🔎 Step 1: Understand the Problem
We need to find all integer values of n for which the expression: \(n^4 + 4^n\) is a prime number.
A prime number is a positive integer greater than \(1\) that has no divisors other than \(1\) and itself.
🔥 Step 2: Try Small Values of n
Let’s check the expression for small integer values of \(n\):
- For n = 0 :
\(0^4 + 4^0\)
\( 0 + 1 = 1\)
\( 1\) is not a prime number.
- For n = 1 :
\(1^4 + 4^1\)
\(1 + 4 = 5\)
\(5\) is a prime number.
- For n = 2 :
\(2^4 + 4^2\)
\(16 + 16 = 32\)
\(32\) is not a prime number.
- For n = 3 :
\(3^4 + 4^3\)
\(81 + 64 = 145\)
\(145\)is not a prime number.
- For n = 4 :
\(4^4 + 4^4\)
\(256 + 256 = 512\)
\(512\) is not a prime number.
🔢 Step 3: Check Negative Values
For negative values of \(n\), the term \(n^4\) is positive (since any even power of a negative number is positive), while\(4^n\) becomes a fraction.
- For n = −1 :
\((-1)^4 + 4^{-1}\)
\(1 + \frac{1}{4} = \frac{5}{4}\)
This is not an integer, hence it cannot be a prime.
- For n = −2:
\((-2)^4 + 4^{-2}\)
\(16 + \frac{1}{16} = \frac{257}{16}\)
This is not an integer.
🚫 Step 4: Conclude the Solution
After testing various integer values, the only valid solution where the expression results in a prime number is: \(n = 1\)
✅ Therefore, the solution to the problem is:
\(\mathbf{n = 1}\)
📝 Final Answer
The only integer value of nnn that makes the expression \(n^4 + 4^n\) a prime number is:
\(\mathbf{n = 1}\)
✅ Verification: Are There Any Other Solutions?
🔎 Step 1: Analyzing the Expression
We are solving the problem: \(n^4 + 4^n\)
We need to determine whether any other integer values of \(n\)make the expression prime.
🔥 Step 2: Check for Larger Values of n
Let’s analyze the behavior of the expression for larger values of n:
- For \(n≥2\), both \(n^4\) and \(4^n\) grow rapidly. Their sum becomes a large even number, making it composite.
For example:
- For n = 5:
\(5^4 + 4^5\)
\( 625 + 1024 = 1649\)
1649 is not a prime number.
- For n = 6:
\(6^4 + 4^6\)
\(1296 + 4096 = 5392\)
\(5392\) is not a prime number.
- For n = 7:
\(7^4 + 4^7\)
\(2401 + 16384 = 18785\)
\(18785\) is not a prime number.
⚠️ For Negative Values of n:
For \(n<0\), the term \(4^n\) becomes a fraction, making the entire expression non-integer, which cannot be prime.
🚫 Step 3: Conclusion
After testing both positive and negative values, we observe:
- For n>1, the sum becomes large and composite.
- For n<0, the expression is non-integer.
- The only valid solution is:
\(\mathbf{n = 1}\)
✅ Therefore, there are no other solutions.
Also Read About: 🧮 Solution to the IMO 2020 Problem – Finding Integer Pairs (a, b)
Khairun
