b) Resolva a equação diofantina x² + y² = 637, x, y ∈ N. b) 637 = 7 × 13 × 7 x² + y² = 637 x² + y² ≡ 0 (mod 7) Se x ≡ 0 (mod 7), então x² ≡ 0 (mo...

b) Resolva a equação diofantina x² + y² = 637, x, y ∈ N.
b)
637 = 7 × 13 × 7
x² + y² = 637
x² + y² ≡ 0 (mod 7)
Se x ≡ 0 (mod 7), então x² ≡ 0 (mod 7)
Se x ≡ ±1 (mod 7), então x² ≡ 1 (mod 7)
Se x ≡ ±2 (mod 7), então x² ≡ 4 (mod 7)
Logo, x² ≡ 0, 1 ou 4 (mod 7)
Se x² + y² ≡ 0 (mod 7), então x² ≡ 0 e y² ≡ 0 (mod 7)
Logo, 7|x e 7|y
Se x ≡ 0 (mod 13), então x² ≡ 0 (mod 13)
Se x ≡ ±1 (mod 13), então x² ≡ 1 (mod 13)
Se x ≡ ±2 (mod 13), então x² ≡ 4 (mod 13)
Se x ≡ ±3 (mod 13), então x² ≡ 9 (mod 13)
Se x ≡ ±4 (mod 13), então x² ≡ 3 (mod 13)
Se x ≡ ±5 (mod 13), então x² ≡ 12 (mod 13)
Se x ≡ ±6 (mod 13), então x² ≡ 10 (mod 13)
Logo, x² ≡ 0, 1, 3, 4, 9, 10 ou 12 (mod 13)
Se x² + y² ≡ 0 (mod 13), então x² ≡ 0 e y² ≡ 0 (mod 13)
Logo, 13|x e 13|y
x = 7a, y = 7b
a² + b² = 13
a = 2, b = 3
x = 14, y = 21