Goldbach Conjecture

In 1754, a mathematician named Goldbach had an idea that all even numbers greater than 2 (4,6,8,10 ...) is the sum of exactly two prime numbers. This idea called a "conjecture" has never been proven to be true, but has never proven to be false. All you would have to do is to find one even number greater than 2 that was not the sum of two prime numbers and the conjecture would be no more. If it is proven to be true, it would be called a "theorem." This is the oldest unproven conjecture.

True mathematicians try to find all of the solutions to an even number. For example, the number 4 has only one solution (2 + 2); whereas, 10 has two solutions (3 + 7 and 5 + 5); and 22 has three solutions (3 + 19, 5 + 17, and 11 + 11). The number 90 has 9 solutions and 114 is the smallest number with 10 solutions. The number 100 only has 6 solutions.

I attached three levels of worksheets: the easiest for the K/1st graders is a worksheet that requires students to add prime numbers vertically to find Goldbach even numbers.The other worksheets require students to find the Goldbach primes for each even number from 4 to 100. The easier of the worksheets gives the first prime of the pair (for example, for the number 16, I give them the numbers 3 + ____ and 5 + ____). In the more difficult worksheet, I give them only the even number 16 = ________________________________, and they have to find the two pairs. In each case, I give them the number of solutions (in the case of 16, there are two solutions).

During class, we discussed a number of strategies including starting with the smallest possible prime number to be used here which is 3 (2 cannot be used because it would need another even number to add up to an even number). So if we are looking at the number 100 =, we try 3 + and see that 97 is a prime solution. Then we test 5 + but this fails because we need 95 (not prime); 7 + fails because 93 is not prime, but 11 + does work because 89 is prime. We continue until we get numbers that are closest to each other (in the case of 100 = 47 + 53, two consecutive primes is the last solution of the six solutions for 100).

Another strategy is to use vertical addition to try to find a number that you must add to your first prime to get the even number sum (for example, when looking for the prime pair that has a sum of 22, when we test 11, we set it up as a vertical sum:

     1 1

   +___

     2 2

Of course, it is easy to see that we need a one in each column to get a sum of 2 giving us 11 as the second prime. Some students realize that this strategy is just another form of subtraction.

The children were so excited when they solved a Goldbach sum. Incidentally, the Goldbach Conjecture has been solved for even numbers as large as 100 million.

The two worksheets that require the children to solve the Goldbach Primes to 100 (and even 114 with 10 solutions), I provided the answers. If the children just want to write down the answers that is OK as long as they do the vertical addition to show me that they understand this critical algorithm.