How To Prove It – Exercise 0.3

Oct 05 2012

Solutions to Exercises in the Introduction of How To Prove It by Daniel J Velleman.

Problem (3): The proof of Theorem 3 gives a method for finding a prime number different from any in a given list of prime numbers.

(a) Use this method to find a prime different from 2, 3, 5, 7.

(b) Use this method to find a prime different from 2, 5, 11.

Solution:

(1) First we find $P_1.P_2.P_3 . . . P_n + 1$

$= 2.3.5.7 + 1$ .

$= 211$ , which is our new prime number.

(2) Lets find $P_1.P_2.P_3 . . . P_n + 1$

$= 2.5.11 + 1$ .

$= 111$ , which is not a prime since $111 = 3 \times 37$

$\therefore 37$ is our new prime.

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