MAT315: Introduction to Number Theory Assignment 2
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Assignment 2
MAT315: Introduction to Number Theory
Due May 26th
Problem 1. Use Euclid’s proof of the infinitude of the primes to give an explicit lower bound of the form C log log x < π(x) on the prime-counting function.
Problem 2. Show the converse to Wilson’s theorem. That is, show that if (m − 1)! ≡m −1, then m must be prime.
Problem 3. a) Let p be a prime and let a be an integer not divisible by p. Determine the number of solutions to the equation x2 − y2 ≡p a.
b) With the same assumptions, determine the number of solutions to the equation x2 + y2 ≡p a.
Problem 4. Let p be a prime and let n ∈ {1, 2, ...,p − 1}. Define the quadratic form Qn (x,y) := x2 + ny2
a) Prove that if the congruence x2 ≡p −n has a solution, then Qn represents at least one of the numbers p,2p, ..., (p − 1)p.
b) Prove that if Q2 represents 2p, then it represents p. Deduce that if x2 ≡p −2 has a solution, then Q2 represents p.
c) In the other direction, prove that if p is represented by Q2 , then x2 ≡p −2 has a solution.
2023-05-25