Question 1

1. Consider the sequence generated by adding the next largest prime to the sequential product at each step. Find the first member of this sequence that is not a prime and factor it completely. Bonus - Find a method for generating odd numbers that continues for more steps before giving a non-prime number.

2+1 = 3

2 x 3+1 = 7

2 x 3 x 5+1 = 31

2 x 3 x 5 x 7+1 = 211

2 x 3 x 5 x 7 x 11+1 = 2,311
and etc.

The first member of this sequence that is not a prime is:

2 x 3 x 5 x 7 x 11 x 13+1 = 30,031; it can be made by multiplying 59 x 509

So.. how do I know this that this is the first member of the sequence that is not a prime? First, I would need to define what a prime number is. A prime number is any number that has no other positive, whole number as divisors except 1 and itself. 

Using this information, I know 3, 7, and 31 are prime numbers. But, how did I figure that 2,311 is a prime number and 30,031 isn't?

First, using information from Dr. Math's forum: http://mathforum.org/library/drmath/view/54371.html, I found out that of all the divisors (factors) of a number, one of the two divisors has to be less than the square root of the number.

                  For example, 12 is the number that I want to find factors for. 

                                              The √12 is approximately 3.46410

                                              Factors of 12 includes: 

                                              1 x 12 = 12, here 1 < 3.46410

                                              2 x 6 = 12, here 2 < 3.46410

                                              3 x 4 = 12, here 3 < 3.46410

So, from this, I know that any pair of divisors of 2311 has to have one that is less than the √2311 = 48.072 and any pair of divisors of 30,031 has to have one that is less than √30,031 = 173.2945.

Then, I can test them using divisibility. If they are prime, any number less than those would not give a remainder of 0. I figured that because all of the numbers made by this sequence (due to the +1 at the end) are not even numbers, I don't have to test 2 or any number like 4, 6, 8, 10, etc. because only even numbers would be divisible by these numbers to give a remainder of 0. Not only that, from knowledge in elementary school, I can go on further to say I don't have to test non-prime numbers because anything that's divisible by 9 can be divided by 3. So, I only have to divide these numbers by prime numbers that are smaller than their square roots.

                  For example, 2311 - √2311 = 48.072 so the divisors (if there are any) has to be less than 48 and prime.

                    Then, using my knowledge of prime numbers, I can figure out and list the ones that are < 48, they are:

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

                                           2311/3 = 770.3333...

                                           2311/5 = 462.2

                                           2311/7 = 330.1428...

                                           2311/11 = 210.09...

                                           2311/13 = 177.769...

                                           2311/17 = 135.941...

                                           2311/19 = 121.632...

                                           2311/23 = 100.478...

                                           2311/29 = 79.689...

                                           2311/31 = 74.548...

                                           2311/37 = 62.459...

                                           2311/41 = 56.365...

                                           2311/43 = 53.744...

                                           2311/47 = 49.170...

From this list, I know that 2311 is a prime number because it has no prime divisors < 48.072 that gives a remainder of 0.

I then repeat this process for 30,031 and found that it's not prime because it can be divided using 59 and the answer will be 509 with 0 as a remainder. This means that 59 and 509 (in fact, 59 x 509 = 30,031) are factors of 30,031 and it is NOT a prime number.