 64, 1], [9, 65, 1], [10, 22, 1], [11, 20, 1], [12, 67, 1], [13, 45, 1], [14, 47, 1], [15, 39, 1], [16, 32, 1], [17, 43, 1], [18, 69, 1], [19, 50, 1], [23, 54, 1], [24, 70, 1], [25, 38, 1], [26, 59, 1], [27, 46, 1], [28, 60, 1], [29, 68, 1], [30, 56, 1], [31, 33, 1], [34, 58, 1], [35, 48, 1], [36, 71, 1], [40, 42, 1], [41, 57, 1], [51, 63, 1], [52, 66, 1], [53, 62, 1]], self_inverse_residues_1_and_p_minus_1, 72, Yes).

(79, [[1, 1], [2, 40], [3, 53], [4, 20], [5, 16], [6, 66], [7, 34], [8, 10], [9, 44], [10, 8], [11, 36], [12, 33], [13, 73], [14, 17], [15, 58], [16, 5], [17, 14], [18, 22], [19, 25], [20, 4], [21, 64], [22, 18], [23, 55], [24, 56], [25, 19], [26, 76], [27, 41], [28, 48], [29, 30], [30, 29], [31, 51], [32, 42], [33, 12], [34, 7], [35, 70], [36, 11], [37, 47], [38, 52], [39, 77], [40, 2], [41, 27], [42, 32], [43, 68], [44, 9], [45, 72], [46, 67], [47, 37], [48, 28], [49, 50], [50, 49], [51, 31], [52, 38], [53, 3], [54, 60], [55, 23], [56, 24], [57, 61], [58, 15], [59, 75], [60, 54], [61, 57], [62, 65], [63, 74], [64, 21], [65, 62], [66, 6], [67, 46], [68, 43], [69, 71], [70, 35], [71, 69], [72, 45], [73, 13], [74, 63], [75, 59], [76, 26], [77, 39], [78, 78]], [1, 78], [[2, 40, 1], [3, 53, 1], [4, 20, 1], [5, 16, 1], [6, 66, 1], [7, 34, 1], [8, 10, 1], [9, 44, 1], [11, 36, 1], [12, 33, 1], [13, 73, 1], [14, 17, 1], [15, 58, 1], [18, 22, 1], [19, 25, 1], [21, 64, 1], [23, 55, 1], [24, 56, 1], [26, 76, 1], [27, 41, 1], [28, 48, 1], [29, 30, 1], [31, 51, 1], [32, 42, 1], [35, 70, 1], [37, 47, 1], [38, 52, 1], [39, 77, 1], [43, 68, 1], [45, 72, 1], [46, 67, 1], [49, 50, 1], [54, 60, 1], [57, 61, 1], [59, 75, 1], [62, 65, 1], [63, 74, 1], [69, 71, 1]], self_inverse_residues_1_and_p_minus_1, 78, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (79, [[1, 1], [2, 40], [3, 53], [4, 20], [5, 16], [6, 66], [7, 34], [8, 10], [9, 44], [10, 8], [11, 36], [12, 33], [13, 73], [14, 17], [15, 58], [16, 5], [17, 14], [18, 22], [19, 25], [20, 4], [21, 64], [22, 18], [23, 55], [24, 56], [25, 19], [26, 76], [27, 41], [28, 48], [29, 30], [30, 29], [31, 51], [32, 42], [33, 12], [34, 7], [35, 70], [36, 11], [37, 47], [38, 52], [39, 77], [40, 2], [41, 27], [42, 32], [43, 68], [44, 9], [45, 72], [46, 67], [47, 37], [48, 28], [49, 50], [50, 49], [51, 31], [52, 38], [53, 3], [54, 60], [55, 23], [56, 24], [57, 61], [58, 15], [59, 75], [60, 54], [61, 57], [62, 65], [63, 74], [64, 21], [65, 62], [66, 6], [67, 46], [68, 43], [69, 71], [70, 35], [71, 69], [72, 45], [73, 13], [74, 63], [75, 59], [76, 26], [77, 39], [78, 78]], [1, 78], [[2, 40, 1], [3, 53, 1], [4, 20, 1], [5, 16, 1], [6, 66, 1], [7, 34, 1], [8, 10, 1], [9, 44, 1], [11, 36, 1], [12, 33, 1], [13, 73, 1], [14, 17, 1], [15, 58, 1], [18, 22, 1], [19, 25, 1], [21, 64, 1], [23, 55, 1], [24, 56, 1], [26, 76, 1], [27, 41, 1], [28, 48, 1], [29, 30, 1], [31, 51, 1], [32, 42, 1], [35, 70, 1], [37, 47, 1], [38, 52, 1], [39, 77, 1], [43, 68, 1], [45, 72, 1], [46, 67, 1], [49, 50, 1], [54, 60, 1], [57, 61, 1], [59, 75, 1], [62, 65, 1], [63, 74, 1], [69, 71, 1]], self_inverse_residues_1_and_p_minus_1, 78, Yes).

