In this article, we discuss the concept of infinity (∞), infinitesimals □ , and a new mathematical operation that expands calculations into the realm of the infinite. This breakthrough allows for the computation of infinite numbers, leading to new possibilities for mathematical research and practical applications. By introducing infinite numbers, numerous advantages arise, such as accurately representing limits, divergences in integrals and infinite series, and extending calculations to infinity. This mathematical advancement opens doors to exploring uncharted territories, offering valuable insights and breakthroughs.

This study delves into a unique binary pattern found within the wellknown 3n+1 problem, or Collatz conjecture. Through careful analysis of the steps in the 3n+1 sequence, we have discovered a special binary representation that captures the behavior of all positive integers undergoing this transformation. With this new understanding, we have provided a solid proof confirming the validity of the 3n+1 problem for all positive integers. Our method goes beyond the need for extensive computational confirmation, providing a simple and elegant resolution to a long-standing mathematical mystery.

The Diophantine equation for which p is prime 2^n = p^2 + 7 and n is integers with p > 0, is examined in this work. It is demonstrated by examining the characteristics of the equation that the only solutions meeting these requirements are p = 3 with n = 5 and p = 5 with n = 7. In order to analyze the above equation, it is necessary to look at the properties of primes and how they relate to powers of two. These results advance our knowledge of prime numbers and how they behave in certain mathematical problems.

One of the most important unresolved mysteries in mathematics is the Riemann Hypothesis, which suggests a fundamental connection between the non-trivial zeros of the Riemann zeta function and the distribution of prime numbers. Here, we explore the fascinating union of trigonometric functions, prime numbers, and the Riemann zeta function through an examination of the zeros in the statement ln(sec(π·n log(n))). We demonstrate a strong mathematical connection between these components, providing information on the mysterious properties of prime numbers and their complex relationships to basic mathematical operations. Our thorough investigation adds to the current discussion of the Riemann Hypothesis by offering possible solutions and deepening our comprehension of the intricate relationship between number theory and analytic functions.

The Collatz conjecture is a well-known number theory puzzle that states that every positive integer would eventually converge to the trivial cycle of 1, 2, 1, 2,... when repeatedly exposed to a particular transformation. In this transformation, even numbers are divided in half, odd numbers are tripled, and one is added. In this study, we present a new method for creating a directed graph and using it to display and analyze Collatz sequences. Our technique creates what we call a Collatz directed graph by joining an endless number of simple directed graphs, each of which corresponds to a natural number. We show by careful mathematical analysis that all positive integers are included in this Collatz directed graph. Moreover, we give an evidence that verifies the Collatz conjecture by showing that the sole cycle in this graph is the trivial cycle of 1, 2, 1, 2,... We also prove that there is no sequence that diverges to infinity in this graph. Our results provide insights into the fundamental structure of Collatz sequences and further our knowledge of the Collatz conjecture .

The Collatz conjecture, a longstanding mathematical puzzle, posits that, regardless of the starting integer, iteratively applying a specific formula will eventually lead to the value 1. This paper introduces a novel approach to validate the Collatz conjecture by leveraging the binary representation of generated numbers. Each transition in the sequence is predetermined using the Collatz conjecture formula, yet the path of transitions is revealed to be intricate, involving alternating increases and decreases for each initial value. The study delves into the global flow of the sequence, investigating the behavior of the generated numbers as they progress toward the termination value of 1. The analysis utilizes the concept of probability to shed light on the complex dynamics of the Collatz conjecture. By incorporating probabilistic methods, this research aims to unravel the underlying patterns and tendencies that govern the convergence of the sequence. The findings contribute to a deeper understanding of the Collatz conjecture, offering insights into the inherent complexities of its trajectories. This work not only validates the conjecture through binary representation but also provides a probabilistic framework to elucidate the global flow of the sequence, enriching our comprehension of this enduring mathematical mystery.

When discussing the Diophantine quintic equation $p(a^5 + b^5) = q(c^5 + d^5)$ where p is a prime and q is an integer, there is a clear gap between the mathematical literature and online forums. Because of the intrinsic complexity of parameterizing fifth-degree equations, this equation re- mains largely unexplored. In this work, we approach this quintic problem through algebraic methods with the goal of providing numerical solutions that illuminate its properties and behaviors.Our research uncovers in- triguing patterns and connections that shed light on the enigmatic nature of Diophantine quintic equations in this particular form.