Similarities Between Landau's Problems: Goldbach's Conjecture and Other Conjectures in Number Theory

By Sr Javier Santos

Is there any relation between the Goldbach's conjecture and the twin primes conjecture? How many times has this question been asked not only by mathematics students but also by amateurs and even professionals in the discipline. As for the other Landau's two problems, the existence of infinite primes in the form of a square plus unity, and the Legendre's conjecture, are they related to each other or to the other two mentioned problems? Is there a statement whose formulation covers all conjectures and whose proof would prove or disprove them? Do their distributions respond to some well-established concrete phenomenon?

This work will give answers to these questions and will open doors to other places already suspected by many addicted to mathematical conjectures. Landau's four problems will be approached by readers with basic knowledge of mathematics (algebra, trigonometry, basic geometry and complex numbers) but without the usual background that characterizes the discipline of number theory. This does not mean that the finding of original conclusions that could shed some light on these conjectures is renounced.

This exhibition in its first part will lead us to focus on Goldbach's strong conjecture in a different way and to generalize the conjecture about the existence of infinite twin primes, clarifying the difference between the Kronecker's conjecture and that of Alphonse de Polignac, often confused. The Landau first problem will also be generalized and the well-known Bertrand's postulate on natural integers will be extended to Gaussian integers. All this to finally synthesize the problems in a more adequate framework, which explains the mechanism to which the distributions of these problems respond. For this, the new term Factorial Pressure related to the primorial numbers is minted, which will help to understand where, in the Goldbach-Kronecker distribution, the relative maxima and minima are located. Likewise, mathematical meaning is given to the primorials of exclusively Pythagorean primes or non-Pythagorean primes, which in this case explain where, in the Bertrand-Landau distribution, the relative maxima and minima are located.

Its second part, through a well-founded speculation on the distribution of primes in two dimensions, either in ℕxℕ or in ℤ i], will deduce how the functions can be that, bounded by intervals, predict the appearance of these prime elements.

• Format: Paperback
• Language: English
• ISBN: B0BCCYMFW2
• ISBN13: 9798849099330
• Release Date: August 2022
• Publisher: Independently Published
• Length: 166 Pages
• Weight: 0.88 lbs.
• Dimensions: 0.3" x 8.5" x 11.0"