Prove It by Velleman, Introduction, Problem 6

Problem

The sequence 3, 5, 7 is a list of three prime numbers such that each pair of adjacent numbers in the list differ by two. Are there any more such “triplet primes”?

Solution

No. Any triplet would have to be three consecutive odd numbers since all even numbers are not prime. The possibilities for the last digits of the triplets are: (1, 3, 5), (3, 5, 7), (5, 7, 9). In each case, there is a 5 in the last digit of one of the numbers. Any such number will be divisible by 5, thus not prime. Therefore, once you get pass 7, there are no more triplets.

Prove It by Velleman, Introduction, Problem 5

Problem

Use the table in Figure 1 and the discussion on p. 5 to find two more perfect numbers.

Solution

Page 5 tells us that if is prime, then is perfect.

Figure 1 shows that is prime for and .

By calculator:

Both 496 and 8128 are perfect numbers.

Prove It by Velleman, Introduction, Problem 4

Problem

Find five consecutive integers that are not prime.

Solution

24, 25, 26, 27, 28

Prove It by Velleman, Introduction, Problem 3

Problem

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

Problem a

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

Solution

The method in Theorem 3 is as follows: Let , , , … ,  be a list of all prime numbers. Let . m will be a prime or a product of primes.

211 is prime.

Problem b

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

Solution

The section on Theorem 3 also discusses Marsenne Primes – if n is prime, then might be prime.

That is one possible solution.

The answer key lists 3 and 37 and solutions, but it’s unclear how to calculate the 37.

Prove It by Velleman, Introduction, Problem 2

Problem

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

Solution

 

n	Prime		Prime		Prime
2	Yes	8	No	7	Yes
3	Yes	26	No	25	No
4	No	80	No	79	Yes
5	Yes	242	No	241	Yes
6	No	728	No	727	Yes
7	Yes	2186	No	2185	No
8	No	6560	No	6559	No
9	No	19682	No	19681	Yes
10	No	59048	No	59047	No

 

Conjecture: is never prime.

Conjecture: may be prime.