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The Goldbach Conjecture

The Goldbach conjecture was introduced in 1742 and has never been proven, though it has been verified by computers for all numbers up to 19 digits. It states that all even numbers above two are the sum of two prime numbers. (Prime numbers are those that are not multiples of any number except 1 and themself.)

For example, 28 = 5 + 23. (5 and 23 are primes that sum to 28, an even number.)

You can explore prime number sums in the grid below.

  1. Select an even number to see all pairs of primes that can be summed to create that number.
  2. Select a prime number (white background) to see all even numbers that are the result of summing with another prime.
Number grid


In the grid below, the numbers are arranged so that the sum of each pair is the same. For example:

28 = 1 + 27
28 = 2 + 26
28 = 3 + 25

The Goldbach conjecture says that if we pick any even number and arrange its pairs this way, at least one of the pairs will always consist of two primes.

Use the slider to select even numbers. The prime numbers that line up in the grid are the Goldbach pairs.

Number grid



We can visualize a lot of this at once by putting the Goldbach pairs for each even number on a plot. Do you see any new patterns?

Zooming much farther out:

The total number of Goldbach pairs for each even integer: