How to Find All Natural Numbers Using the Difference of Squares

The problem of finding all natural numbers that can be expressed as the difference of two squares is a fascinating subject in number theory. This article explores the mathematical techniques and concepts necessary to solve this problem, including the exploration of n-th powers, p-adic integers, and Hensel's lemma. We will delve into the steps and reasoning behind solving such equations.

Understanding the Basic Equation

The starting point for our exploration of natural numbers that can be expressed as the difference of two squares is the equation:

m2 - t2 13n

This equation can be factored as:

(m t) (m - t) 13n

For this to hold true, both factors m t and m - t must be powers of 13, since 13 is a prime number. Let's denote these factors as follows:

mt 13ku

m-t 13lv

where kl and uv 1 in Z13 (the ring of 13-adic integers).

Exploring the Properties of Natural Numbers

We'll start by assuming that n is an odd number. Given this condition, we can deduce the following:

k and l are different and kl. Therefore, we can write:

2m 13ku (13lv)

Given uv 1 and kl n, we have:

4m2 132ku2 132lv2

Since 4m2 13n24

we can equate the two:

12203n 132ku2 132lv2

From this, we deduce that k 0 and l n, giving:

mt u

m-t 13nv

This implies that the 13-adic expansion of m is the same as that of t up to the n-th term.

Dimensions and Exponential Growth

Given the relationship:

m2 13n3

we see that m is approximately 13n/2. This means that for a given value of n, the digits of m are highly constrained.

For instance, if n 9, we know that all mi for i in [5, 8] must be zero. The actual coefficients for t are 4868122199, which means that for n 9 there are no solutions.

The first few indices where coefficients for t are zero include 20, 25, 32, and 43. Moreover, the 100th index is the only occurrence of more than one consecutive zero coefficient among coefficients below 200, specifically at 180 and 181.

Using a table of the first ten thousand coefficients, we can see that the only possible consecutive zeros occur at indices 3107 to 3109, 8213 to 8215, and 8552 to 8554. This strongly suggests that no other solutions exist beyond the observed indices.

Conclusion

In conclusion, the exploration of natural numbers that can be expressed as the difference of two squares involves a deep dive into number theory, including the use of p-adic integers and Hensel's lemma. The constraints imposed by the 13-adic expansion and the behavior of the coefficients in the expansion provide valuable insights into the nature of solutions to these equations.