How do I prove these three statements true/false?

Solution 1:

For the first proposition $n=5$ is a counterexample. $$(2\cdot 5 +1)^2 -2 = 7 \cdot 17$$

For the second proposition, $n=6$ is a counterexample. $$6^3-(6-1)^3=7\cdot 13$$

For the last proposition, $n=6$ is a counterexample.

$$(2\cdot 6)^2=144 \text { and } 147=3\cdot 7^2$$

Solution 2:

You can disprove such a statement by finding a counterexample.

• For small values of $n$ you can do this manually. There are at least 2 counterexamples for $n\le 10$ for each of your claims.
• Your claims are wrong for at least 50% of all $n\le100$. For larger $n$ these values increase. So if you select an arbitrary two digit number you have at least 50% chance to find a counterexample. You can use Wolfram Alpha to check if a number is a prime number.
• To check a larger range of values you can write a program, e.g. in Python or Mathematica.

Solution 3:

Disproving by simple counterexample is the right approach.

In your (1), the quadratic formula result has a sign error. It should be $\sqrt{16+16(1+p)}$, not $16-16(1+p)$.

Building solutions using the quadratic formula requires a lot of steps where there can be errors, so you should always test the result with a simple case, e.g. plug $n=1$ into (1), get $p=7$, and see if those values solve your quadratic.