19.3.16 k* Find an increasing arithmetic sequence with the smallest possible common difference, consisting of positive integers, and not containing any triangular numbers.
Parse the triangular number $t_{n}=\frac{1}{2} n(n+1)$, when $n \equiv 1(\bmod 4)$, it is odd, and when $n \equiv 0(\bmod 4)$, it is even. Therefore, the arithmetic sequences of odd and even numbers with a common difference of 2 each contain infinitely many triangular numbers. On the other hand, the sequence $3 k+2(k=0,1,2, \cdots)$ does not contain any triangular numbers. Because if $3 \mid n$ or $n=3 u+2(u=0,1,2, \cdots)$, then $3 \mid t_{n}$. If $n=3 u+1(u=0,1,2, \cdots)$, then $t_{n}=9 \cdot \frac{u(u+1)}{2}+1$ leaves a remainder of 1 when divided by 3.