The abc conjecture is hot mathematical news at the moment, this problem gives you the opportunity to explore this exciting piece of mathematics. Part 1 gets you started with coprime numbers which gives you the necessary background to talk about and explore the conjecture.
Part 1: All about coprime.
Two natural numbers $a,b >1$ are said to be coprime if, and only if, they share no common factors except $1$. Consider these short questions about coprime numbers, being sure to prove your assertions carefully. It might take you more time to understand what the question is asking than to answer the question itself!
Write down a few pairs $a, b$ of coprime natural numbers
What pair of coprime numbers has the smallest sum?
For which values of $n$ is the pair $n, n+1$ coprime? How about $n, n+2$ or $n, n+3$?
Find a few triples $(a, b, c)$ of natural numbers where $a, b, c$ do not all share a common factor but any two are not coprime to each other.
Find a general method to find triples in the previous question.
Find a set of $6$ natural numbers where each pair share a common factor other than 1, but there is no single common factor other than 1 which divides exactly into each of the numbers.
Suppose that $a$ and $b$ are coprime. For which values of $a$ and $b$ is $a+b$ coprime to $a$ and also coprime to $b$?
Suppose that $a$ and $b$ are coprime natural numbers. Let $c=a+b$ and let $d$ be the product of all of the different prime factors of $a$, $b$ and $c$. Find $d$ for your pairs of coprime numbers in the first part of the question.
Part 2: Easy as $abc$
Suppose $a,b,c,d$ are natural numbers as defined in part 8 above.
The abc conjecture can be stated as follows:
For any positive real number $\epsilon$ there are only finitely many triples $a, b, c$ such that
$$
c>d^{1+\epsilon}
$$
The task:
Write down a few triples of coprime natural numbers $(a, b, c)$. In each case, evaluate $d$. Explore.