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.

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    $\begingroup$ The Collatz conjecture might be one of the most elementary. $\endgroup$ – Thomas Andrews Jun 30 '12 at 15:27
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    $\begingroup$ Can you please state that as an answer? $\endgroup$ – user34522 Jun 30 '12 at 16:56
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    $\begingroup$ See also this MO thread on "Not especially famous, long-open problems which anyone can understand" where "anyone can understand" is explained as "The statement (in some appropriate, but reasonably terse formulation) shouldn't involve concepts beyond (American) K-12 mathematics." $\endgroup$ – t.b. Jun 30 '12 at 17:26
  • $\begingroup$ Flagged this for community wiki. $\endgroup$ – user17762 Jul 1 '12 at 0:24
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    $\begingroup$ The fact that most of the problems listed here are number theory problems is interesting. Does anyone have an intuition as to why this might be the case? Is there a sociological explanation or is there something about number theory that makes it especially rich in easy to state hard to prove conjectures? $\endgroup$ – Seamus Jul 27 '13 at 10:28

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.


This MO thread is also relevant.

  • $\begingroup$ Could you please include the definitions of twin primes, perfect numbers, mersenne primes etc here for those who are not already aware of the names and the statements of the conjectures? $\endgroup$ – user34522 Jun 30 '12 at 15:15
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    $\begingroup$ @user34522 I am sure wikipedia will provide all definitions and more. It actually might be better to have a cold list without too many details, don't you think? $\endgroup$ – Simon Markett Jun 30 '12 at 15:18
  • $\begingroup$ In that case, I am fine with it. :) $\endgroup$ – user34522 Jun 30 '12 at 15:19
  • $\begingroup$ One should also mention Stewart's "Even primes problem" - just for the sake of completness ;-) people.cs.uchicago.edu/~dinoj/crantz.htm $\endgroup$ – Łukasz Maciejewski Jul 3 '12 at 23:52

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.

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    $\begingroup$ It doesn't seem like this one could be false, given that primes grow as ln(n) and squares as n^2, but I see the difficulty in proof. $\endgroup$ – Christian Mann Jul 1 '12 at 0:59
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    $\begingroup$ @ChristianMann: It is not easy. Here is something we can prove: "There is always a prime number between two consecutive cubes." To prove this, you need to use zero density estimates in the critical strip. (Or use Sieve methods as in Baker Harman and Pintz.) $\endgroup$ – Eric Naslund Jul 2 '12 at 20:02
  • $\begingroup$ Isn't it a theorem that there is always a prime between $n,2n$ ? $\endgroup$ – ReverseFlow Jan 2 '17 at 16:37
  • $\begingroup$ @ReverseFlow: Yes, that is known as Bertrand's postulate. $\endgroup$ – daniel Jan 2 '17 at 17:19
  • $\begingroup$ The above conjecture then becomes trivial. What am I missing? $\endgroup$ – ReverseFlow Jan 2 '17 at 17:33

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.

  • $\begingroup$ Can you add the category too? $\endgroup$ – user34522 Jul 1 '12 at 8:05

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.

  • $\begingroup$ Can you please add the category? $\endgroup$ – user34522 Jul 1 '12 at 8:06
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    $\begingroup$ I have no idea what you mean by "add the category". $\endgroup$ – Michael Hardy Jul 1 '12 at 19:42
  • $\begingroup$ ".....classified by categories as on arxiv.org". $\endgroup$ – user34522 Jul 2 '12 at 15:18
  • $\begingroup$ Oh. My first thought was the category tags on math.stackexchange.com. I say this is perhaps "combinatorial number theory"; I'd have to look up the category on arxiv.org. $\endgroup$ – Michael Hardy Jul 2 '12 at 21:01

(math.DS) The existence of periodic orbits of a billiard in a triangle is not known in general -- though some special cases (e.g., acute triangles) are straight forward.

A point travels with unit speed inside the triangle, bouncing off the sides according to the usual rule from geometric optics: angle of incidence equals angle of reflection. The question is does there always exist a trajectory which will be periodic -- loop back on itself.

This simple-sounding question quickly leads deep into research territory with translation surfaces, polygonal billiards, the Veech dichotomy... And uses ideas from Riemann surfaces (and so algebraic geometry and complex analysis) and ergodic theory.

The book "Geometry and Billiards" by Tabachnikov is a good starting point for billiards in general, but probably a bit much for high-school students.


Chromatic Number of the Plane (Hadwiger-Nelson Problem): What is the minimum number of colors required to color the plane so that no two points which are unit distance apart are the same color? The current bounds on $\chi$ (which is commonly used to denote this number) are

$$4\leq \chi \leq 7.$$

The book The Mathematical Coloring Book by Alexander Soifer focuses primarily around this problem, giving a thorough history of the problem and it's creators and presenting the current state of the problem and some of it's generalizations.


An open problem I find surprising, the PAC (Perimeter to Area Conjecture) due to Keleti (1998):

Conjecture: The perimeter to area ratio of the union of finitely many unit squares in the plane does not exceed 4.

See for example Bounded - Yes, but 4? and references therein.


There is a whole collection of relatively easy to state problems without lots of advanced background which are in the general area of geometry and combinatorics collected here: http://maven.smith.edu/~orourke/TOPP/ Many of these problems are quite new and I think very appealing.


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