(83, [[1, 1], [2, 42], [3, 28], [4, 21], [5, 50], [6, 14], [7, 12], [8, 52], [9, 37], [10, 25], [11, 68], [12, 7], [13, 32], [14, 6], [15, 72], [16, 26], [17, 44], [18, 60], [19, 35], [20, 54], [21, 4], [22, 34], [23, 65], [24, 45], [25, 10], [26, 16], [27, 40], [28, 3], [29, 63], [30, 36], [31, 75], [32, 13], [33, 78], [34, 22], [35, 19], [36, 30], [37, 9], [38, 59], [39, 66], [40, 27], [41, 81], [42, 2], [43, 56], [44, 17], [45, 24], [46, 74], [47, 53], [48, 64], [49, 61], [50, 5], [51, 70], [52, 8], [53, 47], [54, 20], [55, 80], [56, 43], [57, 67], [58, 73], [59, 38], [60, 18], [61, 49], [62, 79], [63, 29], [64, 48], [65, 23], [66, 39], [67, 57], [68, 11], [69, 77], [70, 51], [71, 76], [72, 15], [73, 58], [74, 46], [75, 31], [76, 71], [77, 69], [78, 33], [79, 62], [80, 55], [81, 41], [82, 82]], [1, 82], [[2, 42, 1], [3, 28, 1], [4, 21, 1], [5, 50, 1], [6, 14, 1], [7, 12, 1], [8, 52, 1], [9, 37, 1], [10, 25, 1], [11, 68, 1], [13, 32, 1], [15, 72, 1], [16, 26, 1], [17, 44, 1], [18, 60, 1], [19, 35, 1], [20, 54, 1], [22, 34, 1], [23, 65, 1], [24, 45, 1], [27, 40, 1], [29, 63, 1], [30, 36, 1], [31, 75, 1], [33, 78, 1], [38, 59, 1], [39, 66, 1], [41, 81, 1], [43, 56, 1], [46, 74, 1], [47, 53, 1], [48, 64, 1], [49, 61, 1], [51, 70, 1], [55, 80, 1], [57, 67, 1], [58, 73, 1], [62, 79, 1], [69, 77, 1], [71, 76, 1]], self_inverse_residues_1_and_p_minus_1, 82, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (83, [[1, 1], [2, 42], [3, 28], [4, 21], [5, 50], [6, 14], [7, 12], [8, 52], [9, 37], [10, 25], [11, 68], [12, 7], [13, 32], [14, 6], [15, 72], [16, 26], [17, 44], [18, 60], [19, 35], [20, 54], [21, 4], [22, 34], [23, 65], [24, 45], [25, 10], [26, 16], [27, 40], [28, 3], [29, 63], [30, 36], [31, 75], [32, 13], [33, 78], [34, 22], [35, 19], [36, 30], [37, 9], [38, 59], [39, 66], [40, 27], [41, 81], [42, 2], [43, 56], [44, 17], [45, 24], [46, 74], [47, 53], [48, 64], [49, 61], [50, 5], [51, 70], [52, 8], [53, 47], [54, 20], [55, 80], [56, 43], [57, 67], [58, 73], [59, 38], [60, 18], [61, 49], [62, 79], [63, 29], [64, 48], [65, 23], [66, 39], [67, 57], [68, 11], [69, 77], [70, 51], [71, 76], [72, 15], [73, 58], [74, 46], [75, 31], [76, 71], [77, 69], [78, 33], [79, 62], [80, 55], [81, 41], [82, 82]], [1, 82], [[2, 42, 1], [3, 28, 1], [4, 21, 1], [5, 50, 1], [6, 14, 1], [7, 12, 1], [8, 52, 1], [9, 37, 1], [10, 25, 1], [11, 68, 1], [13, 32, 1], [15, 72, 1], [16, 26, 1], [17, 44, 1], [18, 60, 1], [19, 35, 1], [20, 54, 1], [22, 34, 1], [23, 65, 1], [24, 45, 1], [27, 40, 1], [29, 63, 1], [30, 36, 1], [31, 75, 1], [33, 78, 1], [38, 59, 1], [39, 66, 1], [41, 81, 1], [43, 56, 1], [46, 74, 1], [47, 53, 1], [48, 64, 1], [49, 61, 1], [51, 70, 1], [55, 80, 1], [57, 67, 1], [58, 73, 1], [62, 79, 1], [69, 77, 1], [71, 76, 1]], self_inverse_residues_1_and_p_minus_1, 82, Yes).

