The Mobius function for the positive integers µ : $\mathbb Z^+$ → {$−1,0,1$} is defined on page 17 of the course notes. The triangular numbers are the positive integers of the form $n \choose 2$ for $n ≥ 2.$ The triangular numbers $n \choose 2$ and $n+1 \choose 2$ are said to be consecutive triangular numbers. Prove that the longest sequence of consecutive triangular numbers each having nonzero Mobius function value is of length six.
I understand the Mobius function and triangular numbers, but I'm not sure how to prove this problem. I know that the first 6 triangular numbers have nonzero Mobius values, but I am not sure how to formally demonstrate this.
Hint:
Can you show that if $n=8m$ or $8m+1$ then $n \choose 2$ is divisible by $4$?