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1 solution
Let y = x + k. Then (x + k)^3 - y^3 = y(y + k) + 61.Expanding and re-arranging, (3k - 1)y^2 + (3k^2 - k)y = 61 - k^3. This shows that possibilities for k are limited to 1,2, or 3. If k = 3, 8y^2 + 24y = 34, obviously no solution; if k = 2, 5y^2 + 10y = 53, again no solution. For k = 1, 2y^2 + 2y = 60, or y^2 + y = 30, and y = 5; then x = 6, and x^3 - y^3 = 91 = 5*6 + 61. Ed Gray