(89, [[1, 1], [2, 45], [3, 30], [4, 67], [5, 18], [6, 15], [7, 51], [8, 78], [9, 10], [10, 9], [11, 81], [12, 52], [13, 48], [14, 70], [15, 6], [16, 39], [17, 21], [18, 5], [19, 75], [20, 49], [21, 17], [22, 85], [23, 31], [24, 26], [25, 57], [26, 24], [27, 33], [28, 35], [29, 43], [30, 3], [31, 23], [32, 64], [33, 27], [34, 55], [35, 28], [36, 47], [37, 77], [38, 82], [39, 16], [40, 69], [41, 76], [42, 53], [43, 29], [44, 87], [45, 2], [46, 60], [47, 36], [48, 13], [49, 20], [50, 73], [51, 7], [52, 12], [53, 42], [54, 61], [55, 34], [56, 62], [57, 25], [58, 66], [59, 86], [60, 46], [61, 54], [62, 56], [63, 65], [64, 32], [65, 63], [66, 58], [67, 4], [68, 72], [69, 40], [70, 14], [71, 84], [72, 68], [73, 50], [74, 83], [75, 19], [76, 41], [77, 37], [78, 8], [79, 80], [80, 79], [81, 11], [82, 38], [83, 74], [84, 71], [85, 22], [86, 59], [87, 44], [88, 88]], [1, 88], [[2, 45, 1], [3, 30, 1], [4, 67, 1], [5, 18, 1], [6, 15, 1], [7, 51, 1], [8, 78, 1], [9, 10, 1], [11, 81, 1], [12, 52, 1], [13, 48, 1], [14, 70, 1], [16, 39, 1], [17, 21, 1], [19, 75, 1], [20, 49, 1], [22, 85, 1], [23, 31, 1], [24, 26, 1], [25, 57, 1], [27, 33, 1], [28, 35, 1], [29, 43, 1], [32, 64, 1], [34, 55, 1], [36, 47, 1], [37, 77, 1], [38, 82, 1], [40, 69, 1], [41, 76, 1], [42, 53, 1], [44, 87, 1], [46, 60, 1], [50, 73, 1], [54, 61, 1], [56, 62, 1], [58, 66, 1], [59, 86, 1], [63, 65, 1], [68, 72, 1], [71, 84, 1], [74, 83, 1], [79, 80, 1]], self_inverse_residues_1_and_p_minus_1, 88, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (89, [[1, 1], [2, 45], [3, 30], [4, 67], [5, 18], [6, 15], [7, 51], [8, 78], [9, 10], [10, 9], [11, 81], [12, 52], [13, 48], [14, 70], [15, 6], [16, 39], [17, 21], [18, 5], [19, 75], [20, 49], [21, 17], [22, 85], [23, 31], [24, 26], [25, 57], [26, 24], [27, 33], [28, 35], [29, 43], [30, 3], [31, 23], [32, 64], [33, 27], [34, 55], [35, 28], [36, 47], [37, 77], [38, 82], [39, 16], [40, 69], [41, 76], [42, 53], [43, 29], [44, 87], [45, 2], [46, 60], [47, 36], [48, 13], [49, 20], [50, 73], [51, 7], [52, 12], [53, 42], [54, 61], [55, 34], [56, 62], [57, 25], [58, 66], [59, 86], [60, 46], [61, 54], [62, 56], [63, 65], [64, 32], [65, 63], [66, 58], [67, 4], [68, 72], [69, 40], [70, 14], [71, 84], [72, 68], [73, 50], [74, 83], [75, 19], [76, 41], [77, 37], [78, 8], [79, 80], [80, 79], [81, 11], [82, 38], [83, 74], [84, 71], [85, 22], [86, 59], [87, 44], [88, 88]], [1, 88], [[2, 45, 1], [3, 30, 1], [4, 67, 1], [5, 18, 1], [6, 15, 1], [7, 51, 1], [8, 78, 1], [9, 10, 1], [11, 81, 1], [12, 52, 1], [13, 48, 1], [14, 70, 1], [16, 39, 1], [17, 21, 1], [19, 75, 1], [20, 49, 1], [22, 85, 1], [23, 31, 1], [24, 26, 1], [25, 57, 1], [27, 33, 1], [28, 35, 1], [29, 43, 1], [32, 64, 1], [34, 55, 1], [36, 47, 1], [37, 77, 1], [38, 82, 1], [40, 69, 1], [41, 76, 1], [42, 53, 1], [44, 87, 1], [46, 60, 1], [50, 73, 1], [54, 61, 1], [56, 62, 1], [58, 66, 1], [59, 86, 1], [63, 65, 1], [68, 72, 1], [71, 84, 1], [74, 83, 1], [79, 80, 1]], self_inverse_residues_1_and_p_minus_1, 88, Yes).

(97, [[1, 1], [2, 49], [3, 65], [4, 73], [5, 39], [6, 81], [7, 14], [8, 85], [9, 54], [10, 68], [11, 53], [12, 89], [13, 15], [14, 7], [15, 13], [16, 91], [17, 40], [18, 27], [19, 46], [20, 34], [21, 37], [22, 75], [23, 38], [24, 93], [25, 66], [26, 56], [27, 18], [28, 52], [29, 87], [30, 55], [31, 72], [32, 94], [33, 50], [34, 20], [35, 61], [36, 62], [37, 21], [38, 23], [39, 5], [40, 17], [41, 71], [42, 67], [43, 88], [44, 86], [45, 69], [46, 19], [47, 64], [48, 95], [49, 2], [50, 33], [51, 78], [52, 28], [53, 11], [54, 9], [55, 30], [56, 26], [57, 80], [58, 92], [59, 74], [60, 76], [61, 35], [62, 36], [63, 77], [64, 47], [65, 3], [66, 25], [67, 42], [68, 10], [69, 45], [70, 79], [71, 41], [72, 31], [73, 4], [74, 59], [75, 22], [76, 60], [77, 63], [78, 51], [79, 70], [80, 57], [81, 6], [82, 84], [83, 90], [84, 82], [85, 8], [86, 44], [87, 29], [88, 43], [89, 12], [90, 83], [91, 16], [92, 58], [93, 24], [94, 32], [95, 48], [96, 96]], [1, 96], [[2, 49, 1], [3, 65, 1], [4, 73, 1], [5, 39, 1], [6, 81, 1], [7, 14, 1], [8, 85, 1], [9, 54, 1], [10, 68, 1], [11, 53, 1], [12, 89, 1], [13, 15, 1], [16, 91, 1], [17, 40, 1], [18, 27, 1], [19, 46, 1], [20, 34, 1], [21, 37, 1], [22, 75, 1], [23, 38, 1], [24, 93, 1], [25, 66, 1], [26, 56, 1], [28, 52, 1], [29, 87, 1], [30, 55, 1], [31, 72, 1], [32, 94, 1], [33, 50, 1], [35, 61, 1], [36, 62, 1], [41, 71, 1], [42, 67, 1], [43, 88, 1], [44, 86, 1], [45, 69, 1], [47, 64, 1], [48, 95, 1], [51, 78, 1], [57, 80, 1], [58, 92, 1], [59, 74, 1], [60, 76, 1], [63, 77, 1], [70, 79, 1], [82, 84, 1], [83, 90, 1]], self_inverse_residues_1_and_p_minus_1, 96, Yes)

