A miscellaneous compendium of unsolved math problems

Math is a fascinating world. It explains quantitatively almost every process on earth, even in the universe. From traffic patterns to cell differentiation, math is used to forecast future outcomes and explain historical trends.

Personally, I have a great interest in number theory. With that being said, this list won’t cover every single unsolved problem in mathematics; rather, only those that deal with numbers, number sequences, and how numbers relate to each other. Even among unsolved problems in number theory, there are some that are way too complex for a human being to understand. Perhaps that’s why they remain unsolved.

Without further adieu, here is the list:

1. Goldbach’s Conjectures

In 1742, the German mathematician Christian Goldbach wrote a letter to Euler, proposing two conjectures:

  1. Every even integer greater than 2 can be expressed as the sum of two primes.
  2. Every integer greater than 2 can be written as the sum of three primes.
    N.B: 1 is not a prime.

The former is known today as Goldbach’s Strong Conjecture (or simply Goldbach’s Conjecture), and the latter is known as Goldbach’s Weak Conjecture. To demonstrate:

4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7 = 5 + 5
100 = 47 + 53

While mathematicians have been able to “brute-force” prime sums for numbers up to 4 × 1018, there have been no valid proofs submitted. It is currently only assumed true due to the fact that as numbers get larger, the number of ways of expressing them as sums of primes increases (see graph below)

X-axis: The number itself
Y-axis: Amount of possible ways to write said number as two primes

In 2002, a $1,000,000 USD price was offered to anyone who could prove the conjecture. The prize was not claimed.

2. The Collatz Conjecture

First proposed in 1937, the Collatz Conjecture makes for a great party trick. At your next gathering of mathematicians, have a volunteer go through the following steps:

  1. Pick a number greater than zero (e.g. 1, 2, 3, 4, …).
  2. If number is even, divide by 2. If number is odd, multiply by 3 and add 1.
  3. Using the number from Step 2, repeat Step 2 until you get 1.

In fact, no matter which number you choose, you will always get the number 1 at some point, a property called oneness. By intuition, larger numbers should take more time to reach 1. However, as the graph below shows, that’s not always the case. The number of steps it takes to get to 1 varies greatly, though one could argue that there is a slight upward trend.

X-axis: Number
Y-axis: How many iterations (steps) it takes to get to 1

The plots for individual numbers and their “path” to 1 are equally interesting:

Plot for the number 27.
X-axis: Step #
Y-axis: Number at step X

One interesting fact about the Collatz Conjecture is that in 2007, two researchers proved that the Collatz was mathematically unprovable. Now, all we need is someone to prove the proof of the unprovable conjecture.

3. Legendre’s Conjecture

Like #1 on this list, this conjecture deals with primes. It states that between successive positive square integers, there exists at least one prime number. Since that may be an earful, here are some examples:

  • Between 25 (aka 5^2) and 36 (aka 6^2), is the prime number 29. There are other primes, but they are not listed.
  • Between 36 and 49 is the prime number 37.
  • Between 10,000 and 10,201 is the prime number 10939.

It seems logical that, since the gap between successive square integers gets ever larger, that there must be “room” for one prime between those numbers. However, there are also significant gaps between prime numbers when numbers get in the vicinity of 10^8000. Even then, the gaps between successive primes are nowhere near as big as the gaps between successive squares. Who knows though, somewhere way high up there may be a huge prime gap that hammers the nail in Legendre’s coffin.

4. Irrationality of certain numbers

Proofs have been devised that show that π and its cousin e are both irrational constants. However, it is strangely not known whether the following are irrational:

  • π+e, π-e, πe, π/e;
  • πe, π√2, ππ, eπ2,
  • ln π
  • 2e, ee

5. The Sofa Problem

(Quoted from Futility Closet. Original post here)

In 1966, Austrian mathematician Leo Moser asked a pleasingly practical question: If a corridor is 1 meter wide, what’s the largest sofa one could squeeze around a corner?

That was 46 years ago, and it’s still an open question. In 1968 Britain’s John Michael Hammersley showed that a sofa shaped somewhat like a telephone receiver could make the turn even if its area were more than 2 square meters (above). In 1992 Joseph Gerver improved this a bit further, but the world’s tenants await a definitive solution.

Similar problems concern moving ladders and pianos. Perhaps what we need are wider corridors.