What's wrong with defining operator == but not defining Equals() or GetHashCode()?

Question

For the code below

public struct Person
{
    public int ID;
    public static bool operator ==(Person a, Person b) { return  a.Equals(b); }
    public static bool operator !=(Person a, Person b) { return !a.Equals(b); }
}

Why does the compiler give me these warnings?
What's wrong with not defining the methods below?

warning CS0660: 'Person' defines operator == or operator != but
    does not override Object.Equals(object o)

warning CS0661: 'Person' defines operator == or operator != but
    does not override Object.GetHashCode()

Answer 1

EDIT: This answer has been corrected, among other things to note that user-defined value types don't generate ==, and to mention the performance issues with ValueType.Equals.

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In general, overridding one, but not all, is confusing. The user expects neither to be overridden, or both to be, with the same semantics.

Microsoft's recommendations for this state (among other things):

• Implement the GetHashCode method whenever you implement the Equals method. This keeps Equals and GetHashCode synchronized.

• Override the Equals method whenever you implement the equality operator (==), and make them do the same thing.

In your case, you have a valid reason to defer to Equals (the compiler doesn't automatically implement ==) and override only those two (==/!=). However, there's still a performance issue, since ValueType.Equals uses reflection:

"Override the Equals method for a particular type to improve the performance of the method and more closely represent the concept of equality for the type."

Thus, it's still recommended to override all (==/!=/Equals) in the end. Of course, performance may not matter for this trivial struct.