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).
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. |
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:
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:
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
...and
note that it is only 45 digits long! So that largest prime number
is...
...well...
...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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