Euc20205/Sub-Problem 1

Problem

(a) For each positive real number x, define $f(x)$ to be the number of prime numbers $p$ that satisfy $x \le p \le x + 10$ . What is the value of $f(f(20))$ ?

Solution

Because $20$ is a relatively small number, we can just bash this out. We first need to calculate $f(20)$ , so the numbers in this interval are $20$ to $30$ . The only prime numbers in this range are $23$ and $29$ , so $f(20)=2$ . Then, we need to find $f(2)$ . The range we have is $2$ to $12$ , and the prime numbers that are in this interval are $2$ , $3$ , $5$ , $7$ , and $11$ , so $f(2)=\boxed{5}$ .

~Baihly2024

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Euc20205/Sub-Problem 1

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