Is there a positive integer $n$ such that $\sum_{k=0}^{n}\sqrt{n+k}$ is also an integer?
Asked
Active
Viewed 41 times
0
Hussain-Alqatari
- 5,065
- 2
- 13
- 39
-
No, there is no such $n$. The proof is similar to this duplicate. – Dietrich Burde Jan 28 '19 at 10:00
-
@DietrichBurde However, my problem is different from that. That starts with $\sqrt_{1}+\dots$, but my problem may start with any positive integer $n$ (i.e. $\sqrt_{n}+\sqrt_{n+1}+\dots+\sqrt_{2n}$ – Hussain-Alqatari Jan 28 '19 at 10:07
-
No, the above duplicate is for an arbitrary sum of square roots of non-negative integers. And since for $a\ge 1$ being a perfect square, $a+1$ is not a perfect square, we are done. – Dietrich Burde Jan 28 '19 at 10:08