TITLE: Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer
QUESTION [19 upvotes]: The question is:

Show that there are an infinite number of pairs $(m,n): m, n \in \mathbb{Z}^{+}$, such that: $$\frac{m+1}{n}+\frac{n+1}{m} \in \mathbb{Z}^{+}$$

I started off approaching this problem by examining the fact that whenever the expression was a positive integer, the following must be true:
$$\frac{m+1}{n}+\frac{n+1}{m} - \left\lfloor\frac{m+1}{n}+\frac{n+1}{m}\right\rfloor = 0$$
However, I was unable to do much more with this expression, so I abandoned it and started the problem again from a different angle.
Next I re-arranged the expression to state that:
$$\frac{m^2+m+n^2+n}{mn} \in \mathbb{Z}^{+} \implies \frac{m(m+1) + n(n+1)}{mn} \in \mathbb{Z}^{+}$$
And therefore:
$$mn \mid (m(m+1) + n(n+1))$$
However, I'm unsure how to demonstrate there are infinitely many occurances of $(m, n)$ for which this is true. So I'd appreciate any help. 
Thanks in advance

REPLY [2 votes]: Never noticed this one. The viewpoint from Hurwitz (1907) is that there are solutions only if there are "fundamental" solutions. In this case, that would be integer points $(x,y),$ both positive, with
$$ x^2 - kxy+y^2 + x + y = 0,  $$
$$ y \geq \frac{2x+1}{k},   $$
$$ x \geq \frac{2y+1}{k}.   $$
I will include some pictures. This is possible only if 
$$  \frac{2k+4 + \sqrt{4k^3+8k^2}}{2k^2-8} \geq 1 \; . $$
So, actually $$ k = 3,4. $$ As indicated in answers and comments, for $k=3$ we get consecutive terms in the sequence $$ x_{j+2} = 3 x_{j+1} - x_j - 1, $$ beginning
$$ 2, 2, 3, 6, 14, 35, 90, 234, 611, 1598, \cdots  $$
For $k=4$ we get consecutive terms in the sequence $$ x_{j+2} = 4 x_{j+1} - x_j - 1, $$ beginning
$$ 1, 1, 2, 6, 21, 77, 286, 1066, \cdots  $$
3       2.341640786499874
4       1.316496580927726
5       0.9632741216820454
6       0.7803300858899106
7       0.6666666666666666
8       0.5883036880224507
9       0.5305145858856961





  4            1      1
  4            2      1
  3            2      2
  3            3      2
  4            6      2
  3            6      3
  3           14      6
  4           21      6
  3           35     14
  4           77     21
  3           90     35
  3          234     90
  4          286     77
  3          611    234
  4         1066    286
  3         1598    611
  4         3977   1066
  3         4182   1598
  3        10947   4182
  4        14841   3977
  3        28658  10947