Unformatted text preview: P 1 = 2 , P 2 = 3 , and so on. Now, consider the number C formed by multiplying all of the n prime numbers together. The value C + 1 is not divisible by any of the n prime numbers. C + 1 is a prime number larger than P n , a contradiction. Thus, we conclude that there is no largest prime number. ✷ 2.16 Note: This problem is harder than most sophomore level students can handle. Proof : The proof is by contradiction. Assume that √ 2 is rational. By de f ni-tion, there exist integers p and q such that √ 2 = p q , where p and q have no common factors (that is, the fraction p/q is in lowest terms). By squaring both sides and doing some simple algebraic manipula-tion, we get 2 = p 2 q 2 2 q 2 = p 2 Since p 2 must be even, p must be even. Thus,...
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- Fall '08
- Prime number, largest prime number