Number theory is a branch of mathematics. Number theorists study numbers, especially natural numbers and prime numbers. A frequently asked question is "How many of them (satisfying certain properties) are there?" Since there are other accessible sources describing what number theory is, we give here how one is led to prime numbers starting from a minimal input. This is in part an attempt in describing how number theory expands itself, turning itself into the vast field as it is today. We warn that it is neither a historical nor a mathematically rigorous account.
The starting point is probably that we count in our everyday life. We start counting from either 0 or 1, then 2, 3, and so on. These numbers are called natural numbers. The next thing to do is probably to add two numbers. For example, when going shopping, you would be adding natural numbers quite often. One can also multiply two numbers. This is very useful when one gets tired of adding the same number repeatedly. What else can one do with numbers? Going the opposite of adding, one is led to subtraction. While being practical as it is, the natural question of "What do we do with 3 minus 5?" arises. This is when we invent negative numbers. One can proceed in the same vein, using multiplication instead of addition, and is led to division. The question is then "What is 3 divided by 5?". The invention in this case is the fractions; the collection is the set of rational numbers.
Let us note that to reach the set of rational numbers starting from natural numbers, we were moved by practical reasons (say, shopping) and by "abstract" reasons (say, of negative numbers). Of course nowadays, negative numbers are practical enough (say, in bookkeeping). There are often cases where abstract theories find their applications and where practical purposes propel abstract theories.
Let us define prime numbers. A natural number x is divisible by another natural number y if x divided by y is a natural number. A prime number is a (positive) natural number which is divisible only by 1 and itself (and greater than 1). There are questions concerning prime numbers. For example, it was proved that there are infinitely many primes, but we do not know how often a natural number is a prime number. This question led to the invention of Riemann's zeta function.
Group Members
Shushi HarashitaI am studying stratifications and foliations on the moduli space of polarized abelian varieties in positive characteristic and more generally Shimura varieties. Our researches are expected to contribute to establishing the Langlands correspondence, which is one of the main problems in number theory and arithmetic algebraic geometry.
Satoshi Kondo
My research interest is in arithmetic geometry. We use tools from algebraic geometry to study problems in number theory.
