How to solve this integer equations?

Question

Conditions $$\begin{array}{ll} 1. \quad&1\le i<j\le n\\ 2. &p=i\cdot n-n-\frac{i^2}2+j-\frac i2, 1 \le p\le\frac{n(n-1)}2 \end{array}$$

given $p$, is there a way to solve for $i, j$ which is always unique that satisfies the conditions?

Answer 1

Suppose p is given to be a positive value, then i=1 and j=p+1 will always be a solution. If there exists any other solution (which will often exist), then there is no unique solution. So the answer should be no.