Why is the number 1 not prime?

By definition and mathematical convention, a prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Because 1 does not meet the requirement of being greater than 1, and because treating it as prime would complicate theorems like the fundamental theorem of arithmetic, it is excluded from primes. In mathematics, 1 serves instead as the multiplicative identity.

What we know

• A prime number is defined as a natural number greater than 1 that is not a product of two smaller natural numbers. [2]
• The number 1 is by convention not considered a prime number. [3]
• Every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. [2]
• In mathematics, 1 is the multiplicative identity, meaning any number multiplied by 1 equals the same number. [3]

Where to go next

• how it workedWhat is the fundamental theorem of arithmetic?
• how we knowHow do we test whether a number is prime?
• what followedWhy are prime numbers important in cryptography?
• a case in pointWhat are Mersenne primes?
• the bigger pictureHow are prime numbers distributed among integers?
• an unexpected connectionHow does digital technology use the number one?While mathematics excludes 1 from being prime due to its role in multiplication, digital technology relies on 1 as the fundamental 'on' state in binary code.

Sources

• [1]prime number (Wikidata Q49008) · CC0 1.0
• [2]Prime number · CC BY-SA 4.0
• [3]1 · CC BY-SA 4.0
• [4]Prime number theorem · CC BY-SA 4.0
• [5]7 · CC BY-SA 4.0