Warm Up (Algebra)

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A positive integer $p > 1$ is called prime if its only positive divisors are $1$ and $p$.

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$7$ is a prime number because its only divisors are $1$ and $7$.

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Every integer greater than $1$ is divisible by a prime.

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To claim that every integer greater than $1$ is divisible by a prime.

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If $p$ is a prime and $p$ divides $ab$, then $p$ divides $a$ or $p$ divides $b$.

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There are infinitely many primes.

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Try considering the product of all known primes plus one.

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Let’s analyze the steps of the proof in detail.

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Suppose there are only finitely many primes $p_1, p_2, \dots, p_n$. Consider $Q = p_1p_2\dots p_n + 1$…

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A nice corollary to have.

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This theorem was first proved by Euclid around 300 BC.

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Prove that there are infinitely many prime numbers.

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See the proof above for a constructive argument.

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Here is an exercise.

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There are infinitely many primes.

See Euclid’s Theorem.

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There are infinitely many primes.

See New Theorem.

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Show that the number of primes is infinite.

See Problem 1.

A definition:

Definition (Prime). A positive integer is called a prime if $\dots$.

Theorem (Euclid). Let $P$ denote the set of primes. Then, $$|P| = \infty.$$

Let us try to prove it.

Proof. Consider the