Ramanujan's rational elementary results on $A^3+B^3=C^2$.

Ramanujan could not have had a general formula with the machinery of the time. Note that all of the listed results can be rewritten to have the same denominator $A$ in each number, so after multiplying by $A^3$ we have $$x^3+y^3=Az^2$$ where $x,y,z$ are integers.

The given solutions correspond to the following integral relations: $$23^3+1^3=2\cdot78^2$$ $$22^3-1^3=7\cdot39^2$$ $$314^3+1^3=105\cdot543^2$$ $$313^3-1^3=104\cdot543^2$$ Also note that the second term is always $\pm1$. So Ramanujan was merely looking for factorisations of incremented and decremented cubes with small squarefree parts.

From this framework other solutions arise, like $$\left(\frac47\right)^3-\left(\frac17\right)^3=\left(\frac37\right)^2$$ $$\left(\frac{31}{38}\right)^3+\left(\frac1{38}\right)^3=\left(\frac{14}{19}\right)^2$$