How To Prove It – Exercise 0.2

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

Problem (2): Make some conjectures about the values of for which is prime or the values of for which is prime. (You might start by making a table similar to Figure 1) .

Solution:

Okay, first lets make a table:

	Prime?		Prime?		Prime?
2	Yes	8	No	5	Yes
3	Yes	26	No	19	Yes
4	No	80	No	65	No
5	Yes	242	No	211	Yes
6	No	728	No	665	No
7	Yes	2186	No	2059	No
8	No	6560	No	6305	No

Now let us make (up) some conjectures:

(1) For any value of , is not a prime number.

(2) For any value of , is a prime number if is a prime.

Note that conjecture 1 is true because will always be an even number. Conjecture 2 is not true since it breaks down when