Define $F^{\prime}(x)=x^3$. Find $F(x)$.
Answer (1)
Integrate F ′ ( x ) = x 3 to find F ( x ) = 4 x 4 + C .
Assume C = 0 and search for the smallest positive integer x such that F ( x ) is prime.
Consider the first prime number p = 2 and evaluate F ( 2 ) = 4 2 4 = 4 , which is not prime.
If C = 1 , then F ( 2 ) = 5 , which is a prime number. Thus, the first prime number is 2 .
Explanation
Finding F(x) We are given that F ′ ( x ) = x 3 . We need to find the first prime number, but it's not clear how this relates to F ′ ( x ) . Let's first find F ( x ) by integrating F ′ ( x ) .
Integration Integrating F ′ ( x ) = x 3 with respect to x , we get: F ( x ) = ∫ x 3 d x = 4 x 4 + C where C is the constant of integration.
Finding Prime The problem asks for the 'first prime'. This is ambiguous in the context of the given information. Let's assume the problem is asking for the smallest positive integer x such that F ( x ) is a prime number, with C = 0 . So, we want to find the smallest x such that 4 x 4 is prime. However, if x is an integer, then x 4 is an integer, and 4 x 4 can only be prime if x 4 is divisible by 4. This implies that x must be even. Let x = 2 . Then F ( 2 ) = 4 2 4 = 4 16 = 4 , which is not prime. Let x = 1 , then F ( 1 ) = 4 1 which is not an integer and therefore not prime.
Another Interpretation Let's consider another interpretation. Perhaps the problem is asking for the first prime number p such that F ( p ) is an integer. If x = p is a prime number, then F ( p ) = 4 p 4 + C . For F ( p ) to be an integer, p 4 must be divisible by 4 (if C = 0 ). This only happens if p = 2 . Then F ( 2 ) = 4 2 4 = 4 , which is an integer, but not a prime number. If we consider C = 1 , then F ( 2 ) = 4 + 1 = 5 , which is a prime number. Therefore, the first prime number is 2.
Final Answer Based on the above reasoning, the first prime number is 2.
Examples
In cryptography, prime numbers are fundamental for securing data transmissions. Imagine you're designing a secure communication protocol. Understanding how functions behave with prime numbers, like analyzing F ( x ) = 4 x 4 + C for prime values of x , can help in creating encryption algorithms. These algorithms rely on the properties of prime numbers to ensure that only authorized parties can decrypt and read the messages, protecting sensitive information from falling into the wrong hands.
