Number theory

Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer numbers. In order to understand the proofs of the results similar to the conjecture, and very likely also to prove the conjecture itself, a sound knowledge of number theory is required. But this kind of knowledge rarely is part of a mathematician's curriculum. Indeed number theory is considered a very specialistic subject which can be taken under consideration, as an example, for some university exams selected by the student, or a doctorate, for the most enthusiastic people. But it’s not necessary going this far for undertaking a serious study of number theory. Many notions are within the reach even of a first year student of a scientifical degree course, since they require just a good knowledge of school level Math and of real analysis in one variable.

We started from these considerations for conceiving this site section, with the aim of letting approach to number theory as many people as possible, giving at the same time useful elements to who wants to test himself or herself with the proof of the conjecture.
For make our writings more interesting, we focused the itinerary towards a goal, and as a goal we chose one of the most important theorems about prime numbers, the so called Prime number theorem. A bit like in a detective story - in which several evidences are collected and, thanks to them, the various aspects of the story get more and more clear, up to identifying the culprit - in the way towards the proof of the theorem, we'll introduce more and more notions about number theory, and prove several intermediate results, getting more and more close to the final result.

But what does the prime number theorem state? The statement can be expressed by the following formula:

\pi(x) \sim \frac{x}{\log x}

where \pi(x) is the number of primes less than or equal to x, and \log x is the natural logarithm of x. The symbol \sim is an asymptotic equivalence whose meaning will be clear in one of the next posts but, simplifying, we can read the formula by saying that there are about \frac{x}{\log x} prime numbers less than or equal to x, and this estimation becomes more and more accurate as x increases. We encourage you to verify it with different values of x!

The theorem history is interesting by itself. In fact, for a lot of time the only known proofs were based on complex analysis and other quite advanced notions; moreover matematicians thought that complex analysis was somehow "necessary" for the proof. Instead in 1949 Paul Erdős and Atle Selberg independently conceived an "elementary" proof, that is a proof based only on real analysis, and that inspired astonishment in the mathematical community.
The proof we'll present is just the one by Erdős and Selberg. The source we started from is the text "An introduction to the theory of numbers" by G. H. Hardy and E. M. Write. In particular, the proof is contained in chapter XXII, based in turn on the preliminar notions introduced in chapters I and II. They are about 40 pages in total, but very dense. We are reworking this material, adding new original stuff: we introduced some examples, highlighted the key ideas and techniques, and added some details which were not explicit in the original text; we'll also try to enrich the dissertation with various fun facts.

1. The definition of prime number

We start our study of prime numbers explaining the definition of prime number. It's commonly known that a prime number is a positive integer number divisible only by itself and 1. This is true except for number 1, that according to that should be prime (1 is divisible by 1, which is also the number itself), but indeed it isn't. The number 1 in fact is a priori excluded from prime numbers.
CONTINUE

2. Binomial estimates

Binomial coefficients are important for studying prime numbers. In this post we see in particular how to estimate, both upwards and downwards, a binomial coeffient. The reason we look for estimations is that their definition turns out to be quite difficult to use, both in calculations and in analytical arguments. So, many times it's preferrable to employ an estimation, affected by error but easier to use, rather than the exact value.
CONTINUE

3. The product of the first prime numbers: an overestimation

A way to start investigating the sequence of prime numbers is to consider, starting from the beginning, portions of increasing size. For example, you can start with the study of the prime numbers up to 10, then prime numbers up to 20, up to 40, etcetera, doubling each time the size of the considered portion. This approach will reveal itself to be effective, as it is based on product. In fact, since prime numbers are defined starting from the concepts of divisibility and product, it's not surprising that, while studying them, the most used operation is indeed the product.
CONTINUE

4. Bertrand’s postulate

The goal we establish in this post is to prove the Bertrand's postulate, proposed in 1845 by the French mathematician Joseph Louis François Bertrand as a conjecture, and later proved by the Russian Pafnutij L'vovič Čebyšëv. It states that, for all n > 0, there exists a prime p between n and 2n.
CONTINUE

5. The least common multiple of the first positive integers

We know that a way to compute the least common multiple between two or more integer numbers is based on their decomposition into prime factors. But there exists a particularly interesting case in which decomposition into prime factors can be avoided: it's the computation of the least common multiple of all positive integer numbers up to a certain number x, that is LCM(1, 2, ..., x).
CONTINUE

1. The definition of prime number

We start our study of prime numbers explaining the definition of prime number. It's commonly known that a prime number is a positive integer number divisible only by itself and 1. This is true except for number 1, that according to that should be prime (1 is divisible by 1, which is also the number itself), but indeed it isn't. The number 1 in fact is a priori excluded from prime numbers.
CONTINUE

2. Binomial estimates

Binomial coefficients are important for studying prime numbers. In this post we see in particular how to estimate, both upwards and downwards, a binomial coeffient. The reason we look for estimations is that their definition turns out to be quite difficult to use, both in calculations and in analytical arguments. So, many times it's preferrable to employ an estimation, affected by error but easier to use, rather than the exact value.
CONTINUE

3. The product of the first prime numbers: an overestimation

A way to start investigating the sequence of prime numbers is to consider, starting from the beginning, portions of increasing size. For example, you can start with the study of the prime numbers up to 10, then prime numbers up to 20, up to 40, etcetera, doubling each time the size of the considered portion. This approach will reveal itself to be effective, as it is based on product. In fact, since prime numbers are defined starting from the concepts of divisibility and product, it's not surprising that, while studying them, the most used operation is indeed the product.
CONTINUE

4. Bertrand’s postulate

The goal we establish in this post is to prove the Bertrand's postulate, proposed in 1845 by the French mathematician Joseph Louis François Bertrand as a conjecture, and later proved by the Russian Pafnutij L'vovič Čebyšëv. It states that, for all n > 0, there exists a prime p between n and 2n.
CONTINUE

5. The least common multiple of the first positive integers

We know that a way to compute the least common multiple between two or more integer numbers is based on their decomposition into prime factors. But there exists a particularly interesting case in which decomposition into prime factors can be avoided: it's the computation of the least common multiple of all positive integer numbers up to a certain number x, that is LCM(1, 2, ..., x).
CONTINUE