Investigating a Formula 2: Let’s plug a few things in.

Hello mathematicians, puzzlers, and other curious people!

In a previous post, I suggested that folks try to figure out what claims might have been made about the formula , especially on the subject of whether that formula’s output was prime or composite.

One way of investigating a formula is by plugging numbers in and evaluating it. While that method can’t prove that a particular type of result will always happen, it can generate counterexamples to some opposing claims.

Since this formula has a lot of fours in it, letting to see what happens is a natural place to start. After evaluating, we find out that , a number which is composite (as well as even, and a power of 2). While we can’t say “the formula’s results are always composite” just from this, we know better than to say “the formula’s results are always prime” since this particular one is composite. We can safely say “the formula’s results are at least sometimes composite,” which should prompt us to ask questions like “are they actually always composite?” and “if they’re only sometimes composite, when?”

Another good idea is to be lazy like a mathematician by plugging in easy numbers to see if their results are informative, and there aren’t many numbers that are easier than one. With a little arithmetic we find out that , which is a prime number.

We have enough to say that “the formula’s results are sometimes prime and sometimes composite,” since we have at least one example of each. However, any time a mathematical statement contains a flexible word like sometimes, we would like to be more specific.

So now we can restate our task in investigating this formula: figure out a rule that determines whether the formula’s results for a particular value of n are prime or composite, and prove that the rule we’ve found is right. Keep at that and see what you can find out.

Calc You Later!