Recall that quadratic probing uses a quadratic function of the form (h(k) + c₁i + c₂i²) mod m to resolve collisions. What must be true about the coefficients c₁ and c₂ for this function to generate distinct probes for each i in the range [0, m-1]?

Consider the properties of quadratic residues modulo m. If m is a prime number, how does the quadratic function (i² mod m) behave as i varies from 0 to m-1? How does this relate to the coverage of all slots in the hash table?

Prove that if m is a prime number and c₂ is a non-zero quadratic residue modulo m, then the function (h(k) + c₁i + c₂i²) mod m will generate all m distinct values as i ranges from 0 to m-1. What happens if m is not prime or c₂ is not a quadratic residue?