Prove that for all integers n > 1, n⁴ + 4 is composite.
Understand the Problem
The question is asking us to provide a proof that for every integer n greater than 1, the expression n⁴ + 4 can be shown to be a composite number. A composite number is defined as a positive integer that has at least one positive divisor other than one or itself. In this case, we likely need to manipulate the expression n⁴ + 4 to demonstrate that it can be factored or shown to have divisors.
Answer
For every integer $n > 1$, $n^{4} + 4$ is a composite number.
Answer for screen readers
For every integer $n > 1$, the expression $n^{4} + 4$ is a composite number.
Steps to Solve
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Rewrite the Expression We will use the identity for factoring a sum of squares. The expression $n^{4} + 4$ can be expressed as $n^{4} + 4 = n^{4} + 4n^{2} + 4 - 4n^{2} = (n^{2} + 2)^{2} - (2n)^{2}$.
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Apply the Difference of Squares Formula Using the difference of squares formula, we have $a^2 - b^2 = (a+b)(a-b)$. Here, $a = n^2 + 2$ and $b = 2n$. Thus, we can factor the expression as follows: $$ n^{4} + 4 = (n^{2} + 2 + 2n)(n^{2} + 2 - 2n) = (n^{2} + 2 + 2n)(n^{2} + 2 - 2n). $$
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Simplifying the Factors Next, let's simplify the two factors:
- The first factor is $n^{2} + 2 + 2n$.
- The second factor is $n^{2} + 2 - 2n$.
For integers $n > 1$, both factors can be shown to be greater than 1.
- Show that both factors are greater than 1 For $n = 2$, we calculate:
- First factor: $2^{2} + 2 + 2(2) = 4 + 2 + 4 = 10$.
- Second factor: $2^{2} + 2 - 2(2) = 4 + 2 - 4 = 2$.
Both factors are greater than 1. Similarly, for integers greater than 2, both factors will always be positive and greater than 1.
- Conclusion about the Compositeness Since both factors are greater than 1 for all integers $n > 1$, $n^4 + 4$ can be expressed as a product of two integers greater than 1. Therefore, it is a composite number.
For every integer $n > 1$, the expression $n^{4} + 4$ is a composite number.
More Information
The proof shows that the expression can be factored into two integers greater than 1, confirming it's composite. This identity relies on understanding and applying the difference of squares.
Tips
- Neglecting Factor Conditions: A common mistake is to overlook checking that the factors are positive and greater than 1. Always verify the conditions of the integers you evaluate.
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