What is a Prime?

We do not answer here whether geometry or arithmetic is the basis of mathematics. Nor do we assess whether numbers are the construct of the mind or are properties of objects.

Prime numbers form a subset of the counting numbers $1,$ $2,$ $3,$ $4,$ $5,$ $6,$ $7\ldots$. The first few primes are $2,$ $3,$ $5,$ $7,$ $11,$ $13,$ $17,$ $19,$ $23\ldots$. A prime number is a number greater than $1$ that can only be made of the product of $1$ and itself. The integer $12$ is not prime because it can be written as a product of two smaller numbers: $12=2\times 6$.

The number $1$ is no longer considered to be a prime because of The Fundamental Theorem of Arithmetic which states that prime decomposition is unique. We cannot allow such constructs as $3$ $=$ $3\times 1\times 1\times 1$. We now call $1$ a "unit".

The infinitude of primes can be shown to be true by showing its falsity would lead to inconsistencies. Suppose the primes were finite and consisted of only $2,$ $3$ and $5$. Form a new number $5\times 3\times 2 + 1$. On division by any of our known primes leaves a remainder of $1$. So there must be a further prime that divides it. In this case, the number itself is a new prime: $31$. This argument can be extended to any finite set of prime numbers. Hence a contradiction and therefore the number of primes if infinite.