From my experience, a mathematician is saddled by a curse. When you talk to ordinary people, they will claim that you are very good in adding, subtracting, multiplying, and dividing numbers. When you talk to people in the field of sciences and engineering, they will claim that you can differentiate and integrate the most mind-boggling expressions. As a number theorist, I do not know if I will be elated or offended by such remarks as I rarely do any of those activities. I also cannot count the number of times I have been asked about the real-life application of my publications.

So what is number theory? What is the point of studying this? And what do number theorists do if they are not plain number crunchers? Merriam Webster defines number theory as the study of integers and their properties. This is the old-age definition of number theory. In ancient times, number theorists investigated perfect numbers, or numbers equal to the sum of their proper factors like 6 = 1+2+3; friendly pairs or number pairs where one is equal to the sum of the proper factors of the other like 220 =1+2+4+71+142 and 284=1+2+4+5+10+11+20+22+44+55+110; twin primes, or pairs of numbers differing by 2 with each one divisible only by 1 and itself like 3 and 5 or 11 and 13; Pythagorean triples or numbers which appear as lengths of the sides of a right triangle such as 3,4 and 5; and the list goes on.

Most of these problems are anchored on a more general problem considered by number theorists: the factorization or decomposition of numbers into irreducible factors. This problem is very fundamental in various disciplines. In chemistry, a prime interest is the determination of the elements which make up a compound. For instance, we know that salt is NaCl , one sodium and one chlorine; water is H2O, two hydrogen and one oxygen; and carbon dioxide is CO 2, one carbon and two oxygen. In number theory, we deal with integers instead of compounds. For instance, 12 =22 X 3, two 2’s and one 3, 15=3 X 5, one 3 and one 5. In chemistry the indecomposable are what they call elements whereas in number theory, the indecomposable are called primes. Fortunately, for a chemist there are only a finite number of elements that have been discovered and thus everything can be written in the periodic table. For number theorists, it has been known since the time of Euclid (circa 365-300BC) that there are infinitely many primes. Thus an analog of the periodic table is not possible.

Over the years, the definition of number theory has remained the same but the scope of the word integers has expanded. Four hundred years of mathematical researches revealed that other than the usual numbers 1,2, etc., there are objects which behave like integers; among these are the polynomials and the Gaussian integers. Carl Friedrich Gauss (circa 1798) initiated investigation of numbers of the form a+b\sqrt{d}, where a,b, and d are integers and d being negative and not divisible by a square factor. These numbers behave like the integers however not all of them have the unique factorization property, a property that means any number can be written as a product of its irreducible factors in one and only one way. The integers that most of us are familiar with satisfy the unique factorization property. There is no other way of expressing 30 as a product of primes other than 2 X 3 X 5. It was natural, therefore, to expect that even the Gaussian integers satisfy the unique factorization property but Gauss himself conjectured that the property is satisfied only when d is one of -1, -2, -3, -7, -11, -19, -43, -67 and -163. It took until 1952 for this conjecture to be verified; that indeed these are the only cases where the Gaussian integers satisfy the unique factorization property.

The insatiable curiosity of number theorists was aroused rather than quenched by these findings. They began to investigate more structures which behave like integers and test the unique factorization properties. Thus came the concepts on algebraic integers, number theory on function fields, and algebraic geometry. Then the problem shifted from determining what you have and do not have if you have integers which do not satisfy the unique factorization property. And the simple ancient problem snowballed into the problems that modern-day number theorists are solving. Today, among the seven most difficult mathematical problems of the millennium identified by the Clay institute, two belong to number theory, the Riemann hypothesis and the Birch and Swinnerton-dyer conjecture.

Over the years, number theorists and real mathematicians in particular are becoming more and more engrossed with the problems born out of mathematical curiosities and thus becoming more and more detached from the problems of real life. But this is not just true of mathematics. It is worth noting that Isaac Newton made revolutionary advances in mathematics and physics at the height of the bubonic plague. The works of Guglielmo Marconi were initially judged as toys and devoid of any real life application by his contemporaries. When asked of what good was his invention, Michael Faraday replied, “What good is a new-born baby?” Thus, the beauty and value of mathematics is not judged by its application and its impact on society but rather by its logical consistencies. This reminds me of the most published mathematician Paul Erdos’ reply to the question why numbers are beautiful. He said it is like asking why Beethoven’s Symphony No. 9 is beautiful. If you do not see why, someone cannot tell you. The prominent English mathematician Godfrey Harold Hardy also had a say on this.

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. A painter makes patterns with shapes and colors, a poet with words. A painting may embody an idea but the idea is usually commonplace and unimportant....Poetry is not the thing said but the way of saying it.

Fortunately, with advances in information science, numerous applications of long-known results in mathematics have been found. Just as we have learned to appreciate the works of Newton, Marconi, and Faraday, number theory concepts now are highly used in scrambling messages so that only the intended recipient can unscramble and make sense of the message; in devising schemes for sharing secrets in a group so that the cooperation of everyone is required for the secret to be revealed; in digitally signing documents; in verifying the authenticity of documents; and in time stamping. The baby that number theorists gave birth to has now grown.