Students learn about prime numbers early in their careers, but the true, quirky nature of these numbers isn’t really explored unless kids go on to become math majors.

But there are many fun prime explorations suitable for young students. Here are three examples. Each is a conjecture: something that appears true, but hasn’t been proven or disproven.

### Twin Prime Conjecture

Twin primes are pairs of prime numbers with only one integer between them. We’d write them as *n* and *n* + 2. Examples include:

- 5 and 7
- 17 and 19
- 29 and 31

The Twin Prime conjecture tells us that there are an *infinite number* of twin primes. Keep going up and you’ll never run out! Of course, since it’s a conjecture, this is a strong belief but not a proven fact. See how many Twins students can find!

### Cousin Primes Conjecture

Cousin Primes are close, but not *as* close as Twins. We’re looking for *n* and *n* + 4, two primes that are four integers apart. Some examples:

- 3 and 7
- 7 and 11
- 43 and 47

Mathematicians *think* there are an infinite number of pairs of cousin primes, but haven’t proven it yet.

There’s also a name for pairs of primes that are six apart (*n* and *n* + 6), but it isn’t Byrdseed-appropriate ;)

### Legendre’s Conjecture

Legendre’s Conjecture tells us that there *should* always be a prime number between *n*^{2} and (*n* + 1)^{2}, where *n* is a positive integer. This means that, for any whole number, take its square and the square of the next integer. Somewhere in between lies a prime number.

For example:

- For n = 1, find 1
^{2}and 2^{2}. There should be a prime between 1 and 4. There is! It’s 3. - For n = 2, find 2
^{2}and 3^{2}. There should be a prime between 4 and 9. There are two: 5 and 7! - For n = 5, find 5
^{2}and 6^{2}. There should be a prime between 25 and 36. There are two: 29 and 31!

Notice that we have some twin primes in there! Like all conjectures, Legendre’s Conjecture remains unproven.

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