One of the oldest and most famous unsolved mathematical problems is the Goldbach conjecture. This is
Every even number greater than 2 can be expressed as the sum of two prime numbers.
This problem was first posed in 1742 by the German mathematician Christian Goldbach and nearly three hundred years later no one has managed to prove whether or not it is true.
Christian Goldbach
Examples
A prime number (or prime) is a positive number, which only has factors of itself and 1. For example: 2, 3, 5, 7, 11, 13, 17, 19 etc. By convention 1 is a not considered to be a prime number. Clearly all prime numbers other than 2 must be odd.
I’ve illustrated the Goldbach conjecture for some even numbers below:
4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 5 + 5 OR 3 + 7
100 = 3 + 97 OR 11 + 89 OR 17 + 83 OR 29 + 71 OR 41+ 59 OR 47 +53
In general, the larger the even number the more different ways it can be split between two primes.
Proving the Goldbach Conjecture
Shortly after the Goldbach stated the conjecture one of the greatest eighteenth century mathematicians Leonhard Euler said:
‘That … every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it’
Leonard Euler one of the greatest ever mathematicians – Image from Wikimedia Commons
For the last 300 years mathematicians have tried without success to prove it, or to disprove it by finding an even number which cannot not be expressed a sum of two primes. Before the advent of computers, to try to disprove the conjecture, mathematicians had to laboriously check even numbers by hand to look for a number which could not be expressed as the sum of two primes. In 1938 the mathematician Nils Popping showed that all even numbers up to 100,000 could be expressed as the sum of two primes. This must have been an incredible tedious task.
Today, using super computers the Goldbach conjecture has been verified to be true for all even numbers less than four million trillion (4 000 000 000 000 000 000). However, this impressive feat does not prove the conjecture is true for all numbers. All it shows is that the conjecture is true for all even numbers less than four million trillion. It is possible that there exists an even larger number which cannot be split into two primes. What has never been found is a mathematical proof that the conjecture is true for all even numbers
The Bloomberg Prize
In a 1992 novel Uncle Petros and Goldbach’s Conjecture by Apostolos Doxiadis the anonymous narrator describes his fascination with his reclusive Uncle Petros, who is considered a failure by his family. When his nephew shows an interest in mathematics, Petros offers him a problem to solve
prove that any even number greater than 2 is the sum of two primes.
The narrator works hard on trying crack this problem all summer without success. He later learns that it is more than 250 years old and has remained unsolved by the greatest mathematicians.
Enraged and frustrated, he confronts his uncle, only to discover that Petros has been psychologically crippled by the Goldbach conjecture for decades. When he was a young scholar determined to pursue distinction in the world of mathematics, Petros decided to tackle the proof. But as the proof revealed itself to be beyond him, his pursuit became a nightmare in which he imagined that numbers had taken human form and were speaking to him. As Petros lost hope of finding the proof, he lost his grip on his sanity and his livelihood as a professor at the University of Munich.
In March 2000 the publishers of the book (Bloomberg in the USA and Faber and Faber in the UK) offered a prize of one million dollars anyone who could prove the Goldbach conjecture and whose proof was accepted by fellow mathematicians. The prize was kept open for two years but nobody claimed it.
Other unsolved problems
In the year 2000 the Clay Mathematics Institute produced a list of of seven mathematical problems for which they would give a million dollar prize for the person(s) who solved them The Goldbach Conjecture was not among them. I have listed the seven problems below and I think that most of my readers will agree that they cannot be understood by anyone without a high-level knowledge of mathematics. To me the great attraction of the Goldbach conjecture is its simplicity to state and its impossibility to prove!
From http://www.ams.org/notices/200008/comm-millennium.pdf
I hope you have enjoyed this article. To find out more about Explaining Science, click here or at the Explaining Science Home link at the top of this page.
For another unsolved (but less famous) problem see https://explainingscience.org/2017/08/12/196-an-unsolved-problem/
Explaining Science 
I had been pushing the envelope, looking for a solution to GBC, and had been using the paper by Jason South for inspiration, when wham, I understood is proof, clear as day. Beautiful job Jason! Clean proof. Below is the arXiv address. This guy is a genius.
https://arxiv.org/abs/2206.01179?context=math
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Three Theorems on the Goldbach Conjecture, and the Intimate Prime-Pairs
http://researchmathsci.org/APAMtoc.html
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Dear Steve,
I’m Adriko from Arua Uganda.
