Exercises from MNT chapter 1, pages 14-16
- MNT, 1.
- MNT, 15,
- MNT, 16
- MNT, 30
- The function pi(x) counting the number of primes up to x is called PrimePi in WolframAlpha. Use WolframAlpha or other means to calculate pi(x)/(x / ln x), for x = 10^k, for k=2,3,4,5,6. Does it seem that the limit might approach 1, as predicted by the prime number theorem?
- The ring of Gaussian integers is a Euclidean domain. Show directly that the ring is an integral domain. Apply a division algorithm to divide 3+4i by 1+i and work out its remainder.