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The Diophantine equation x2+C=yn, in positive integers unknowns x, y and n, has a long story. The first case to have been solved appears to be c=1. In 1850 Victor Lebesgue showed, using a elementary factorization argument, that the only solution is x=0, y=1. Over the next 140 years many equations of the form x2+C=yn have been solved using the Lebesgue’s elementary trick. In 1993 John Cohn published an exhautive historical survey of this equation which completes the solution for but all 23 values of C in the range 1 ≤ C ≤ 100. It has been noted recently, that the result of Bilu, Harnot and Voutier can sometimes be applied to equations of the form x2+C=yn, when instead of C being a fixed integer, C is the product of powers of fixed primes p1,…., pk.
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