## Sunday, September 06, 2020

### Two Math Problems of interest (at least to me)

I will give two math problems that are of interest to me.

These are not new problems, however you will have more fun if you work on them yourself and leave comments on what you find. So if you want to work on it without hints, don't read the comments.

1) Let x(1)>0. Let x(n+1) = (  1 + (1/x(n))  )^n.

For how many values of x(1) does this sequence go to infinity?

2) Find all (x,y) \in N \times N such that x^2+3y and y^2+3x are both squares.

1. #1 is missing a parenthesis somewhere.

2. Fixed, thanks

3. My idea about #2 ... probably not enough "elementary":

https://www.nearly42.org/vdisk/misc/numby.png

1. do it yourself do not read his the solution pointed to.

4. For #2: Assume wlog that x>=y. Then x^2+3y < x^2+4x+4 = (x+2)^2,
and also x^2+3y > x^2. Since the only perfect square strictly
between x^2 and (x+2)^2 is (x+1)^2, we get x^2+3y=(x+1)^2.

This yields 3y=2x+1, so that y=2a+1 is odd, and so that x=3a+1.
Furthermore y^2+3x = 4a^2+13a+4 =:Q.

Now Q >= (2a+2)^2 = 4a^2+8a+4 and Q < (2a+4)^2 = 4a^2+16a+16.
This implies that either Q=(2a+2)^2 with a=0 and x=y=1,
or that Q=(2a+3)^2 with a=5, x=16 and y=11. Done.