Let’s agree to say that a positive integer is prime-like if it is not divisible by 2, 3, or 5. How
many prime-like positive integers are there less than 100? less than 1000? A positive integer is very prime-like if it is not divisible by any prime less than 15. How many very primelike positive integers are there less than 90000? Without giving an exact answer, can you
say approximately how many very prime-like positive integers are less than 1010? less than
10100? Explain your reasoning as carefully as you can.
To determine the number of prime-like positive integers less than a given number, we need to check the divisibility of each positive integer within that range. Let's break down the process for each scenario:
1. Prime-Like Integers less than 100:
- Start with a counter initialized at zero.
- Iterate through positive integers from 1 to 99.
- For each number, check if it is divisible by 2, 3, or 5. If not, increment the counter by one.
- The final value of the counter will give us the number of prime-like positive integers less than 100.
2. Prime-Like Integers less than 1000:
- Repeat the same process as above but iterate from 1 to 999 instead.
- Count the numbers that are not divisible by 2, 3, or 5.
- This will give us the number of prime-like positive integers less than 1000.
3. Very Prime-Like Integers less than 90000:
- We need to check divisibility by primes less than 15 (2, 3, 5, 7, 11, 13).
- Start with a counter set to zero.
- Iterate through positive integers from 1 to 89999.
- For each number, check if it is divisible by any prime less than 15. If not, increment the counter.
- The final value of the counter will give us the number of very prime-like positive integers less than 90000.
4. Approximating the number of Very Prime-Like Integers:
- We can use the concept of prime density to estimate the number of very prime-like positive integers.
- The prime density is defined as the number of primes less than n divided by n.
- As n increases, the prime density gets closer to 1 / ln(n) (where ln is the natural logarithm).
- We can use this approximation to estimate the number of primes less than a given number.
- For example, for n = 1010, we can approximate the number of very prime-like positive integers using the formula: 1010 / ln(1010).
- Similarly, for n = 10100, we can approximate the number using the formula: 10100 / ln(10100).
- These approximations will provide an estimate of the number of very prime-like positive integers less than the given values without giving an exact answer.
It's important to note that while these methods provide approximations, they may not be completely accurate as prime distribution doesn't follow a strict pattern.