List every nonzero residue and its inverse, identify self-inverse residues and pair each remaining unordered inverse pair once. For p=2,1 and−1 are the same residue, not two distinct ones. Retain the complete factorial congruence for this modulus. The exact result, retaining the requested order, is (97, [[1, 1], [2, 49], [3, 65], [4, 73], [5, 39], [6, 81], [7, 14], [8, 85], [9, 54], [10, 68], [11, 53], [12, 89], [13, 15], [14, 7], [15, 13], [16, 91], [17, 40], [18, 27], [19, 46], [20, 34], [21, 37], [22, 75], [23, 38], [24, 93], [25, 66], [26, 56], [27, 18], [28, 52], [29, 87], [30, 55], [31, 72], [32, 94], [33, 50], [34, 20], [35, 61], [36, 62], [37, 21], [38, 23], [39, 5], [40, 17], [41, 71], [42, 67], [43, 88], [44, 86], [45, 69], [46, 19], [47, 64], [48, 95], [49, 2], [50, 33], [51, 78], [52, 28], [53, 11], [54, 9], [55, 30], [56, 26], [57, 80], [58, 92], [59, 74], [60, 76], [61, 35], [62, 36], [63, 77], [64, 47], [65, 3], [66, 25], [67, 42], [68, 10], [69, 45], [70, 79], [71, 41], [72, 31], [73, 4], [74, 59], [75, 22], [76, 60], [77, 63], [78, 51], [79, 70], [80, 57], [81, 6], [82, 84], [83, 90], [84, 82], [85, 8], [86, 44], [87, 29], [88, 43], [89, 12], [90, 83], [91, 16], [92, 58], [93, 24], [94, 32], [95, 48], [96, 96]], [1, 96], [[2, 49, 1], [3, 65, 1], [4, 73, 1], [5, 39, 1], [6, 81, 1], [7, 14, 1], [8, 85, 1], [9, 54, 1], [10, 68, 1], [11, 53, 1], [12, 89, 1], [13, 15, 1], [16, 91, 1], [17, 40, 1], [18, 27, 1], [19, 46, 1], [20, 34, 1], [21, 37, 1], [22, 75, 1], [23, 38, 1], [24, 93, 1], [25, 66, 1], [26, 56, 1], [28, 52, 1], [29, 87, 1], [30, 55, 1], [31, 72, 1], [32, 94, 1], [33, 50, 1], [35, 61, 1], [36, 62, 1], [41, 71, 1], [42, 67, 1], [43, 88, 1], [44, 86, 1], [45, 69, 1], [47, 64, 1], [48, 95, 1], [51, 78, 1], [57, 80, 1], [58, 92, 1], [59, 74, 1], [60, 76, 1], [63, 77, 1], [70, 79, 1], [82, 84, 1], [83, 90, 1]], self_inverse_residues_1_and_p_minus_1, 96, Yes).

(4, No, [[1, 1], [2, 2], [3, 2]], 2, 3, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (4, No, [[1, 1], [2, 2], [3, 2]], 2, 3, Yes, finite_composite_instance_not_converse_proof).

(6, No, [[1, 1], [2, 2], [3, 0], [4, 0], [5, 0]], 0, 5, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (6, No, [[1, 1], [2, 2], [3, 0], [4, 0], [5, 0]], 0, 5, Yes, finite_composite_instance_not_converse_proof).

(8, No, [[1, 1], [2, 2], [3, 6], [4, 0], [5, 0], [6, 0], [7, 0]], 0, 7, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (8, No, [[1, 1], [2, 2], [3, 6], [4, 0], [5, 0], [6, 0], [7, 0]], 0, 7, Yes, finite_composite_instance_not_converse_proof).

(9, No, [[1, 1], [2, 2], [3, 6], [4, 6], [5, 3], [6, 0], [7, 0], [8, 0]], 0, 8, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (9, No, [[1, 1], [2, 2], [3, 6], [4, 6], [5, 3], [6, 0], [7, 0], [8, 0]], 0, 8, Yes, finite_composite_instance_not_converse_proof).

(10, No, [[1, 1], [2, 2], [3, 6], [4, 4], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0]], 0, 9, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (10, No, [[1, 1], [2, 2], [3, 6], [4, 4], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0]], 0, 9, Yes, finite_composite_instance_not_converse_proof).

(12, No, [[1, 1], [2, 2], [3, 6], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0]], 0, 11, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (12, No, [[1, 1], [2, 2], [3, 6], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0]], 0, 11, Yes, finite_composite_instance_not_converse_proof).

(14, No, [[1, 1], [2, 2], [3, 6], [4, 10], [5, 8], [6, 6], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0]], 0, 13, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (14, No, [[1, 1], [2, 2], [3, 6], [4, 10], [5, 8], [6, 6], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0]], 0, 13, Yes, finite_composite_instance_not_converse_proof).

(15, No, [[1, 1], [2, 2], [3, 6], [4, 9], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0]], 0, 14, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (15, No, [[1, 1], [2, 2], [3, 6], [4, 9], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0]], 0, 14, Yes, finite_composite_instance_not_converse_proof).

