Unsolves Math Conjectures

Math is a minefield of surprises, mistakes, and impossible challenges. Contrary to popular belief, math is unsolved and may forever be. In this article, I will dive into some of math’s simple yet nearly impossible-to-prove statements.

First is the Goldbach Conjecture. (The word conjecture means a conclusion made on incomplete information.) The Goldbach Conjecture states that every even integer more than two can be expressed as the sum of two primes. A computer has verified the conjecture works for all numbers less than 4 × 1018 (4 followed by 18 zeros). I polled math club members about how likely, on a scale of 1-10, they think this conjecture is. Averaging the responses and adding in my own opinion, we get about 8.3/10. 

Next is the Twin Prime Conjecture. The Twin Prime Conjecture states that there are infinitely many twin primes, where twin primes are pairs of primes with a difference of 2. For example, 3 and 5 are twin primes (as 5-3 = 2). The largest twin prime found to date is 2,996,863,034,895 × 21290000 + 1, with 388,342 digits.  We polled the math club on this conjecture, and this time we got 9.1/10. 

Next is the Collatz Conjecture, aka 3x+1 or 3n+1. This conjecture is complicated, so stay with me. Start with a number. If it is odd, multiply by three and add one. If it is even, halve it. For example, take 37. It is odd, so the next number is 3 × 37 + 1 = 112. That is even, so we halve it to get 56. Continuing, we get the sequence 37, 112, 56, 28, and so on, ending in 5, 16, 8, 4, 2, 1, 4, 2, 1, ... The Collatz Conjecture states that the sequence of any integer more than one will eventually reach the cycle of 4, 2, 1, … This conjecture has been tested up to about 265, which is a twenty-digit number, but no formal proof has been achieved. The math club poll on this conjecture was divided, with responses ranging from 3 to 10, and the average was 7.8.

Lastly, we have the Odd Perfect Number Conjecture. A perfect number is a number such that the sequence sum of all the proper factors (factors of a number that are not the number itself) of the number is itself. For example, six is perfect because the sum of the proper factors of 6 is 1+2+3=6. The conjecture states that no odd perfect numbers exist. So far, mathematicians have proven that if an odd perfect number exists, it is more than 102200. This poll was more controversial than the last, with a range of nine. Still, the average was 8.2.

In my opinion, Goldbach feels true, because every integer can be expressed as the product of primes, so why not the sum? Twin prime makes sense because we’ve found really large twin primes, and it only makes sense that there will be more to come. Collatz, however, feels unintuitive, but it feels more unintuitive to have a cycle that defies this. Odd Perfect Number also makes sense because it feels unlikely that there will be a really big number that defies it. 

These questions are old mysteries. Some date back hundreds or even thousands of years. No proof or counterexample exists for any of them, and whether they are true is up to debate. Maybe we'll know someday - and maybe a Lakeside student will find the proof! It is equally possible that we will never know, and these will remain unsolved for eternity.