A rational number is a number that can be written as a quotient of two whole numbers. Examples are 2/3 , 1/2 and 5 (note that 5=5/1). Not all numbers are rational. In fact, there are many more numbers that are not rational (called irrational numbers) than there are rational. Probably the best known example of an irrational number is √ ̅2 (the first known proof of the irrationality of √ ̅2 dates back to around the sixth century B.C.)
One of the oldest mathematical problems is a problem known as the congruent math problem. This problem appeared in an Arab manuscript written before 972 A.D. Before stating the problem, recall that a right triangle is a triangle in which one of its angles is 90°. Also recall that the area of a triangle is equal to 1/2 x base x height. A number n is said to be a congruent number if n is equal to the area of a right triangle where the lengths of all of its sides are rational numbers. The congruent number problem is: given a rational number n determine if n is a congruent number.
In the tenth century A.D. the Arabs already had tabulated a list of many congruent numbers. For example, they have shown that the number 6 is a congruent number: we have a right triangle with (rational) sides 3,4,5. The base is equal to 3, the height is 4 and the hypotenuse is 5. The area of this triangle is 1/2 x 3 x 4=6. Another example known to the Arabs is that 5 is a congruent number. Here one uses the triangle with sides 3/2, 20/3, 41/6. Is the number 1 congruent? This was an open question for many years. Not until the 17th century was this question answered: the French mathematician Pierre de Fermat in 1640 showed that 1 was not congruent.
Although it has been over a thousand years since this problem was postulated, we have not been able to solve it. The difficulty with trying to determine whether a given number is congruent is that one cannot simply list all rational triangles because there are an infinite number of them. Also, some of the techniques used by Fermat and others to prove that a number is or is not congruent cannot be applied to all numbers.
The congruent number problem has an interesting link with an important modern mathematical problem called the Birch and Swinnerton-Dyer conjecture (BSD conjecture, for short). The BSD conjecture is a very important problem in mathematics. The problem is so important, that an $1,000,000 reward is offered (by the Clay Mathematical Institute) for a correct solution.
It has been shown that a solution to the BSD conjecture would ultimately solve the congruent number problem. So are we far from finally resolving the latter problem? Probably yes. Many mathematicians are working on the BSD conjecture, but due to its difficulty it will likely be many years (if not decades) before this problem is solved.
