Simple Conjectures, Difficult Proofs

The general trend in mathematics is that the simpler the conjecture, the simpler the proof. One only needs to know the basic properties of integers in order to prove the infinitude of primes, while only professional mathematicians with the appropriate specialization can prove theorems about modular forms, Lie algebras, p-adic numbers, and other such esoteric objects.

However, sometimes the opposite is true.

The popular example is Fermat's Last Theorem, that there are no solutions to \(a^n+b^n=c^n\) for \(n\geq3\) and integers \(a, b, c\). It took 358 years for Andrew Wiles to finally bring down this mathematical monster, which any middle school student could understand, but which has defeated mathematicians for centuries.

But this is not the end! Take the Collatz sequence:

Take any arbitrary positive integer, and execute the following operations:

  • If the number is even, divide it by \(2\).

  • If the number is odd, multiply it by \(3\) and add \(1\).

  • Repeat.

Anyone with basic mathematical knowledge can understand these rules. However, no mathematicians have been able to tackle the nightmarish Collatz's Conjecture:

Does the above sequence reach \(1\) for all possible seed values?

Generalizations of Collatz have been proved undecidable: unfortunately, these proofs do not apply to Collatz.


Another two such conjectures are the Goldbach conjectures, proposed by amateur mathematician Christian Goldbach to his friend Leonard Euler in 1742:

Goldbach's 'Weak' Conjecture: Every odd number greater than \(5\) can be expressed as the sum of three primes. Proven by Harald Helfgott in 2013.

Then there is its far more elusive sister, proposed in a letter from Goldbach to Euler:

Goldbach's 'Strong' Conjecture: Every even integer greater than \(2\) can be expressed as the sum of two primes. Proven for \(n<4\times 10^{18}\).

Euler stated that he regarded the second conjecture to be true, although he had no way of proving it. 'Almost all' even numbers have been proven to satisfy the conjecture (Chudakov, Estermann, Van der Corput), and it has been shown that every sufficiently large even number can be written as either a sum of two primes, or the sum of a prime and the product of two primes (Chen).


On a completely different note (topology, this time), consider the square peg problem:

Does every Jordan curve (closed, non-intersecting loop on the plane) contain all four vertices of a square?

Despite being easy to understand, no one has ever come close to proving its truth. A weaker generalization, the rectangular peg problem (replacing the square with an arbitrary rectangle), has been proven (H. Vaughan, 1977). The conjecture has also been proven for many different types of Jordan curves, but there's no general proof.

What other simple-to-understand, not-so-simple-to-prove conjectures do you know? Feel free to share them in the comments below.

Note by Andrei Li
3 months, 2 weeks ago

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How about cubics, quartics and quintics. It took centuries to get cubics

Mohmmad Farhan - 3 months, 2 weeks ago

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And to think that the breakthrough in the study of quintics came with two teenage mathematicians, Évariste Galois and Niels Henrik Abel...

Andrei Li - 3 months, 2 weeks ago

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That's … That's .. AMAZING

Mohmmad Farhan - 3 months, 2 weeks ago

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Twin Primes Conjecture? Hadwiger's Conjecture (from graph theory)? I think the former is a notable omission...

Look, there are many conjectures out there; I'm sure you can find a whole list of them on Wikipedia.

P.S. I highly doubt a middle school student would be able to fully understand Wiles' proof of Fermat's Last Theorem. There's a reason why universities hire postdoctorals and PhD students to work on projects in arithmetic geometry...

Gennady Notowidigdo - 3 months, 2 weeks ago

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I meant middle school students understanding Fermat's Last Theorem. Sorry for not clarifying! ;)

Andrei Li - 3 months, 2 weeks ago

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How do you section the note

Mohmmad Farhan - 3 months, 2 weeks ago

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For quotes, write (> Stuff). This appears as

Stuff

For the rest, see here

Andrei Li - 3 months, 1 week ago

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Thank you but I mean the big lines that creates it into sections

Mohmmad Farhan - 3 months, 1 week ago

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@Mohmmad Farhan For those, write three (*). This appears as


Andrei Li - 3 months, 1 week ago

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@Andrei Li Did you see my note: \(\pi\) is a beautiful number

Mohmmad Farhan - 3 months, 1 week ago

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@Andrei Li I just figured out today that you can also write 3 underscores( _ ) in a row to make a section break

Mohmmad Farhan - 3 months, 1 week ago

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How about odd perfect numbers

Mohmmad Farhan - 3 months, 2 weeks ago

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The statistics of finding one is scary

Mohmmad Farhan - 3 months, 2 weeks ago

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I searched for Fermat's Last Theorem (too scary) and they said it was CONCISE AND SIMPLE!

Mohmmad Farhan - 3 months, 2 weeks ago

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How about the notorious square root of negative 1 . (Oh, That evil number)

Mohmmad Farhan - 3 months, 2 weeks ago

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