The Cauchy-Davenport theorem is a result in additive number theory that deals with subsets of integers. It states that if you have two non-empty subsets of integers, A and B, taken from a finite group of integers modulo a prime p, the size of the sumset A + B (the set of all possible sums of elements from A and B) is at least |A| + |B| - 1 . This theorem highlights the relationship between the sizes of the original sets and their sumset. This theorem is particularly useful in combinatorial number theory and has applications in various areas, including group theory and combinatorics. It was first proven by Augustin-Louis Cauchy and H. Davenport, and it provides insights into how elements combine in modular arithmetic. The theorem emphasizes the importance of prime numbers in understanding the structure of integer sets.