The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.
Use Exercise 1 to show that
Exercise 1
The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.
Use Exercises 2 and 3 to show that
Exercise 2
The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.
Use Exercise I to show that
Exercise I
The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.
Let p be a prime and let n be a positive integer. Show that p divides exactly
times, where t = [logp 2n]Conclude that if pr divides , then pr ≤2n.
Exercise 3
The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.
Show that the product of all primes between n and 2n is between aud nπ(2n)−π(n). (Hint:Use the fact that every prime between n and 2n divides (2n)! but not (n!)2.)
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