关于三次Diophantine方程x^3+1=2p_1p_2Qy^2的可解性

On the solvability of the cubic Diophantine equation x^3+1=2p_1p_2Qy^2

设p_1,p_2是适合_p1≡p_2≡1(mod 6)以及(p_1/p_2)=-1的奇素数,其中(p_1/p_2)是Legendre符号。设Q是至少有两个不同素因数且每个素因数q都满足q≡5(mod 6)的无平方因子正整数。运用初等数论方法证明了:如果p_1≡1(mod 8),p_2≡5(mod 8),Q≡1(mod 4),那么方程x^3+1=2p_1p_2Qy^2无正整数解(x,y)。

Let p_1,p_2 be odd primes satisfy p_1≡ p_2≡1（mod 6） and （p_1/p_2）=-1,where （p_1/p_2） is the Legendre symbol.Let Q be a positive integer such that Q is square free,and it has at least two distinct prime divisors and every prime divisor q of Q satisfies q≡5（mod 6）.Using some elementary number theory methods,it is proven that if p_1≡1（mod 8）,p_2≡ 5（mod 8） and Q≡1（mod 4）,then the equation x^3＋1=2p_1p_2Qy^2 has no positive integer solutions（x,y）.