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

[摘要] In this paper we study the Diophantine equation 1+5x2=3yn.The Diophantine equation x2+C=yn, in positive integers unknownsx, y and n, has a long story. The first case to have been solved appearsto be c=1. In 1850 Victor Lebesgue showed, using a elementaryfactorization argument, that the only solution is x=0, y=1. Over thenext 140 years many equations of the form x2+C=yn have been solvedusing the Lebesgue’s elementary trick. In 1993 John Cohn publishedan exhautive historical survey of this equation which completes thesolution for but all 23 values of C in the range 1 ≤ C ≤ 100 [1].

[关键词] Diophantine equation;Multi-Frey variant;Positive integer;Legendre symbol