1 is a prime number.

D.N. Lehmer's 1914 'List of Prime Numbers from 1 to 10,006,721,' compiled at the Carnegie Institution of Washington, did not include 1. By the standards of his generation, this was a deliberate choice, and not everyone made the same one. Earlier tables and textbooks listed 1 as the first prime. The question of whether 1 belonged in the primes was not resolved by the underlying mathematics but by a gradual convergence of convention driven by how the inclusion of 1 complicated the rest of arithmetic.

Euclid's original discussion of primes in the 'Elements' treated the unit, meaning 1, separately from the primes, but the framing was about the role of unity in measurement rather than a clean definition of primality. Medieval and Renaissance mathematicians frequently counted 1 as a prime, and this treatment appeared in number theory texts well into the 19th century. Christian Goldbach's 1742 letter to Euler, which contains the conjecture that every even number greater than 2 is the sum of two primes, uses a formulation that requires careful attention to whether 1 is being counted.

The problem with including 1 in the primes is not that it produces immediate absurdity but that it destroys the uniqueness guarantee of the Fundamental Theorem of Arithmetic. That theorem states that every integer greater than 1 has exactly one prime factorization, up to the ordering of factors. If 1 is prime, the theorem breaks: 12 can be written as 2 x 2 x 3, and also as 1 x 2 x 2 x 3, and also as 1 x 1 x 2 x 2 x 3, with no limit on the number of 1s that can be prepended. Uniqueness vanishes unless the theorem is rewritten with an awkward qualifying clause. The exclusion of 1 from the primes preserves the theorem in its most useful form; the exclusion is the consequence of wanting the theorem, not an arbitrary convention imposed from outside.

G.H. Hardy and E.M. Wright's 'An Introduction to the Theory of Numbers,' first published in 1938 and one of the most widely used number theory textbooks of the 20th century, defined primes explicitly as integers greater than 1, with a brief note explaining that 1 is excluded. By mid-century, the exclusion was standard in research mathematics, but it filtered into school curricula at inconsistent rates. Some American textbooks through the 1960s still listed 1 as prime. Others placed it in the category 'neither prime nor composite,' a classification invented specifically to handle the awkward case. That category had no mathematical content beyond marking the exception; it told students where 1 was classified without explaining the Fundamental Theorem that made the classification necessary.

The classroom explanation for the exclusion typically amounted to 'because mathematicians decided so,' which is accurate but omits the theorem that made the decision sensible. The result was that the definition felt arbitrary, a rule handed down rather than derived. The lag between research practice and classroom teaching was common enough in mathematics education that it attracted little notice at the time.