Open math problems which high school students can understand

I request people to list some moderately and/or very famous open problems which high school students,perhaps with enough contest math background, can understand, classified by categories as on arxiv.org. Please include statement of the theorems,if possible, and if there are specific terms, please state what they mean.

Thank you.I am quite inquisitive to know about them and I asked this question after seeing how Andrew J.Wiles was fascinated by Fermat's last theorem back in high school.

Solution 1:

Goldbach's conjecture (math.NT)

An even integer is a positive integer, which is divisible by $2$.

Goldbach's conjecture states that $$\text{"Every even integer greater than $2$ can be expressed as the sum of two primes."}$$

For instance, $4= 2 + 2$, $6 = 3 + 3$, $8 = 5 + 3$, $10 = 7 + 3$, $12 = 7 + 5$ and so on.

Twin prime conjecture (math.NT)

A prime positive integer is one, which is divisible only by $1$ and itself. Twin primes are the primes which differ by $2$. For instance, $(5,7)$, $(11,13)$, $(17,19)$, $(101,103)$ are all examples of twin primes.

The twin prime conjecture asks the following question $$\text{"Are there infinitely many twin primes?}"$$

Mersenne prime (math.NT)

A Mersenne prime is a prime of the form $2^n-1$. For instance, $31$ is a Mersenne prime, since $31 = 2^5-1$. Similarly, $127 = 2^7-1$ is also a Mersenne prime.

It is easy to show that if $2^n-1$ is a prime, then $n$ has to be a prime. However, the converse is not true.

The Mersenne prime conjecture asks the following question

$$\text{"Are there infinitely many Mersenne primes?"}$$

Perfect numbers (math.NT)

A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. The first few perfect numbers are $$6 = 1 + 2 + 3$$ $$28 = 1 + 2 + 4 + 7 + 14$$ $$496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248$$ There are two interesting conjectures on perfect numbers. The first one asks $$\text{"Are there infinitely many perfect numbers?"}$$ The second one asks $$\text{"Are there any odd perfect numbers?"}$$ Euler proved that $2^{p-1} \left(2^p-1 \right)$, where $2^p-1$ is a Mersenne prime, generates all the even perfect numbers. Note that if one proves that the Mersenne prime conjecture is true, then this will also imply the first conjecture on perfect numbers.

EDIT

This MO thread is also relevant.

Solution 2:

You might include Legendre's Conjecture:(Math.NT)

There is at least one prime number between any two consecutive square numbers.

I think this is easy to understand, and engaging because it is not hard to verify for small numbers.

Solution 3:

The most elementary open math problem I know is the Collatz conjecture.

Start with a natural number. If it is even, divide by two, if it is odd, multiply by three and add one. Repeat this operation, stopping only when you get a value of one.

The conjecture is that you always reach 1 eventually, no matter what your starting number is.

Solution 4:

Singmaster's conjecture says there is a finite upper bound on the number of times a number (other than the $1$s on the edge) can appear in Pascal's triangle. The upper bound may be as low as $8$. If so, then no number (besides those $1$s) appears more than eight times in Pascal's triangle. Only one number is known to appear that many times: $$ \binom{3003}{1} = \binom{78}{2} = \binom{15}{5} = \binom{14}{6} $$

It has been proved that infinitely many numbers appear twice; similarly three times, four times, and six times. It is unknown whether any number appears five times or seven times.

Singmaster states that Erdős said the conjecture is probably true but probably difficult to prove.