Math

The Goldbach Conjecture Is A Simple Problem That's Never Been Solved

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The Goldbach conjecture is an old mathematical problem that seems easy to prove, but has remained one of the most stubborn conundrums in modern mathematics. Not a math person? We promise this is an easy one to wrap your head around.

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A 275-Year Old Dilemma

In June of 1742, Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (the mathematician so influential that they named a number after him) in which he laid out this famous conjecture. It was refined a bit since that fateful day to say this: All even whole numbers greater than two are the sum of two prime numbers.

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A prime number, to refresh your memory, is a number that can only be divided by 1 and itself. Let's try out the conjecture on the even number 4. 4 can be expressed as 2 + 2. 2 is a prime, so the conjecture holds up. Same goes for a larger number like 28. You can express 28 as the sum of the prime numbers 5 and 23, or 11 and 17. You can go even higher—why not try something crazy like 12,345,678? That can be expressed as the sum of prime number 20,297 and 12,325,381. (You can try it yourself on this Goldbach calculator.) Hey, that all worked! We did it. A nearly 300-year-old math problem solved right here, right now. Well, not so fast.

Why Hasn't It Been Solved?

Just because every number you try works doesn't mean Goldbach's conjecture will hold up for every even number in the universe. For the problem to be solved, a mathematician has to come up with a way to prove that there will never be an even number that doesn't work. As The Guardian pointed out, "A variation of Fermat's Last Theorem that looked certain to be true, failed on the number 61,917,364,224. Just checking numbers is never enough: showing that something will always, undoubtedly work is known as a proof in maths and a problem isn't finished until one is found." (A spin-off known as the weak Goldbach conjecture was actually proven in 2013.)

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The Goldbach conjecture has plagued mathematicians so incessantly that in the year 2,000, the publisher Faber & Faber announced a million-dollar prize for anyone who could prove it. Two years later, no one had managed to do so.

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Key Facts In This Video

  1. There don't seem to be any clear patterns in the sequences generated by the simplest impossible problem. 01:19

  2. To date, all numbers up to 5 x 2^60 have been tested with the 3x+1 problem. 02:00

  3. Speaking about the 3x+1 problem, famous mathematician Paul Erdős said, "Mathematics is not yet ready for such problems." 03:23

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