4/n = 1/a + 1/b + 1/c
Notice that this is a very simple type of problem. It does indeed involve only arithmetic-- no algebra, trigonometry, calculus, or weird stuff you haven't heard of yet. It's saying that if I pick some integer, let's say 3, then I should be able to find a, b, and c, such that:4/3 = 1/a + 1/b + 1/c
Fractions which have 1 in the denominator are known as "Egyptian fractions" in the world of number theory, since the people of ancient Egypt wrote all fractions that way. So I need to find a way to represent four-thirds as the sum of three Egyptian fractions. How about this?4/3 = 1/2 + 1/2 + 1/3
That was fairly easy. Luckily, I was able to use the same number twice. The Erdos-Strauss conjecture is stating that I can do this not only for four-thirds but also for four-fourths, four-fifths, four-sixths, and so on. As you can see, this conjecture is fairly easy to state, but could be extremely difficult to prove. Number theory problems often work that way. Mathematicians have checked the Erdos-Strauss Conjecture up to an n value of about one trillion, and so far it does hold true, but it is yet to be proved.Erdos, by the way, was this famous Hungarian mathematician of the 20th Century. You might enjoy learning more about him. He was never a professor anywhere. Instead, he was constantly itinerant, traveling around the world, lecturing and posing problems. Erdos would stay with various other mathematicians. I read one article by someone who had hosted Erdos for a few months. The host reported that Erdos slept very little and worked on math problems constantly. One night, the host awoke to the sound of Erdos banging pots and pans in the kitchen to wake up his host so that they could discuss one of Erdos' new ideas. I don't know who Strauss is, but my guess is that the two of them came up with their conjecture in Strauss' kitchen.
Another thing I really like about number theory is that it always makes you think of new problems. After spending a little time with the Erdos-Strauss Conjecture, I started wondering about other rational numbers besides those with four in the numerator. What about 5/n? Can I write 5/n as a sum of three Egyptian fractions, no matter what n is? Maybe I will have to use more than three of them in order to get it to work. Come to think of it, just how well did those Egyptian fractions work for the Egyptians, anyway? Could they express any rational number, m/n, as a sum of Eygyptian fractions? Of course, one could always add 1/n + 1/n + 1/n..., a total of m times, but is there also another method available? Suppose I impose the condition that all of the Egyptian fractions have to have different denominators. Can I still find a way to make any m/n I want to make?
If you do your own internet search on Number Theory, you will find many
resources. Even if you don't understand all of the conversation at one
of the sites, just seeing the problems that they are looking at will give
you an idea of what number theory is all about. It will also give you a
picture of what it is that mathematicians do all day long. Here are a few sites that I have found:
Just to help you read those pages, I should tell you that the topic of modular arithmetic is often referred to as "Congruences" and that the word "mod" that we have been using is actually short for the word "modulo" that you may see in other documents.
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