Edmund Georg Hermann
Landau (1877-1938) was a German mathematician who wrote more
than 250 articles on number theory, which had a major impact
on the development of the subject. From 1909, he was a professor
at the University of Göttingen.

At the International Congress of Mathematicians in 1912 (Cambridge,
England), Landau listed four problems (conjectures) about prime
numbers that had been unproven for centuries, and argued that
they were unlikely to be proved in the 20th century by known
mathematical methods. His claim has been borne out by life,
as these conjectures remained unproven in the 20th century,
and are unresolved even today. These four problems have been
known as **Landau's problems**
since the 1912 Congress of Mathematicians:

**1.**
Christian Goldbach (1690-1764) - **Even Goldbach's conjecture:
**

Each even number which greater than two is the sum of two prime
numbers.

**2.** *Twin Primes conjecture:*
There are infinitely many p primes, where p+2 is also a prime.

**3.**
Adrien-Marie Legendre (1752-1833) - *Legendre's conjecture:
*

There is always at least one prime number between two consecutive
square numbers.

**4.** *There exists infinitely
many prime numbers in the form n2+1.*

I dedicate the
present volume, in which I publish the proofs of these four
conjectures, to mark the **110th anniversary**

*of the formulation of Landau's problems*, and as
a tribute to all the great mathematicians who have made enormous
efforts (using traditional number theory) to prove Landau's
problems up to the present day. I would like to express my particular
respect to Edmund Georg Hermann Landau's insight that **"these
problems are unlikely to be provable in the 20th century by
known mathematical methods"!**

In his 1993 lecture
held in Budapest, Pál Erdõs commented Landau's
1912 problems the following way:

**"They may be solved in the next century."**

In the beginning
of the 2000s, Professor János Pintz wrote a rather detailed
analysis of the results achieved regarding the problems enumerated
by Landau, including a 9-page bibliography listing the sources
used for the analysis.

The proofs presented
below are based on my **"complementary prime sieve"
**number theorem published at the dawn of the 21st century,
in 2001 (see APPENDIX), and the new methodology based on this
theorem (*"Dénes type Symmetric Prime Number theorem"*).

To illustrate the
further successful application of this methodology, in the two
concluding chapters of the volume,

I present - *as a kind of mathematical dessert* - the proof
of the conjecture concerning *Mersenne primes*, which has

not been proved for 400 years, and *the proof of the twin
primes theorem generalized by myself*.

All these results,
-despite the failure to publish the papers- included in this
volume,

have deeply confirmed my belief in A. Turing's idea that:

*"Sometimes it is the people no
one can imagine anything of who *

do the things no one can imagine." (A.
M. Turing)

**C O
N T E N T**

**FOREWORD**

1. Dénes
type Symmetric Prime Number theorem and its application to*
*proof of the* Even Goldbach conjecture*

2. Proof of the
*Twin Primes conjecture*

3. Proof of the
*existence of prime number between successive squares*

4. Application
of the Dénes type Symmetric Prime Number theorem to proof
of there *exist infinitely many primes of the form n2+1*

5. Proof of the
existence of *infinite number of Mersenne primes*

6. *Generalized
Twin Primes Theorem*

**APPENDIX**

Original journal copy of the [Dénes
2001] article.

Budapest, February
2022------------------ ---- ------------------------
The Author