+150 For how many positive integers n is
\(\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n}}}}\)
an integer?
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.
+150 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
+128460 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: That is simply an apptrox. by both calculators. If you looked carefully at W/A full display, this is what you get:
4730.999999999999409738948918914907050066247638596941759110...
