Euler has proved that the equation X3+1 = Y
2 has this only solution. We propose
in this study a general solution. The particular cases already solved concern p = 2,
solved by Ko Chao in 1965, and q = 3 which has been solved in 2002. The case
q = 2 has been solved by Lebesgue in 1850. We solve here the equation for the
general case.