March 18, 2013 - Happy Birthday, Christian Goldbach

How would you like to be known for a conjecture?

Some of you may be thinking, “I don't know, what's a conjecture?”

A conjecture in mathematics is what a hypothesis is in science: 

a good, educated “guess,” a possible explanation of how the universe works (in science) or how a number system works (in math).

Another way of defining conjecture is that it is a conclusion reached on the basis of incomplete information.

A conjecture is not a proof. Absolute proof is impossible in science, but in mathematics a proof is a logical argument that establishes a fact or truth. And Goldbach's Conjecture (the thing he is most famous for) has never been proved. It is one of the oldest and best-known unsolved problems in all of mathematics.

The conjecture is that every even integer (counting number) larger than 2 can be expressed as the sum of two prime numbers.
The even numbers from 4 to 28 as
sums of two primes.

To understand this conjecture...

You have to know what even numbers are: they are numbers that can be evenly divided by 2. The even numbers larger than 2 start 4, 6, 8, 10, 12 and go on and on to infinity. Because there are an infinite number of integers larger than 2, we can never just test them all and prove Goldbach's Conjecture.

You also have to know what prime numbers are: they are numbers that have no factors other than one and themselves. In other words, there aren't two numbers that can be multiplied together to form a prime number—other than (obviously) the number itself times one.

So numbers like 9 and 10 are not prime because they have factors like 3 and 2 and 5. Check it out:

9 = 3 x 3                    10 = 2 x 5
AND (obviously)       AND (obviously)
9 = 1 x 9                    10 = 1 x 10

Numbers like 5 and 7 are prime because they only have the obvious factors:

          5 = 1 x 5                     7 = 1 x 7

The number 2 is also considered prime because it only has the obvious factors:

          2 = 1 x 2
There doesn't appear to be a
pattern to easily find prime
numbers. Notice, however, that
except for the first few prime
numbers, all prime numbers
end with a 1, 3, 7, or 9.

But the number 1 is special and is not considered prime. (If you want to know why, see PrimePages.) 

There are lots of other prime numbers—in fact, an infinite number of prime numbers. Here is a partial list of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...and on and on. 

The largest prime number we know so far (as of February 2013) has 17,425,170 digits. It isn't the number 17,425,170, itself (obviously), but that is how many numerals are in the number. That's a pretty big number, and it took a computer to come up with it. 

Just see how large this number is:


...and note that it is only 45 digits long! So that largest prime number is...


...really, really large!

Okay, back to the conjecture:

4 is an even number larger than 2.
Testing to see if it can be expressed as the sum of 2 prime numbers, we come up with:

         4 = 2 + 2

How about 6 and 8 and 10?

         6 = 3 + 3                 8 = 3 + 5
       10 = 3 + 7     AND  10 = 5 + 5
This graph shows that, the higher
the even number, the more sets of
prime numbers that add up to
that number.

Higher even numbers can be expressed as the sum of several sets of prime numbers. Check out all the ways we can add two prime numbers to make 100:

  3 + 97 = 100               11 + 89 = 100
17 + 83 = 100               29 + 71 = 100
41 + 49 = 100               47 + 53 = 100

Did I mention it was Goldbach's birthday today? Christian Goldbach was born on this date in 1690 in Prussia (which later became Germany). After he finished his university studies, he continued his education by taking a tour of Europe and meeting and talking to other mathematicians. He ended up settling down to do the bulk of his work in Russia, where he was tutor to Peter II before the latter became the czar, and where he also came up with his famous conjecture.

Also on this date:

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