When Is 4 X 5 − 7 4x^5-7 4 X 5 − 7 And 4 X 13 − 7 4x^{13}-7 4 X 13 − 7 Both A Perfect $ \square $ For $ X \in \mathbb Z^+ $

Finding integer solutions to problems involving polynomials and perfect squares is a classic topic in number theory. This article delves into a fascinating problem originating from the German Mathematical Olympiad, exploring the conditions under which both $4x^5 - 7$ and $4x^{13} - 7$ are perfect squares for positive integers x. We'll journey through the initial observations, explore potential solution strategies, and ultimately uncover the unique solution to this intriguing problem.

Problem Statement: Unveiling the Perfect Squares

The core of our exploration lies in the following question:

Find all positive integers x such that both $a = 4x^5 - 7$ and $b = 4x^{13} - 7$ are perfect square numbers.

This problem immediately invites us to consider the interplay between polynomial expressions and the properties of square numbers. The exponents 5 and 13 add a layer of complexity, suggesting that a direct algebraic approach might be challenging. Instead, we'll need to employ a blend of clever observations, number-theoretic principles, and perhaps a touch of computational exploration to crack this puzzle.

Initial Explorations: A Glimpse of the Solution

The most natural starting point in any number theory problem is to test small values. By substituting x = 1, we find that $4(1)^5 - 7 = -3$ and $4(1)^{13} - 7 = -3$, which are not perfect squares. However, when we try x = 2, we discover that:

• $a = 4(2)^5 - 7 = 4(32) - 7 = 128 - 7 = 121 = 11^2$
• $b = 4(2)^{13} - 7 = 4(8192) - 7 = 32768 - 7 = 32761 = 181^2$

This is a significant breakthrough! We've found a solution: x = 2. But does this mean we're done? Absolutely not. The challenge now is to prove that x = 2 is the only solution. This requires a more rigorous approach.

Diving Deeper: Number Theory Insights

To demonstrate the uniqueness of the solution, we need to delve into the properties of perfect squares and how they interact with the given polynomial expressions. Let's assume that for some positive integer x, both $4x^5 - 7$ and $4x^{13} - 7$ are perfect squares. This means we can write:

• $4x^5 - 7 = m^2$ for some integer m
• $4x^{13} - 7 = n^2$ for some integer n

Our goal is to show that the only integer x that satisfies both equations is x = 2. Subtracting the two equations gives us:

$4x^{13} - 4x^5 = n^2 - m^2$

Factoring out common terms, we have:

$4x^5(x^8 - 1) = (n - m)(n + m)$

This equation is a crucial stepping stone. It connects the powers of x to the difference of squares. We can further factor the term $(x^8 - 1)$ as a difference of squares:

$4x^5(x^4 - 1)(x^4 + 1) = (n - m)(n + m)$

And again:

$4x^5(x^2 - 1)(x^2 + 1)(x^4 + 1) = (n - m)(n + m)$

One more time:

$4x^5(x - 1)(x + 1)(x^2 + 1)(x^4 + 1) = (n - m)(n + m)$

This factorization is powerful because it exposes the building blocks of the equation. We now have a product of several terms involving x on one side, and a product of two terms related to the square roots on the other side. The challenge now is to extract useful information from this equation.

Analyzing the Factors: A Path to Uniqueness

Let's analyze the factors we've obtained. We know that x is a positive integer. The terms $(x - 1)$, $(x)$, and $(x + 1)$ represent three consecutive integers. This observation is significant because consecutive integers often have special divisibility properties. Furthermore, the terms $(x^2 + 1)$ and $(x^4 + 1)$ are always positive for any integer x.

The right-hand side of the equation, $(n - m)(n + m)$, represents the product of two integers. Since $n^2$ and $m^2$ are both of the form $4x^k - 7$, they leave a remainder of 1 when divided by 4 (as any odd square leaves a remainder of 1 when divided by 4). This means both m and n are odd. Consequently, $(n - m)$ and $(n + m)$ are both even, and their product is divisible by 4.

We can consider the cases based on the value of x. We already know x = 2 is a solution. Let's consider x > 2. In this case, we can analyze the growth rates of the functions $4x^5 - 7$ and $4x^{13} - 7$. As x increases, $4x^{13} - 7$ grows much faster than $4x^5 - 7$. This suggests that the difference between n and m will become significant, which could lead to constraints on the possible values of x.

Another key observation is to consider the equation $4x^5 - 7 = m^2$ modulo some suitable integer. For instance, considering the equation modulo 11 can provide valuable insights. If $4x^5 - 7$ is a perfect square modulo 11, then it must be congruent to one of the quadratic residues modulo 11, which are 0, 1, 3, 4, 5, and 9. We can check the values of $4x^5 - 7$ modulo 11 for different values of x and see if any patterns emerge.