(16, No, [[1, 1], [2, 2], [3, 6], [4, 8], [5, 8], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0]], 0, 15, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (16, No, [[1, 1], [2, 2], [3, 6], [4, 8], [5, 8], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0]], 0, 15, Yes, finite_composite_instance_not_converse_proof).

(18, No, [[1, 1], [2, 2], [3, 6], [4, 6], [5, 12], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0]], 0, 17, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (18, No, [[1, 1], [2, 2], [3, 6], [4, 6], [5, 12], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0]], 0, 17, Yes, finite_composite_instance_not_converse_proof).

(20, No, [[1, 1], [2, 2], [3, 6], [4, 4], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0]], 0, 19, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (20, No, [[1, 1], [2, 2], [3, 6], [4, 4], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0]], 0, 19, Yes, finite_composite_instance_not_converse_proof).

(21, No, [[1, 1], [2, 2], [3, 6], [4, 3], [5, 15], [6, 6], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0]], 0, 20, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (21, No, [[1, 1], [2, 2], [3, 6], [4, 3], [5, 15], [6, 6], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0]], 0, 20, Yes, finite_composite_instance_not_converse_proof).

(22, No, [[1, 1], [2, 2], [3, 6], [4, 2], [5, 10], [6, 16], [7, 2], [8, 16], [9, 12], [10, 10], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0]], 0, 21, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (22, No, [[1, 1], [2, 2], [3, 6], [4, 2], [5, 10], [6, 16], [7, 2], [8, 16], [9, 12], [10, 10], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0]], 0, 21, Yes, finite_composite_instance_not_converse_proof).

(24, No, [[1, 1], [2, 2], [3, 6], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0]], 0, 23, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (24, No, [[1, 1], [2, 2], [3, 6], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0]], 0, 23, Yes, finite_composite_instance_not_converse_proof).

(25, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 20], [6, 20], [7, 15], [8, 20], [9, 5], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0]], 0, 24, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (25, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 20], [6, 20], [7, 15], [8, 20], [9, 5], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0]], 0, 24, Yes, finite_composite_instance_not_converse_proof).

(26, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 16], [6, 18], [7, 22], [8, 20], [9, 24], [10, 6], [11, 14], [12, 12], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0]], 0, 25, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (26, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 16], [6, 18], [7, 22], [8, 20], [9, 24], [10, 6], [11, 14], [12, 12], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0]], 0, 25, Yes, finite_composite_instance_not_converse_proof).

(27, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 12], [6, 18], [7, 18], [8, 9], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0]], 0, 26, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (27, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 12], [6, 18], [7, 18], [8, 9], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0]], 0, 26, Yes, finite_composite_instance_not_converse_proof).

(28, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 8], [6, 20], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0]], 0, 27, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (28, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 8], [6, 20], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0]], 0, 27, Yes, finite_composite_instance_not_converse_proof).

(30, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0]], 0, 29, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (30, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0]], 0, 29, Yes, finite_composite_instance_not_converse_proof).

(32, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 24], [6, 16], [7, 16], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0]], 0, 31, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (32, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 24], [6, 16], [7, 16], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0]], 0, 31, Yes, finite_composite_instance_not_converse_proof).

(33, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 21], [6, 27], [7, 24], [8, 27], [9, 12], [10, 21], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0]], 0, 32, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (33, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 21], [6, 27], [7, 24], [8, 27], [9, 12], [10, 21], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0]], 0, 32, Yes, finite_composite_instance_not_converse_proof).

(34, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 18], [6, 6], [7, 8], [8, 30], [9, 32], [10, 14], [11, 18], [12, 12], [13, 20], [14, 8], [15, 18], [16, 16], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0]], 0, 33, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (34, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 18], [6, 6], [7, 8], [8, 30], [9, 32], [10, 14], [11, 18], [12, 12], [13, 20], [14, 8], [15, 18], [16, 16], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0]], 0, 33, Yes, finite_composite_instance_not_converse_proof).

(35, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 15], [6, 20], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0], [34, 0]], 0, 34, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (35, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 15], [6, 20], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0], [34, 0]], 0, 34, Yes, finite_composite_instance_not_converse_proof).

(36, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 12], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0], [34, 0], [35, 0]], 0, 35, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (36, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 12], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0], [34, 0], [35, 0]], 0, 35, Yes, finite_composite_instance_not_converse_proof).

(38, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 6], [6, 36], [7, 24], [8, 2], [9, 18], [10, 28], [11, 4], [12, 10], [13, 16], [14, 34], [15, 16], [16, 28], [17, 20], [18, 18], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0], [34, 0], [35, 0], [36, 0], [37, 0]], 0, 37, Yes, finite_composite_instance_not_converse_proof)

Compute all running factorial residues and show the final residue differs from n−1. This exact finite instance does not replace the converse proof for all composite integers, and it is not Fermat little theorem. The exact result, retaining the requested order, is (38, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 6], [6, 36], [7, 24], [8, 2], [9, 18], [10, 28], [11, 4], [12, 10], [13, 16], [14, 34], [15, 16], [16, 28], [17, 20], [18, 18], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0], [34, 0], [35, 0], [36, 0], [37, 0]], 0, 37, Yes, finite_composite_instance_not_converse_proof).

(p2_self_inverse, 2, [1], 1_equals_minus_1_mod_2, one_not_two_distinct_self_inverse_residues, Yes)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (p2_self_inverse, 2, [1], 1_equals_minus_1_mod_2, one_not_two_distinct_self_inverse_residues, Yes).

(invalid_n, -1, No, Wilson_domain_requires_n>1)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (invalid_n, -1, No, Wilson_domain_requires_n>1).

