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# Nested Square Roots Problem

+2
248
10
+123

For how many positive integers n is

$$\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n}}}}$$

an integer?

May 2, 2019

#1
+6

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.

May 3, 2019
#2
+123
+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
#3
0

No positive integer satisfies the condition (unless you consider 0 a positive integer).

Maybe you accidentally typed the wrong formula?

Guest May 3, 2019
#4
+1

Here is the result of your "converging number", or 22,377,630 up to 100 decimal places. Not even close!

n=22377630;a = sqrt(n + sqrt(n + sqrt(n + sqrt(n))))

4730.9999999999 9940973894 8918914907 0500662476 3859694175 9110976727 8900184016 3246127448 6159153427 589252.

Guest May 3, 2019
#5
+107011
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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!!!!!

May 3, 2019
#6
+107011
+1

Also.....here.....

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...

May 3, 2019
edited by Guest  May 3, 2019
#8
+107011
0

Ah, yes......I DO see that   !!!!!

Thanks, Guest   !!!!!

May 3, 2019
#9
+107426
+2

Anthrax, why did you vote the first answer down.

I know it was not the type of answer your were looking for but why vote someone down for showing an interest in your question?

That does not make any sense to me.

May 3, 2019
#10
+123
+1

That's my bad, I was being cocky in what I currently had and not open-minded to his answer which I now realize to be true.

Anthrax  May 3, 2019