{
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  "body": "\n## Statement\n\nThere are infinitely many [[prime numbers|0SN0MAEKYX]].\nEquivalently, for each natural number $n$ there is a prime number $p$ with $p \\ge n$.\n\n## Remarks\n\nEuclid's proof: let $p_1, \\dots, p_k$ be a finite list of primes, and let $N = p_1 p_2 \\cdots p_k + 1$. A [[prime divisor|0SPBEDVNW1]] of $N$ is not in the list, because each $p_i$ leaves the remainder $1$ when it divides $N$.\nThe [[divergence of the sum of the reciprocals of the primes|00RA98X2CP]] is a stronger result.\n[[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]] gives infinitely many primes in each suitable arithmetic progression.\n\n## History\n\nThe theorem is Proposition 20 of Book IX of the Elements of Euclid.\n\n## Explanations\n\n### L1 human\n\nNo finite list holds all the primes. Multiply all the primes of the list and add $1$.\nThe new number leaves the remainder $1$ on division by each prime of the list, so its prime factors are new.\nFor example, $2 \\cdot 3 \\cdot 5 \\cdot 7 \\cdot 11 \\cdot 13 + 1 = 30031 = 59 \\cdot 509$, and $59$ and $509$ are not in the list.\n",
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