How To Prove It - Introduction

Chapter : Intro

1.
(a) 2^15 - 1 = 32767
4681×7 is a possible answer, trying out divisors from 3 onwards.

2.

n	Is n prime?	3^n – 1	Prime?	3^n – 2^n	Prime?
2	Yes	8	No	5	Yes
3	Yes	28	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
9	No	19682	No	19171	No

Conjectures -
(a) 3^n - 1 is not a prime irrespective of what n is
(b) If n is a prime, so is 3^n - 2^n, and if n is not, neither is 3^n - 2^n

3.
(a) m = 2 * 3 * 5 * 7 = 210
From the above table, we know 211 (m + 1) is a prime!

(b) m = 2 * 5 * 11 = 110
m + 1 = 111, which is divisible by 3 (as the sum of the digits is divisible by 3)
m + 1 = 37 * 3, both of which are primes.


4.
For n = 5,
x = (n + 1)! + 2 = 6! + 2 = 722
Non primes are - 722, 723, 724, 725, 726

5.
If 2^n − 1 is prime, then 2^(n−1) (2^n − 1) is perfect (Euclid).

Using the table in the book, 2^n - 1 is prime for n = 5, 7

P1 = 16 * 31 = 496
P2 = 64 * 127 = 8128

6. There are no more such triplets - http://en.wikipedia.org/wiki/Prime_triplet