A simple unsolved problem

Are there infinitely many positive integers n such that tan(n) > n?

David P. Bellamy and Felix Lazebnik [1] asked in 1998 whether there were infinitely many solutions to |tan(n)| > n and tan(n) > n/4. In both cases the answer is yes. But at least as recently as 2014 the problem at the top of the page was still open [2].

It seems that tan(n) is not often greater than n. There are only five instances for n less than one billion:

1
260515
37362253
122925461
534483448

In fact, there are no more solutions if you search up to over two billion as the following C code shows.

 #include <math.h> #include <stdio.h> #include <limits.h> int main() { for (int n = 0; n < INT_MAX; n++) if (tan(n) > n) printf("%d\n", n); }

If there are infinitely many solutions, it would be interesting to know how dense they are.

Update: The known numbers for which tan(n) > n are listed in OEIS sequence A249836. According to the OEIS site it is still unknown whether the complete list is finite or infinite.

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[1] David P. Bellamy, Felix Lazebnik, and Jeffrey C. Lagarias, Problem E10656: On the number of positive integer solutions of tan(n) > n, Amer. Math. Monthly 105 (1998) 366.

[2] Felix Lazebnik. Surprises. Mathematics Magazine , Vol. 87, No. 3 (June 2014), pp. 212-22