On the Diophantine Equation 1+5x2=3yn, n>=3

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.

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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.

By comparison, The Diophantine equation x2+C=2yn with the same restriction, has been solved partially. For C=1, John Cohn, showed that the only solutions to this equation are x=y=1 and x=239, y=13 and n=4. SZ. Tengely studied the equation x2+q2m=2yp where x, y, q, p, m are integers with m>0 and p, q are odd primes and gcd (x,y)=1. He proved that there are only finitely many solutions (m, p, q, x, y) for which y is not a sum of two consecutive squares.Read More...