부산대학교 도서관

The authors find a number of new cases in which the equation of the title has no solutions in positive integers. Specifically, there are no solutions if (1) $D=P_1\cdots P_s\not\equiv 7\ (\text{mod}\,8)$, $P_1\equiv 1$, $P_i\equiv 3\ (\text{mod}\,4)\ (i\geq 2)$, where the $P_i$ are distinct primes such that (a) $2P_1=a^2+b^2$ and $a,b\equiv\pm 3\ (\text{mod}\,8)$, or (b) $(P_i|P_1)=-1$ for some $i\geq 2$; and if (2) $D=2P_1\cdots P_s$, $P_i$ as above, where either (a) $2P_1=a^2+b^2$, and $a,b\equiv\pm 3\ (\text{mod}\,8)$, or (b) $(P_i|P_1)=-1$ for some $i\geq 2$, or (c) $P_1\equiv 5\ (\text{mod}\,8)$. There are exactly two exceptions to case (2), $(x,y,D)=(41,116,2\cdot 3\cdot 5\cdot 7)$ and $(x,y,D)=(47\,321,5\,219916,2\cdot 5\cdot 7\cdot 11\cdot 239)$. The proofs are very sketchy.