Puzzles, Number

The study of number puzzles has fascinated mathematicians and others for thousands of years. These studies reveal much about the way our number system is constructed. Studies of number puzzles also often produce new methods of mathematical thinking and new computer programming techniques. These classic number puzzles are explained in the following paragraphs.

Triangles

Many people know that 3² + 4² = 5². There are many other combinations of three whole numbers that have the same property, such as 5² + 12² = 13² and 7² + 24² = 25². These numbers are called Pythagorean triples because of the property of right triangles known as the Pythagorean Theorem (after the Greek mathematician Pythagoras).

Since the time of the early Greeks, mathematicians have known of a way of generating any number of Pythagorean triples. If m and n are any two integers , then a triangle with sides X, Y, and Z can be "generated" by these formulas:

m ²- n ² = X

2mn = Y

and

m ² + n ² = Z. The illustration below gives a graphical depiction.

It is obvious that there are infinitely many different sets of Pythagorean triples that can be generated using this formula.

Fermat's Last Theorem

The equation x ² + y ²= z ² has many different solutions which can be generated by the technique in the paragraph above. What about the equation x ³ + y ³= z ³? Or the equation x ⁴ + y ⁴= z ⁴? Do these equations have integral solutions or even rational solutions? (A rational number is any number that can be written as where a and b are integers and b ≠ 0.) In general, does the equation x ⁿ+ y ⁿ= z ⁿ have rational solutions for values of n greater than 2? The surprising answer is no.

The statement that "x ⁿ+ y ⁿ= z ⁿ has no rational solutions for values of n greater than 2" is referred to as Fermat's Last Theorem because the famous mathematician Pierre de Fermat wrote in the margin of a book, "I have discovered a truly marvelous proof of this, which, however, the margin is not large enough to contain."

Fermat's reputation as a mathematician was so great that everyone assumed he had found a proof. However, no one has ever been able to rediscover it. The proof of Fermat's Last Theorem, which was published by Andrew J. Wiles in the Annals of Mathematics in 1995, is based on other theorems that would not have been available to Fermat.

So, did Fermat actually have a proof? Probably not. Although he claimed to have found a proof of the theorem, he spent much time and effort proving the cases n = 4 and n = 5. Had he derived a proof to his theorem earlier, there would have been no need for him to study these specific cases.

Beyond Infinity

Can anything be bigger than infinity? At first glance, the question seems ridiculous. What is normally meant by infinity, ∞, is the limit of a mathematical operation. If and n is allowed to get progressively smaller and smaller, approaching zero, we say that x approaches ∞. Since n never reaches zero, x never reaches ∞. So ∞ is a "potential" infinity, approached but never reached.

There are also actual infinities. Suppose you went to a football game and looked around and saw that every seat was taken. You know the stadium holds 50,000 people, so you could say with certainty that there were 50,000 people at the game, even though you had not counted them. The people and the seats had a one-to-one correspondence.

In the same way, any set of numbers that can be placed in one-to-one correspondence with the set of whole numbers has the same size as the set of whole numbers. That means that the set of even numbers is the same size as the set of whole numbers. This is one of the curious properties of transfinite numbers discovered by the mathematician Georg Cantor. He also proved that the set of rational numbers can be counted (placed in one-to-one correspondence with the whole numbers) and is also the same size.

However, the set of real numbers cannot be counted. Real numbers contain numbers such as pi (π) and that cannot be written as the ratio of two whole numbers. There is no way to place the set of real numbers into one-to-one correspondence with the whole numbers because there will always be numbers left over. Hence, the set of real numbers is larger than the set of rational numbers. In other words, the size of the set of real numbers is a larger infinity! Cantor discovered an entire class of transfinite numbers of ever-larger size.

see also Fermat's Last Theorem; Infinity; Numbers, Rational; Numbers, Real; Numbers, Whole.

Elliot Richmond

Bibliography

Beiler, Albert H. Recreations in the Theory of Numbers—The Queen of Mathematics Entertains. New York: Dover Publications, 1964.

Leiber, Lillian R. and Hugh Gray Lieber. Infinity. New York: Rinehart and Company, 1953.

Tietze, Heinrich. Famous Problems of Mathematics. Baltimore: Graylock Press, 1965.