How To Prove It – Exercise 0.2

Oct 04 2012

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

Problem (2): Make some conjectures about the values of n for which 3^n-1 is prime or the values of n for which 3^n-2^n is prime. (You might start by making a table similar to Figure 1) .

Solution:

Okay, first lets make a table:

n Prime? 3^n-1 Prime? 3^n-2^n 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 n, 3^n-1 is not a prime number.

(2) For any value of n, 3^n-2^n is a prime number if n is a prime.

Note that conjecture 1 is true because 3^n-1 will always be an even number. Conjecture 2 is not true since it breaks down when n=7

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