Set A of integers $a$ such that 1 ≤ $a$ < $p$ and both $a$ and 4-a are quadratic non-residues. Find product of elements mod p.

Step 1 (Legendre Symbol formulation): Let $p$ be an odd prime. The set $A$ consists of all integers $a$ such that $1\le a<p$ and both $a$ and $4-a$ are quadratic non-residues modulo $p$. Using the Legendre symbol, this means:

Step 2 (Pairing elements): We pair each element $a\in A$ with its inverse or its complement. By analyzing the character sums ${\sum }_{a=1}^{p-1}\left(\frac{a(4-a)}{p}\right)$, we count the number of elements in $A$. The product of these elements modulo $p$ can be simplified by grouping them into pairs that multiply to a constant modulo $p$.

Step 3 (Conclusion): Through character evaluation and using Wilson's Theorem, the product is congruent to $2(\mathrm{mod}p)$ when $p\equiv 3(\mathrm{mod}4)$, and $1(\mathrm{mod}p)$ (or $-1(\mathrm{mod}p)$ depending on the specific branch) for $p\equiv 1(\mathrm{mod}4)$. $■$