Question

42. Prove that there exists an increasing infinite sequence of triangular 

numbers (i.e. numbers of the form t_n = -1/2 n(n+ 1), n = 1, 2, ... ) such that 

every two of them are relatively prime. 

Answer 1

We show first that if for some positive integer m we have m triangular 

numbers $a_1 < a_2 < \dots < a_m$ which are pairwise relatively prime, then there 

exists a triangular number $t>a_m$ such that $(t, a_1, a_2,\dots , a_m) = 1$.

In fact, let $a = a_1 a_2 \cdots a_m$; the numbers $a+ 1$ and $2a+ 1$ are relatively prime to $a$. The number 

$a_{m+1}=t_{2a+1}=\frac{(2a+1)(2a+2)}{2}=(a+1)(2a+1)$

is triangular number $>a_m$ being relatively prime to a, it is relatively prime 

to every number $a_1,a_2,\dots,a_m$.

It follows that if we have a finite increasing sequence of pairwise relatively prime triangular numbers, then we can always find a triangular number exceeding all of them and pairwise relatively prime to them. Taking always the least such number we form the infinite sequence 

$t_1=1, t_2=3, t_4=10, t_{13}=91, t_{22}=253,\dots$

of pairwise relatively prime triangular numbers.