(finite_testing, 4, 3, checked_moduli_2_through_4, No, finite_cutoff_does_not_prove_the_universal_equivalence)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (finite_testing, 4, 3, checked_moduli_2_through_4, No, finite_cutoff_does_not_prove_the_universal_equivalence).

(fermat_confusion, 7, [[1, 1], [2, 2], [3, 6], [4, 3], [5, 1], [6, 6]], 6, 6, Yes, Wilson:(n-1)!==-1_mod_7, Fermat:a^(7-1)==1_mod_7_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (fermat_confusion, 7, [[1, 1], [2, 2], [3, 6], [4, 3], [5, 1], [6, 6]], 6, 6, Yes, Wilson:(n-1)!==-1_mod_7, Fermat:a^(7-1)==1_mod_7_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed).

(invalid_n, -4, No, Wilson_domain_requires_n>1)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (invalid_n, -4, No, Wilson_domain_requires_n>1).

(finite_testing, 7, 6, checked_moduli_2_through_7, No, finite_cutoff_does_not_prove_the_universal_equivalence)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (finite_testing, 7, 6, checked_moduli_2_through_7, No, finite_cutoff_does_not_prove_the_universal_equivalence).

(fermat_confusion, 17, [[1, 1], [2, 2], [3, 6], [4, 7], [5, 1], [6, 6], [7, 8], [8, 13], [9, 15], [10, 14], [11, 1], [12, 12], [13, 3], [14, 8], [15, 1], [16, 16]], 16, 16, Yes, Wilson:(n-1)!==-1_mod_17, Fermat:a^(17-1)==1_mod_17_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (fermat_confusion, 17, [[1, 1], [2, 2], [3, 6], [4, 7], [5, 1], [6, 6], [7, 8], [8, 13], [9, 15], [10, 14], [11, 1], [12, 12], [13, 3], [14, 8], [15, 1], [16, 16]], 16, 16, Yes, Wilson:(n-1)!==-1_mod_17, Fermat:a^(17-1)==1_mod_17_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed).

(invalid_n, -7, No, Wilson_domain_requires_n>1)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (invalid_n, -7, No, Wilson_domain_requires_n>1).

(finite_testing, 10, 9, checked_moduli_2_through_10, No, finite_cutoff_does_not_prove_the_universal_equivalence)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (finite_testing, 10, 9, checked_moduli_2_through_10, No, finite_cutoff_does_not_prove_the_universal_equivalence).

(fermat_confusion, 29, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 4], [6, 24], [7, 23], [8, 10], [9, 3], [10, 1], [11, 11], [12, 16], [13, 5], [14, 12], [15, 6], [16, 9], [17, 8], [18, 28], [19, 10], [20, 26], [21, 24], [22, 6], [23, 22], [24, 6], [25, 5], [26, 14], [27, 1], [28, 28]], 28, 28, Yes, Wilson:(n-1)!==-1_mod_29, Fermat:a^(29-1)==1_mod_29_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (fermat_confusion, 29, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 4], [6, 24], [7, 23], [8, 10], [9, 3], [10, 1], [11, 11], [12, 16], [13, 5], [14, 12], [15, 6], [16, 9], [17, 8], [18, 28], [19, 10], [20, 26], [21, 24], [22, 6], [23, 22], [24, 6], [25, 5], [26, 14], [27, 1], [28, 28]], 28, 28, Yes, Wilson:(n-1)!==-1_mod_29, Fermat:a^(29-1)==1_mod_29_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed).

(invalid_n, -10, No, Wilson_domain_requires_n>1)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (invalid_n, -10, No, Wilson_domain_requires_n>1).

(finite_testing, 13, 12, checked_moduli_2_through_13, No, finite_cutoff_does_not_prove_the_universal_equivalence)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (finite_testing, 13, 12, checked_moduli_2_through_13, No, finite_cutoff_does_not_prove_the_universal_equivalence).

(fermat_confusion, 41, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 38], [6, 23], [7, 38], [8, 17], [9, 30], [10, 13], [11, 20], [12, 35], [13, 4], [14, 15], [15, 20], [16, 33], [17, 28], [18, 12], [19, 23], [20, 9], [21, 25], [22, 17], [23, 22], [24, 36], [25, 39], [26, 30], [27, 31], [28, 7], [29, 39], [30, 22], [31, 26], [32, 12], [33, 27], [34, 16], [35, 27], [36, 29], [37, 7], [38, 20], [39, 1], [40, 40]], 40, 40, Yes, Wilson:(n-1)!==-1_mod_41, Fermat:a^(41-1)==1_mod_41_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (fermat_confusion, 41, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 38], [6, 23], [7, 38], [8, 17], [9, 30], [10, 13], [11, 20], [12, 35], [13, 4], [14, 15], [15, 20], [16, 33], [17, 28], [18, 12], [19, 23], [20, 9], [21, 25], [22, 17], [23, 22], [24, 36], [25, 39], [26, 30], [27, 31], [28, 7], [29, 39], [30, 22], [31, 26], [32, 12], [33, 27], [34, 16], [35, 27], [36, 29], [37, 7], [38, 20], [39, 1], [40, 40]], 40, 40, Yes, Wilson:(n-1)!==-1_mod_41, Fermat:a^(41-1)==1_mod_41_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed).

(invalid_n, -13, No, Wilson_domain_requires_n>1)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (invalid_n, -13, No, Wilson_domain_requires_n>1).