Modular Arithmetic: A Powerful Tool

Let's explore the use of modular arithmetic more systematically. Consider the equation $4x^5 - 7 = m^2$ modulo 4. We have:

$4x^5 - 7 ext{ mod } 4 ext{ ≡ } m^2 ext{ mod } 4$

Since $4x^5$ is divisible by 4, it's congruent to 0 modulo 4. Also, $-7$ is congruent to 1 modulo 4. Therefore:

$1 ext{ ≡ } m^2 ext{ mod } 4$

This tells us that $m^2$ leaves a remainder of 1 when divided by 4, which is consistent with m being an odd integer. Now, let's consider the equation modulo 11:

$4x^5 - 7 ext{ mod } 11 ext{ ≡ } m^2 ext{ mod } 11$

We need to find the possible values of $x^5$ modulo 11. By Fermat's Little Theorem, we know that $x^{10} ext{ ≡ } 1 ext{ mod } 11$ for any x not divisible by 11. The possible values of $x^5$ modulo 11 are 0, 1, and 10. Let's analyze the cases:

• If $x^5 ext{ ≡ } 0 ext{ mod } 11$, then $4x^5 - 7 ext{ ≡ } -7 ext{ ≡ } 4 ext{ mod } 11$, which is a quadratic residue.
• If $x^5 ext{ ≡ } 1 ext{ mod } 11$, then $4x^5 - 7 ext{ ≡ } 4 - 7 ext{ ≡ } -3 ext{ ≡ } 8 ext{ mod } 11$, which is not a quadratic residue.
• If $x^5 ext{ ≡ } 10 ext{ mod } 11$, then $4x^5 - 7 ext{ ≡ } 40 - 7 ext{ ≡ } 33 ext{ ≡ } 0 ext{ mod } 11$, which is a quadratic residue.

From this analysis, we see that $x^5$ must be congruent to either 0 or 10 modulo 11 for $4x^5 - 7$ to be a perfect square. A similar analysis can be performed for the equation $4x^{13} - 7 = n^2$ modulo 11. The combination of these modular constraints significantly narrows down the possible values of x.

Proving Uniqueness: A Chain of Reasoning

While modular arithmetic provides strong constraints, a complete proof of uniqueness often requires a more intricate argument. Let's revisit our equation:

$4x^5(x - 1)(x + 1)(x^2 + 1)(x^4 + 1) = (n - m)(n + m)$

We know that x = 2 is a solution. Assume there exists another solution x > 2. Then, the left-hand side of the equation will grow very rapidly. We need to show that the right-hand side cannot grow at the same rate while maintaining the perfect square conditions.

The key insight here is to analyze the relative sizes of m and n. We have:

• $m = ext{√}(4x^5 - 7)$
• $n = ext{√}(4x^{13} - 7)$

As x increases, n becomes significantly larger than m. This means that $(n + m)$ will be much larger than $(n - m)$. The difference $(n - m)$ can be expressed as:

$n - m = ext{√}(4x^{13} - 7) - ext{√}(4x^5 - 7)$

Multiplying and dividing by the conjugate, we get:

n - m = rac{(4x^{13} - 7) - (4x^5 - 7)}{ ext{√}(4x^{13} - 7) + ext{√}(4x^5 - 7)} = rac{4x^5(x^8 - 1)}{ ext{√}(4x^{13} - 7) + ext{√}(4x^5 - 7)}

Now, comparing this expression to our factored equation, we can see how the terms relate. The denominator grows at a rate between $x^{13/2}$ and $x^{5/2}$, while the numerator contains a term of $x^{13}$. This suggests that the difference $(n - m)$ will be relatively small compared to the other factors.

However, a rigorous proof often involves bounding the terms and showing that the equation cannot hold for x > 2. This might involve techniques such as analyzing the prime factorization of the terms or using inequalities to constrain the possible values.

Conclusion: The Unique Solution

Through our exploration, we've discovered that x = 2 is a solution to the problem of finding positive integers x such that both $4x^5 - 7$ and $4x^{13} - 7$ are perfect squares. While we've outlined the key steps and insights towards proving the uniqueness of this solution, a complete and rigorous proof often requires advanced number theory techniques.

This problem showcases the beauty and challenge of number theory, where seemingly simple questions can lead to deep and intricate mathematical investigations. The interplay between polynomial expressions, perfect squares, and modular arithmetic provides a rich landscape for mathematical exploration. The German Mathematical Olympiad problem serves as an excellent example of how these concepts can be combined to create a compelling and thought-provoking challenge.

Further research and exploration in this area might involve investigating similar problems with different polynomial expressions or exploring alternative approaches to proving uniqueness, such as using elliptic curves or other advanced techniques in number theory. The world of number theory is vast and fascinating, and problems like this one serve as a gateway to deeper mathematical understanding.