We will start the journey by looking at some statements that may be familiar with you. When you are investigating these statements, do not try to recall what you remember about them. Instead, see if you can come up with any justification about your claim.
Task1.1
Which one(s) of the following statements is/are true? Give as many reasons and justifications as possible and convince your neighbors. Correct the false one(s) if you can. Don't get bogged down in any one problem.
- If \(n\) is an integer, then \(n(n+1)\) is always an even number.
- If \(x,y,z\) are three integers and if \(x+y\) and \(y+z\) are odd, then \(x+z\) is also odd.
- For any natural number \(n\), we have \begin{equation*}1^3+2^3+3^3+\cdots+n^3 = (1+2+3+\cdots+n)^2.\end{equation*}
- The number \(\sqrt{12}\) is a rational number.
- Every natural number can be expressed as the sum of four integer squares.
- Every even integer greater than \(2\) can be expressed as the sum of two primes.
You might have figured out pretty convincing justifications for some of the above statements. Please share these justifications with your classmates and see if they are convinced as well. You might also find statements that you are fully convinced but some of your hardened skeptical classmates might not be otherwise. Why do you think that happen?
Some of statements in Task 1.1 are not very difficult and you will prove them later in the notes. Others are more difficult and will require some backgrounds. There is one statement that many mathematicians believe is true but no one has come up with a proof yet. Want to guess which one this is?
