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To solve the equation 16p+1=x3 where p is a prime number, we need to find the value of x. Let's break down the solution step-by-step.
Step 1: Understand the Equation
The given equation is:
16p+1=x3
Rearranging it, we get:
16p=x3−1
The expression x3−1 can be factored using the identity for the difference of cubes:
x3−1=(x−1)(x2+x+1)
Thus, the equation becomes:
16p=(x−1)(x2+x+1)
Step 2: Analyze the Factors
We know that p is a prime number, which implies that the product (x−1)(x2+x+1) divided by 16 should result in a prime number. Let's analyze the factors:
Step 3: Solve for x
Given that x−1 must be a multiple of 16, let's assume:
x−1=16k
where k is a positive integer. Therefore, x=16k+1.
Substituting back into the equation:
16p=(16k)(x2+x+1)
Since p is a prime number, k must be 1 to keep p prime. Therefore:
x−1=16 x=16+1=17
Step 4: Verify the Solution
Substituting x=17 back into the original equation:
16p+1=173 16p+1=4913 16p=4912 p=164912=307
Check if 307 is a prime number:
Thus, the solution is consistent, and the value of x is:
17