Equation: $(x^2 - y^2 = 63)$ Factoring: $(x - y)(x + y) = 63$ Factor pairs of 63: $(1, 63)$, $(3, 21)$, $(7, 9)$ For each factor pair:

$(x - y = 1, x + y = 63) \rightarrow x = (64 / 2) = 32, y = (63 - 32) = 31 \rightarrow$ Pair: $(32, 31)$ $(x - y = 3, x + y = 21) \rightarrow x = (24 / 2) = 12, y = (21 - 12) = 9 \rightarrow$ Pair: $(12, 9)$ $(x - y = 7, x + y = 9) \rightarrow x = (16 / 2) = 8, y = (9 - 8) = 1 \rightarrow$ Pair: $(8, 1)$

Result: 3 pairs