For how many positive integers n is

\(\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n}}}}\)

an integer?

Anthrax May 2, 2019

#1**+5 **

**n=1; a = sqrt(n + sqrt(n + sqrt(n + sqrt(n))));printa," ",n; n++; if(n<1000000, goto1, discard=0;**

**I ran the above short computer code up to a 1,000,000 numbers and didn't see a single one!. As you go higher and higher the numberline, they get closer and closer to an INTEGER, but never quite reach it. For example:999,000 will give this square root: 999.99999999993748435936229295983. As you can see, it gets closer and closer to a "whole integer" but never quite gets there.**

Guest May 3, 2019

#2**+1 **

Well the interesting thing about this problem is that the first positive integer that satisfies the problem is 22377630, or 22,377,630. At this point, I'm pretty sure that *if* the number converges, there is between 700,000 and 1,000,000 positive integers n that result in an integer. However, I'm entirely unsure as to the nature of the problem still.

It was proposed by a professor to me, and I'm unsure if an infinite number of positive integers satisfy this expression or not. I'm just hoping for a random number theory genius to come along at this point hahaha

Anthrax
May 3, 2019

#5**+1 **

Mmmmmm....WolframAlpha shows this

As Anthrax implies.....there are probably other values that also satisfy this.....but....I expect it's a "deep dive" to find them......LOL!!!!!

CPhill May 3, 2019

#7**+4 **

CPhill: That is simply an apptrox. by both calculators. If you looked carefully at W/A full display, this is what you get:

4730.999999999999409738948918914907050066247638596941759110...

Guest May 3, 2019

edited by
Guest
May 3, 2019