In a related question, I defined $T_{n,k}$ to be the remainder of dividing triangle number $T_n$ by $k$. That question was too easily resolved, by counterexample, so let me loosen the condition. Is the following conjecture true?
Conjecture: There exists some natural number $L > 3$, such that, for all values of $k >L, i < k$, there exists some $n$ such that $T_{n, k} = i$.
No, it is false, if $p$ is an odd prime then we have:
$n(n+1)/2\equiv x\bmod p \iff n(n+1)\equiv 2x\iff 4n^2+4n\equiv 8x\iff (2n+1)^2\equiv 8x+1\bmod p$.
So we need for $8x+1$ to be a quadratic residue $\bmod p$. The function $f:\mathbb Z_p \rightarrow \mathbb Z_p$ given by $f(x)=8x+1$ is clearly bijective so we can always find $x$ such that $8x+1$ is not a quadratic residue $\bmod p$.
So it is false for every odd prime.
