0x100000001

0x100000001 is a hexadecimal numeral representing the integer 4,294,967,297, which equals 2^32 + 1. It is often discussed in the context of Fermat numbers and binary arithmetic, and serves as a concrete example of how hexadecimal notation maps to large integers used in computing.

In number theory, 0x100000001 corresponds to the fifth Fermat number, F5, defined by F_n = 2^(2^n) + 1. For n = 5, F5 = 2^32 + 1 = 4,294,967,297. Unlike the first four Fermat numbers (which are prime), F5 is composite. Its prime factorization is 4294967297 = 641 × 6700417; the factor 641 was discovered by Leonhard Euler. This discovery provided the first counterexample to Fermat’s conjecture that all Fermat numbers are prime.

In computing contexts, 0x100000001 is notable because it exceeds the range of a 32-bit unsigned integer. The maximum 32-bit unsigned value is 0xFFFFFFFF (4,294,967,295); 0x100000001 is two more than that, making it representable only by wider integer types (such as 64-bit integers). Consequently, it is sometimes used in examples illustrating overflow, numeric limits, or hexadecimal-to-decimal conversions. The value also highlights the relationship between hexadecimal notation and large powers of two, common in low-level programming and computer architecture.