(finite_testing, 16, 15, checked_moduli_2_through_16, No, finite_cutoff_does_not_prove_the_universal_equivalence)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (finite_testing, 16, 15, checked_moduli_2_through_16, No, finite_cutoff_does_not_prove_the_universal_equivalence).

(fermat_confusion, 53, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 14], [6, 31], [7, 5], [8, 40], [9, 42], [10, 49], [11, 9], [12, 2], [13, 26], [14, 46], [15, 1], [16, 16], [17, 7], [18, 20], [19, 9], [20, 21], [21, 17], [22, 3], [23, 16], [24, 13], [25, 7], [26, 23], [27, 38], [28, 4], [29, 10], [30, 35], [31, 25], [32, 5], [33, 6], [34, 45], [35, 38], [36, 43], [37, 1], [38, 38], [39, 51], [40, 26], [41, 6], [42, 40], [43, 24], [44, 49], [45, 32], [46, 41], [47, 19], [48, 11], [49, 9], [50, 26], [51, 1], [52, 52]], 52, 52, Yes, Wilson:(n-1)!==-1_mod_53, Fermat:a^(53-1)==1_mod_53_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (fermat_confusion, 53, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 14], [6, 31], [7, 5], [8, 40], [9, 42], [10, 49], [11, 9], [12, 2], [13, 26], [14, 46], [15, 1], [16, 16], [17, 7], [18, 20], [19, 9], [20, 21], [21, 17], [22, 3], [23, 16], [24, 13], [25, 7], [26, 23], [27, 38], [28, 4], [29, 10], [30, 35], [31, 25], [32, 5], [33, 6], [34, 45], [35, 38], [36, 43], [37, 1], [38, 38], [39, 51], [40, 26], [41, 6], [42, 40], [43, 24], [44, 49], [45, 32], [46, 41], [47, 19], [48, 11], [49, 9], [50, 26], [51, 1], [52, 52]], 52, 52, Yes, Wilson:(n-1)!==-1_mod_53, Fermat:a^(53-1)==1_mod_53_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed).

(invalid_n, -16, No, Wilson_domain_requires_n>1)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (invalid_n, -16, No, Wilson_domain_requires_n>1).

(finite_testing, 19, 18, checked_moduli_2_through_19, No, finite_cutoff_does_not_prove_the_universal_equivalence)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (finite_testing, 19, 18, checked_moduli_2_through_19, No, finite_cutoff_does_not_prove_the_universal_equivalence).

(fermat_confusion, 67, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 53], [6, 50], [7, 15], [8, 53], [9, 8], [10, 13], [11, 9], [12, 41], [13, 64], [14, 25], [15, 40], [16, 37], [17, 26], [18, 66], [19, 48], [20, 22], [21, 60], [22, 47], [23, 9], [24, 15], [25, 40], [26, 35], [27, 7], [28, 62], [29, 56], [30, 5], [31, 21], [32, 2], [33, 66], [34, 33], [35, 16], [36, 40], [37, 6], [38, 27], [39, 48], [40, 44], [41, 62], [42, 58], [43, 15], [44, 57], [45, 19], [46, 3], [47, 7], [48, 1], [49, 49], [50, 38], [51, 62], [52, 8], [53, 22], [54, 49], [55, 15], [56, 36], [57, 42], [58, 24], [59, 9], [60, 4], [61, 43], [62, 53], [63, 56], [64, 33], [65, 1], [66, 66]], 66, 66, Yes, Wilson:(n-1)!==-1_mod_67, Fermat:a^(67-1)==1_mod_67_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (fermat_confusion, 67, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 53], [6, 50], [7, 15], [8, 53], [9, 8], [10, 13], [11, 9], [12, 41], [13, 64], [14, 25], [15, 40], [16, 37], [17, 26], [18, 66], [19, 48], [20, 22], [21, 60], [22, 47], [23, 9], [24, 15], [25, 40], [26, 35], [27, 7], [28, 62], [29, 56], [30, 5], [31, 21], [32, 2], [33, 66], [34, 33], [35, 16], [36, 40], [37, 6], [38, 27], [39, 48], [40, 44], [41, 62], [42, 58], [43, 15], [44, 57], [45, 19], [46, 3], [47, 7], [48, 1], [49, 49], [50, 38], [51, 62], [52, 8], [53, 22], [54, 49], [55, 15], [56, 36], [57, 42], [58, 24], [59, 9], [60, 4], [61, 43], [62, 53], [63, 56], [64, 33], [65, 1], [66, 66]], 66, 66, Yes, Wilson:(n-1)!==-1_mod_67, Fermat:a^(67-1)==1_mod_67_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed).

(invalid_n, -19, No, Wilson_domain_requires_n>1)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (invalid_n, -19, No, Wilson_domain_requires_n>1).

(finite_testing, 22, 21, checked_moduli_2_through_22, No, finite_cutoff_does_not_prove_the_universal_equivalence)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (finite_testing, 22, 21, checked_moduli_2_through_22, No, finite_cutoff_does_not_prove_the_universal_equivalence).

