Goldbach's Conjecture

Goldbach's conjecture is that every whole number greater than 3 can be written as the sum of two or three primes. This idea is almost 300 years old, first being suggested in 1742. This online calculator will find two or three primes that add to any given number. Just type your number in the box and click the Submit button.

Some Questions About This Calculator

Is this true for every number greater than 3?

Probably. no-one knows for sure if the conjecture is true or not, but it is true for every number smaller than 4,000,000,000,000,000,000 (four million trillion). It is true for all odd numbers.

Is there only one set of primes that adds up to any given number?

In general, no. Some small numbers, like 5 have only one set of primes. Other numbers have more ways. (10=5+5 and 10=7+3 for example.) This program only gives you one way, generally the way with the smallest prime number. If the number is odd, it always gives a solution that includes the number 2 or 3.

Are there any other ways to write the conjecture?

Yes, many. Several more common ways are:

• Every even number greater than 2 can be written as the sum of two primes. (This is how the problem is usually written in math books.)
• Every odd number greater than or equal to 9 can be written as the sum of three primes (sometimes called the weak version of the conjecture).
• Every integer greater than 5 can be written as the sum of three primes.

Why is nobody sure that this conjecture is true? Shouldn't it be easy to prove?

It turns out that some easy looking problems are hard, and some hard looking problems are easy. Computers have been used to check the conjecture for some numbers (see above) but we can never check every number. Personally I think that is is a good thing that this has been hard to prove, as looking for the proof has helped us learn other things along the way.

In fact, since I first wrote this page, we have come closer to solving this problem, in 2013 Helfgott proved the weak version of this conjecture (true for odd numbers). Before that, we only knew that the conjecture was true for some even numbers (less than four million trillion and odd numbers larger than 1 with 43 zeros after it). We also know that every even number can be written as the sum of no more than 30,000 primes. That's a few more than 2!

The calculator doesn't do very large numbers.

This calculator uses an algorithm that needs a list of prime numbers to work correctly. This make the program fast to run and easy to write, but means that it can't do very large numbers unless I give it a very large list. A large list would take more space on the webserver, and make the page load more slowly. This calculator should do large enough numbers for the casual user.