Goldbach’s Conjecture

It looks complex, it sounds complex. It is complex. Since 1742, Goldbach’s Conjecture has been fought to prove and still stands, yet unproven. Is this at the fault of our hardworking mathematicians? Can Goldbach’s Conjecture even be solved?

Before we begin, what actually is a conjecture?

The definition of conjecture is: “an opinion or conclusion formed on the basis of incomplete information” (Google, 2021). In this sense, this mathematical conjecture is actually an unsolved mathematical theory.

What actually is Goldbach’s Conjecture and where did it come from?

Goldbach’s Conjecture is the theory that “Every even number (greater than two) is the sum of two primes.” (Popular Mechanics, 2019). Smaller numbers can be checked in our heads, such as 18 = 13 + 5 and 42 = 23 + 19, and computers have attempted to prove this conjecture by increasing the scope further through larger numbers. Unfortunately, no luck.

This led to the unsolved conjecture being present since 1742 – that’s two hundred and seventy-nine years. The famous Goldbach’s Conjecture theory remains unsolved, and when it will be solved in future is unclear. Who it will be solved by in future is unclear.

Maybe it will be you.

The Large Cardinal Project

When the prospect of infinity came to rise in mathematics, it was regarded as a placeholder for an infinite range of real numbers. Real numbers are all numbers between 0 and positive infinity (positive numbers) and 0 and negative infinity (negative numbers).

It was suggested that rather than infinity being a placeholder for an infinite range of real numbers, infinity would contain individual sets of real numbers (Popular Mechanics, 2019). These sets of real numbers – at this stage in the Large Cardinal Project – have the ability to overlap and are infinite; however these sets create an infinite set of finite numbers.

The numbers within the Cardinal Number Sets are represented by the symbols ZFC (ncatLab, 2020). To represent and prove the Large Cardinal Axioms, the following formulas are used:

ZFC+A1 and ZFC+A2

Assuming the theory is correct, ZFC+A1, proves the accuracy of ZFC+A2, and ZFC+A2 proves the accuracy of ZFC+A1. These formulas are regarded as mutually exclusive sets of cardinal numbers when regarding the Large Cardinal Project. To prove these sets of numbers as mutually exclusive would contribute to proving the theory; however, these sets of numbers cannot be proven using the ZFC formula.

To prove the ZFC formula would be to prove infinity.

Google (2021). Dictionary: Conjecture. [Accessed 2 April 2021]. Available at:

https://www.google.com/search?q=conjecture+definition&rlz=1CAVUZL_enAU942&oq=conjecture+defin&aqs=chrome.1.69i57j0l9.5801j1j1&sourceid=chrome&ie=UTF-8

ncatLab (2020). Large Cardinal. [Accessed 12 April 2021]. Available at:

https://ncatlab.org/nlab/show/large+cardinal

Popular Mechanics (2019). The Large Cardinal Project. [Accessed 12 April 2021]. Available at:

https://www.popularmechanics.com/science/math/g29251596/impossible-math-problems/

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