The Intersection of Primes, Pseudoprimes, and the Mathematical Universe

Introduction

The question you have posed delves into a fascinating interplay between number theory and the fabric of the mathematical universe. Let's explore what it would mean if every number in the sequences (6n-1) and (6n 1) started out as prime, and whether pseudoprimes and primes might have different extensions across the cosmos.

Primes and Pseudoprimes: A Mathematical Journey

Prime numbers are the building blocks of numbers, and the study of primes is one of the most fundamental and enduring areas of number theory. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A pseudoprime, on the other hand, is a composite (non-prime) number that has properties similar to primes in certain contexts. For example, a Fermat pseudoprime is a composite number (n) that satisfies the congruence (a^{n-1} equiv 1 pmod{n}) for some integer (a) that is not a multiple of (n).

Exploring the Sequences (6n-1) and (6n 1)

Consider the sequences defined by (6n-1) and (6n 1). If every number in these sequences started out as a prime, this would be a highly unusual and significant statement. The Ulam spiral is a graphical representation that has highlighted prime numbers in these kinds of sequences, and it has been observed that there are long diagonals composed entirely of prime numbers, although this is not a strict rule.

Logical Consistency

From a logical perspective, if every number in the sequences (6n-1) and (6n 1) were to be prime, this would contradict our current understanding of number theory. Prime numbers are the exceptions and not the rule, and the density of primes decreases as we move further along the number line. This means that any assumption that all numbers in these sequences are prime leads to a logical inconsistency. Such a scenario would require us to rethink our foundational concepts in number theory, specifically the Peano Arithmetic system, which is the basis for our current understanding of natural numbers.

The Mysterious Pseudoprimes

Pseudoprimes, despite their mathematical nature, have a certain enigmatic quality to them. They exist in a shadowy realm between primes and composites, challenging our understanding of number properties. For pseudoprimes to 'stop at the edge of the universe' is a concept that defies conventional mathematical reasoning and ontological frameworks. Mathematical objects like pseudoprimes exist in a completely abstract and axiomatic context, separate from the physical universe. Philosophically, one might entertain the idea that the universe has constraints beyond our current understanding, but from a purely mathematical standpoint, pseudoprimes extend as far as the proofs and theorems can take us.

Time, Space, and the Mathematical Continuum

The relationship between time and space in the context of mathematics provides a fascinating layer to consider. In number theory, the concept of cyclical math can be seen through patterns that repeat and extend infinitely. The Riemann Hypothesis, for instance, deals with the distribution of prime numbers, which exhibit a kind of periodicity despite the apparent randomness. The idea of a 'cyclical math' extends to the cyclical nature of modular arithmetic, where operations repeat in a regular pattern.

Interpreting the Question

The question about primes and pseudoprimes stopping at the edge of the universe is more fitting for a philosophical debate than a mathematical one. In mathematics, the concept of the 'edge of the universe' is not applicable. Mathematical objects don't have physical boundaries; they exist purely as logical constructs. Therefore, while we can entertain the notion of cyclical math and the complex behavior of primes and pseudoprimes in various contexts, the idea that pseudoprimes or primes would 'stop' in any physical sense is not meaningful within the axiomatic framework of number theory.

Conclusion

The exploration of primes, pseudoprimes, and the sequences (6n-1) and (6n 1) leads us to a deeper understanding of the intricate nature of number theory. While the question you've raised is significant and thought-provoking, the logical and ontological framework of mathematics does not support the idea that primes and pseudoprimes would have different extensions based on physical constraints. Instead, they exist and behave as we have defined them, extending infinitely within the mathematical continuum.