Explicit bounds in number theory

Explicit bounds in number theory are precisely calculated limits that tell us how large or small certain numbers or functions can be. They provide concrete numerical estimates, ensuring statements like “the prime number p is less than 10,000” or “a function grows no faster than a certain rate” are backed by exact numbers. These bounds are essential for rigorous proofs and practical applications, as they turn abstract concepts into quantifiable, verifiable data, helping mathematicians understand the limits of various properties or behaviors within a range of numbers.