The Probability and Density of Consecutive Integer Pairs with Specific Characteristics

In this article, we will delve into the probability and density of specific pairs of consecutive natural numbers, focusing on a mathematical sequence generated using the square root of 3. We will explore a brute force approach using the J programming language and then move on to a more theoretical analysis using probability theory.

Brute Force Solution Using J Programming Language

The J programming language is a powerful suite of utilities that allows for concise and efficient programming. In our example, we aim to generate a set of one million consecutive pairs of natural numbers and analyze them to understand the probability of a specific characteristic. Here is how we proceed:

Generate the first one million consecutive pairs of natural numbers. Multiply both integers in each pair by the square root of 3 (sqrt 3). Take the floor of each number in the pair after multiplying by 3. Subtract the two floors of each pair. Mark each result with a 1 if the pair difference is one; otherwise, mark it with a 0. Find the average of the one million marks.

The result of this process is as follows:

avg 1-/
0.267949

This indicates that the probability that an adapted consecutive pair (where the difference of the floors is 1) is approximately 0.267949, or 26.79%.

Understanding the Floor Functions and Their Impact

When the floor functions are applied, the differences between the two numbers in the pair can be either 1 or 2. The floor functions 'round' the product of each number and the square root of 3 to the nearest integer. The fraction of cases where the difference is 1 is denoted as p, and the fraction of cases where the difference is 2 is denoted as 1-p. The weighted average of these differences can be calculated as follows:

sqrt 3 ≈ 1.73
1 × p 2 × (1-p) 2 - sqrt{3}

Hence, the probability p that a pair of consecutive natural numbers differs by 1 is:

p 2 - sqrt{3}

Theoretical Analysis: Density of Adapted Pairs

Applying probability to nonnegative integers doesn't have an inherent meaning since there is no uniform distribution over the set of nonnegative integers. Instead, we consider the density of the set of nonnegative integers n that satisfy the condition that the floor of ( n sqrt{3} - 1 ) equals the floor of ( n sqrt{3} ). Let B represent the set of nonnegative integers less than B.

Let a be the number of integers in B that are 'adapted,' meaning ( lfloor n sqrt{3} - 1 rfloor lfloor n sqrt{3} rfloor ), and let m be the number of integers in B that are 'maladapted,' meaning ( lfloor n sqrt{3} - 1 rfloor 2 lfloor n sqrt{3} rfloor ).

We derive the following equations:

( a B ) ( a 2m lfloor B sqrt{3} rfloor )

From these equations, we can solve for a and m:

m ( lfloor B sqrt{3} rfloor - B )
a ( B - m 2B - lfloor B sqrt{3} rfloor lceil 2 - sqrt{3} B rceil )

The density of the set of positive integers n such that ( lfloor n sqrt{3} - 1 rfloor lfloor n sqrt{3} rfloor ) is given by:

lim_{B to infty} frac{lceil 2 - sqrt{3} B rceil}{B} 2 - sqrt{3}

This calculation gives us the theoretical probability of the specific characteristic in a large set of consecutive natural numbers.

Conclusion

In conclusion, through a combination of brute force computation and theoretical analysis, we have determined the probability and density of pairs of consecutive natural numbers with specific characteristics. This study demonstrates how mathematical insights can be applied to understand patterns and probabilities in number sequences involving the square root of 3.