6. [Marks 3] Suppose p and q are distinct primes. Find the general solution to the set of equations: x= -1 mod p x = -1 mod q. Show all the steps/details.
Problem 4.51. Let p and q be distinct primes. Find the number of generators of Zpa
6. (a) In the dozenal base, ind all primes up to the dozenal number 200. Use whatever symbol you want for the digits ten and eleven (b) In the octal base, find all the primes up to the octal number 400. You just may want to use the sieve of Eratosthenes for this problem!
6. (a) In the dozenal base, ind all primes up to the dozenal number 200. Use whatever symbol you want for the digits ten and eleven...
Suppose that pı, P2, ..., P, are the only primes congruent to 1 (mod 4). Prove that 4p?p, ... p, + 1 is divisible only by primes congruent to 3 (mod 4). Assuming that all odd prime factors of integers of the form x2 +1 are congruent to 1 (mod 4), use Exercise 6 to prove that there exist infinitely many primes congruent to 1 (mod 4).
5. Find all twin primes less than 100, and find π(100).
8) (Problem 17 (a) on page 49) Let p and q be two distinct primes. Show that for any integer a, pq|(a p+q − a p+1 − a q+1 + a 2 ). Hint: Find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo p, and then find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo q. After that, use the following result: Suppose x,...
2. Primes [2 marks] A prim e p > 1 has no factors other than 1, so p%m 0 for all m є {2.3, p-1). Test all the integers greater than 1 that you can, and identify the primes (again, write your own code). How far can you get in 10 minutes (use a clock or watch to approximately time this)? Print (or write) the last three primes that you find, and also check them using a web tool (Google...
4. Use quadratic reciprocity to find a congruence describing all odd primes for which 5 is a quadratic residue.
4. Use quadratic reciprocity to find a congruence describing all odd primes for which 5 is a quadratic residue.
Determine the odd primes p for which −26 is a square mod p
6. Let n be any positive integer which n = pq for distinct odd primes p. q for each i, jE{p, q} Let a be an integer with gcd(n, a) 1 which a 1 (modj) Determine r such that a(n) (mod n) and prove your answer.