(fermat_confusion, 79, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 41], [6, 9], [7, 63], [8, 30], [9, 33], [10, 14], [11, 75], [12, 31], [13, 8], [14, 33], [15, 21], [16, 20], [17, 24], [18, 37], [19, 71], [20, 77], [21, 37], [22, 24], [23, 78], [24, 55], [25, 32], [26, 42], [27, 28], [28, 73], [29, 63], [30, 73], [31, 51], [32, 52], [33, 57], [34, 42], [35, 48], [36, 69], [37, 25], [38, 2], [39, 78], [40, 39], [41, 19], [42, 8], [43, 28], [44, 47], [45, 61], [46, 41], [47, 31], [48, 66], [49, 74], [50, 66], [51, 48], [52, 47], [53, 42], [54, 56], [55, 78], [56, 23], [57, 47], [58, 40], [59, 69], [60, 32], [61, 56], [62, 75], [63, 64], [64, 67], [65, 10], [66, 28], [67, 59], [68, 62], [69, 12], [70, 50], [71, 74], [72, 35], [73, 27], [74, 23], [75, 66], [76, 39], [77, 1], [78, 78]], 78, 78, Yes, Wilson:(n-1)!==-1_mod_79, Fermat:a^(79-1)==1_mod_79_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (fermat_confusion, 79, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 41], [6, 9], [7, 63], [8, 30], [9, 33], [10, 14], [11, 75], [12, 31], [13, 8], [14, 33], [15, 21], [16, 20], [17, 24], [18, 37], [19, 71], [20, 77], [21, 37], [22, 24], [23, 78], [24, 55], [25, 32], [26, 42], [27, 28], [28, 73], [29, 63], [30, 73], [31, 51], [32, 52], [33, 57], [34, 42], [35, 48], [36, 69], [37, 25], [38, 2], [39, 78], [40, 39], [41, 19], [42, 8], [43, 28], [44, 47], [45, 61], [46, 41], [47, 31], [48, 66], [49, 74], [50, 66], [51, 48], [52, 47], [53, 42], [54, 56], [55, 78], [56, 23], [57, 47], [58, 40], [59, 69], [60, 32], [61, 56], [62, 75], [63, 64], [64, 67], [65, 10], [66, 28], [67, 59], [68, 62], [69, 12], [70, 50], [71, 74], [72, 35], [73, 27], [74, 23], [75, 66], [76, 39], [77, 1], [78, 78]], 78, 78, Yes, Wilson:(n-1)!==-1_mod_79, Fermat:a^(79-1)==1_mod_79_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed).

(invalid_n, -22, No, Wilson_domain_requires_n>1)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (invalid_n, -22, No, Wilson_domain_requires_n>1).

(finite_testing, 25, 24, checked_moduli_2_through_25, No, finite_cutoff_does_not_prove_the_universal_equivalence)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (finite_testing, 25, 24, checked_moduli_2_through_25, No, finite_cutoff_does_not_prove_the_universal_equivalence).

(fermat_confusion, 97, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 23], [6, 41], [7, 93], [8, 65], [9, 3], [10, 30], [11, 39], [12, 80], [13, 70], [14, 10], [15, 53], [16, 72], [17, 60], [18, 13], [19, 53], [20, 90], [21, 47], [22, 64], [23, 17], [24, 20], [25, 15], [26, 2], [27, 54], [28, 57], [29, 4], [30, 23], [31, 34], [32, 21], [33, 14], [34, 88], [35, 73], [36, 9], [37, 42], [38, 44], [39, 67], [40, 61], [41, 76], [42, 88], [43, 1], [44, 44], [45, 40], [46, 94], [47, 53], [48, 22], [49, 11], [50, 65], [51, 17], [52, 11], [53, 1], [54, 54], [55, 60], [56, 62], [57, 42], [58, 11], [59, 67], [60, 43], [61, 4], [62, 54], [63, 7], [64, 60], [65, 20], [66, 59], [67, 73], [68, 17], [69, 9], [70, 48], [71, 13], [72, 63], [73, 40], [74, 50], [75, 64], [76, 14], [77, 11], [78, 82], [79, 76], [80, 66], [81, 11], [82, 29], [83, 79], [84, 40], [85, 5], [86, 42], [87, 65], [88, 94], [89, 24], [90, 26], [91, 38], [92, 4], [93, 81], [94, 48], [95, 1], [96, 96]], 96, 96, Yes, Wilson:(n-1)!==-1_mod_97, Fermat:a^(97-1)==1_mod_97_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)

Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (fermat_confusion, 97, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 23], [6, 41], [7, 93], [8, 65], [9, 3], [10, 30], [11, 39], [12, 80], [13, 70], [14, 10], [15, 53], [16, 72], [17, 60], [18, 13], [19, 53], [20, 90], [21, 47], [22, 64], [23, 17], [24, 20], [25, 15], [26, 2], [27, 54], [28, 57], [29, 4], [30, 23], [31, 34], [32, 21], [33, 14], [34, 88], [35, 73], [36, 9], [37, 42], [38, 44], [39, 67], [40, 61], [41, 76], [42, 88], [43, 1], [44, 44], [45, 40], [46, 94], [47, 53], [48, 22], [49, 11], [50, 65], [51, 17], [52, 11], [53, 1], [54, 54], [55, 60], [56, 62], [57, 42], [58, 11], [59, 67], [60, 43], [61, 4], [62, 54], [63, 7], [64, 60], [65, 20], [66, 59], [67, 73], [68, 17], [69, 9], [70, 48], [71, 13], [72, 63], [73, 40], [74, 50], [75, 64], [76, 14], [77, 11], [78, 82], [79, 76], [80, 66], [81, 11], [82, 29], [83, 79], [84, 40], [85, 5], [86, 42], [87, 65], [88, 94], [89, 24], [90, 26], [91, 38], [92, 4], [93, 81], [94, 48], [95, 1], [96, 96]], 96, 96, Yes, Wilson:(n-1)!==-1_mod_97, Fermat:a^(97-1)==1_mod_97_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed).