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The prime counting function is the function giving the number of primes less than or equal to a given number (Shanks 1993, p. 15). For example, there are no primes , so . There is a single prime (2) , so . There are two primes (2 and 3) , so . And so on.
The notation for the prime counting function is slightly unfortunate because it has nothing whatsoever to do with the constant . This notation was introduced by number theorist Edmund Landau in 1909 and has now become standard. In the words of Derbyshire (2004, p. 38), "I am sorry about this; it's not my fault. You'll just have to put up with it."
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What is the last prime number?
The is no last prime number - unless you are counting down, in which case, it is 2.
What counting number is nethier a prime or composite?
The number 1.
What counting number is not a prime nor composite number?
1
What is meant by there is at least one prime number between any counting number and its double?
Take any counting number greater than one. 2, 3, 4, 5 and so on. Double it. Between the number and twice the number, there will be at least one prime number. 3, a prime number, is in between 2 and 4.
Are there any number that are neither prime nor composite?
Yes. All prime numbers and composite numbers are positive integers, or whole counting numbers. That leaves infinitely many numbers that are neither prime nor composite. If you intended to narrow the scope of your question to the whole counting numbers or to the positive integers, then there are NO such numbers that are neither. A counting number, however large, will be either prime or composite.
