Question

Task 8.11 We are now in position to prove an extraordinary classical fact. Proposition (Euclid on the Infinitude of Primes) There are infinitely many prime numbers. Euclid proved this by contradiction by assuming there were finitely many primes. Let's suppose there are finitely many primes, and call them $p_1, p_2, \dots, p_k$, for some $k \in \mathbb{Z}^+$. Consider the number $a = p_1p_2 \dots p_k + 1$. (a) Explain why there must be some $i$ between 1 and $k$ such that $p_i | a$. (b) Show how this leads to the conclusion that $p_i | 1$. (c) Explain why this is a contradiction.

          Task 8.11 We are now in position to prove an extraordinary classical fact.
Proposition (Euclid on the Infinitude of Primes) There are infinitely many prime numbers.
Euclid proved this by contradiction by assuming there were finitely many primes. Let's suppose there are
finitely many primes, and call them $p_1, p_2, \dots, p_k$, for some $k \in \mathbb{Z}^+$. Consider the number $a = p_1p_2 \dots p_k + 1$.
(a) Explain why there must be some $i$ between 1 and $k$ such that $p_i | a$.
(b) Show how this leads to the conclusion that $p_i | 1$.
(c) Explain why this is a contradiction.

Task 8.11 We are now in position to prove an extraordinary classical fact.
Proposition (Euclid on the Infinitude of Primes) There are infinitely many prime numbers.
Euclid proved this by contradiction by assuming there were finitely many primes. Let's suppose there are
finitely many primes, and call them p1, p2, …, pk, for some k ∈ℤ^+. Consider the number a = p1p2 … pk + 1.
(a) Explain why there must be some i between 1 and k such that pi | a.
(b) Show how this leads to the conclusion that pi | 1.
(c) Explain why this is a contradiction.

Added by Jamie P.

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