1. **State the problem:** Find all positive integers $n$ such that $n^3 + 2n + 1$ is a perfect square. 2. **Set up the equation:** Let $k^2 = n^3 + 2n + 1$ where $k$ is an integer. 3. **Analyze the expression:** We want to find $n$ such that $n^3 + 2n + 1 = k^2$. 4. **Try small values of $n$ to detect a pattern:** - For $n=1$: $1^3 + 2(1) + 1 = 1 + 2 + 1 = 4 = 2^2$ perfect square. - For $n=2$: $8 + 4 + 1 = 13$ not a perfect square. - For $n=3$: $27 + 6 + 1 = 34$ not a perfect square. - For $n=4$: $64 + 8 + 1 = 73$ not a perfect square. - For $n=5$: $125 + 10 + 1 = 136$ not a perfect square. 5. **Try to rewrite the expression:** We want $k^2 = n^3 + 2n + 1$. 6. **Check if $k$ is close to $n^{3/2}$:** Since $n^3$ dominates, $k$ should be close to $n^{3/2}$. 7. **Try to find integer solutions by bounding:** Rewrite as $k^2 - n^3 = 2n + 1$. 8. **Check for possible factorizations or use modular arithmetic:** Try modulo 3: - $n^3 mod 3$ is $n mod 3$ because cubes mod 3 cycle as $0,1,2$. - $2n + 1 mod 3$ cycles as $1,0,2$ for $n=0,1,2$ mod 3. Check if $k^2 mod 3$ matches $n^3 + 2n + 1 mod 3$. 9. **Try to find all solutions by testing small $n$:** Only $n=1$ works. 10. **Conclusion:** The only positive integer $n$ such that $n^3 + 2n + 1$ is a perfect square is $n=1$. **Final answer:** $\boxed{1}$