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.

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Simple Conjectures, Difficult Proofs

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