Solving Diophantine Equations: The Unique Positive Integer Solution to (x^5 - y! 2736)

Diophantine equations are a fascinating branch of mathematics that deal with finding integer solutions to polynomial equations. In this article, we explore a specific problem: solving (x^5 - y! 2736). We will show that ((x, y) (6, 7)) is the only positive integer solution to this equation. This article is divided into clear sections to provide a step-by-step understanding of the solution process, complete with detailed explanations and analytical insights.

Introduction to the Equation

Consider the equation (x^5 - y! 2736). Here, (x) and (y) are positive integers, and (y!) denotes the factorial of (y). A Diophantine equation specifically seeks solutions in the integers. We start by breaking down the given equation and exploring its properties before proceeding to find the solution.

Analyzing the Factorial Component

The factorial function, denoted (y!), has special properties, particularly regarding its last digit. For (y geq 5), (y!) ends in either 0, 2, 4, 6, or 8. This is due to the presence of factors 2 and 5 in (y!) for (y geq 5). Therefore, the expression (y!) must satisfy certain modular conditions.

Modular Arithmetic Constraints

For (x^5 - y! 2736), (y!) must be congruent to 2736 modulo 10. Since (2736 equiv 6 mod 10), we can deduce that (y! equiv 6 mod 10). This limits the values of (y) to (y ) such that (y! 6, 72, 120, 720, ldots).

Furthermore, (x^5 equiv y! mod 10), and given (x^5 equiv x mod 10), (x) can only take values that make (x^5 equiv 6 mod 10). The possible values are 6 and 4 (since (4^5 1024 equiv 4 mod 10) and (6^5 7776 equiv 6 mod 10)). However, (y 1, 2, 3, 4, 5) yield different last digits for factorials, specifically 1, 2, 6, 4, and 0, respectively. Thus, (x 6) or (x 4), but we will test the case where (x 6).

Exploring Larger Values of (y)

When (y geq 5), (y!) must be divisible by 10. This suggests (x 6). We now check if (x 6) fits the equation. We solve for (6^5 - y! 2736): [begin{align*} 6^5 7776, 6^5 - y! 2736 implies 7776 - y! 2736 implies y! 5040. end{align*}]

(5040) is indeed (7!), so (y 7). This gives us a potential solution ((x, y) (6, 7)).

Verification and Uniqueness of the Solution

To ensure that ((6, 7)) is the only solution, we reduce the equation modulo 11. Note that (2736 equiv 8 mod 11). Consider the equation modulo 11:

[begin{align*} x^5 - y! equiv 8 y! equiv x^5 - 8 mod 11. end{align*}]

The fifth powers modulo 11 are limited to (0, 1, -1). Since (x^5 - 8 eq 0, 1, -1), there are no integer values of (x) and (y) that satisfy the equation. Therefore, the only solution is ((x, y) (6, 7)).

Conclusion

The solution to the equation (x^5 - y! 2736) is unique, and it is ((x, y) (6, 7)). This conclusion is supported by the properties of factorials, modular arithmetic, and the unique fifth power residues modulo 11. Through careful analysis, we have shown that there are no other positive integer solutions to this Diophantine equation.