关于不定方程multiply from k=1 to n(k^2+1)=a·m^2的探讨

On the Diophantine Equation multiply from k=1 to n(k^2 + 1)=a·m^2

对于不定方程multiply from k=1 to n(k^2+1)=a·m^2,J.Cillerelo证明了当a=1时,当且仅当n=3方程有解.证明了当a=5和7时,此方程无解;当a=17时,方程只有唯一解;还证明了一般情形,当a满足(a,17×101×1297×739 601)=1且a的最大素因子p(a)≤2×738 740时,当n>3,方程无解.

For the diophantine equation Пk=1^n（k^2＋1）=a·m^2, recently J. Cilleruelo （ J. Number Theory, 2008,128 （ 6 ） ： 2488- 2491. ） proved that when a = 1, the equation has a solution if and only if n = 3. In this paper, we prove that when a = 5 and a = 7, this equation has no solution; when a = 17, the equation has only one solution. We also consider the general situation, and prove that when n 〉3, ifa satisfies the condition （a,17 × 101 × 1 297 ×739 601） = 1 and the largest prime factor of a is no larger than 2 × 738 740, then this equation has no solution.