I’m happy to find this page but frustrated because i’ve failed to announce to the world my discoveries in Mathematics after coming across them in May 2020 due to the COVID-19 lockdown restrictions and the hard journal publication restrictions.
The following are what i have done just in about one week, a surprise.
1. I have solved the Beal’s conjecture.
2. Fermat’s Last conjecture.
3.Goldbach’s conjecture.
4.Riemann Hypothesis (prime distribution).
5.The Twin prime conjecture.
6. I derived some formulae for solving Diophantine equations e.g
(i) xⁿ+yⁿ+zⁿ=k
(ii) x³+y³+z³=42
(iii) x³+y³+z³=33
(iv)x³+y³+z³=8
etc have been solved by me.
7. I derived a relation between Pi(π) and Euler’s constant, e. (Deal between pi and constant, e.
8. I derived another formula for finding arithmetic mean and Geometric mean.
Please help me on how i can submit my journals to publishing companies for peer-reviews. I believe i have done something to the world by solving some of the world’s millennium Mathematics problems.
Best regards.
Adriko.
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You have been busy 😉
But sorry I am unable to help you as my field is astronomy rather than pure mathematics.
You could try writing to some top Universities and see what they think of your work!!
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“The Definitive Proof of the Goldbach Conjecture* by Kenneth A. Watanabe, PhD was posted on Arxiv in 2018. He sums up all pairs that have composites. eg. n pairs have n/3 that are divisible by 3. Of pairs left, NOT divisible by 3, a fifth are divisible by 5. After he sums up all pairs not divisible by a smaller prime, he proves total pairs exceed pairs with composites in proof of Goldbach’s Conjecture. He does not remove 3, n-3, 5, n-5, 7, n-7 …Sqrt(n),n -Sqrt(n). His small but acceptable error is that he does not know how to separate out of that residue, prime, prime pairs from prime, composite pairs. That’s still good enough in my opinion but it never got published as far as I know. Where he uses a
crude meat cleaver, I use a laser in my attempt, “The Stepladder Proof of the Goldbach Conjecture.” Subject to minor defined adjustments, every even n can be expressed as n/4 unique and mandatory pairs. Unlike Watanabe’s proof, the pair p, n-p has no mirror image duplicate n-p, n.
200 has 50 unique even number pairs. All primes are embedded. My excess pairs algorithm isolates 37 pairs with at least one composite. Prime,prime pairs = 50 -37 = 13 exactly. No margin of error. I posted my simple proof, “The Stepladder Proof of the Goldbach Conjecture” on Academia.edu. I would welcome questions and comments on my new blog, htt219390965.wordpress.com or email gregorymazur02@gmail.com.
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In the example for “100 = 3 + 97 OR 7 + 93 OR 11 + 89 OR 13 +87…”, the numbers 93 and 87 are not prime.
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Thank you for spotting this. Clearly I should have checked this post more thoroughly before publishing 😉
I have removed the two incorrect examples !!
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I was studying the Goldbach conjecture last year, but completely forgot the name! Thanks for reminding me.
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I’m leaning towards ‘flawedman’s final paragraph. 😀 … but this question did occur to me, if it was proved, or disproved, what would the consequences be, both practically (day-to-day life) and theoretically (science)?
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Hi Widdershins,
The Goldbach conjecture is a problem in pure mathematics. So the impact on day to day life and Science would be zero !!
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😀
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that is not true…to solve goldbach`s conjecture one needs to develop a theory on coherent forms of quadratic and non quadratic residues over certain elliptic curves,which are vital in cryptography.
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I looked up the last two prize problems are they not Physics problems ? I know physicists use mathematics a lot but physics is concerned with the real world mathematics is a world of its own.
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I would tend to agree with you,
the last two are really mathematical physics problems rather than pure mathematics problems
However, the Clay Mathematics Institute considered them to mathematics problems when offering the $1 000 000 prize 😉
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Is it not curious that simple numbers can display properties that are so difficult to prove.
I would mention Fermat’s theorem which most of us know as Pythagoras and learnt in one way or another at school.
The list you give is beyond me but I did understand that Fermat’s has at last been proved.
From what I understand Pythagoras was proved by geometry but it would be too difficult for me as my algebra and geometry are very elementary.
Goldbach seems true to me especially as computers have checked it for millions of cases.
It seems almost absurd to the layman that obvious truths need absolute proof but I’m told the absolute proof is mathematically possible.
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