1. Let p be a prime number which is neither 2 nor 3. Prove that either...

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1. Let p be a prime number which is neither 2 nor 3. Prove that either p = 1 mod 6 or p = -1 mod 6. 2. Prove that there are infinitely many primes p such that p = -1 mod 6. Hint: Suppose on the contrary that there are finitely many, say P1, N = 6p₁ Pk1, and consider a prime divisor of N. Pk. Let 3. Why does the same proof fail to show that there are infinitely may primes p such that p = 1 mod 6? 4. Dirichlet's theorem on primes in arithmetic progressions, which is way beyond the scope of this course, states that for all coprime positive integers a and b, there are infinitely many primes p such that p = a mod b; in particular, there are in fact infinitely many primes p such that p = 1 mod 6. Why, in the statement of this theorem, is it necessary to assume that a and b are coprime? 1. Let p be a prime number which is neither 2 nor 3. Prove that either p = 1 mod 6 or p = -1 mod 6. 2. Prove that there are infinitely many primes p such that p = -1 mod 6. Hint: Suppose on the contrary that there are finitely many, say P1, N = 6p₁ Pk1, and consider a prime divisor of N. Pk. Let 3. Why does the same proof fail to show that there are infinitely may primes p such that p = 1 mod 6? 4. Dirichlet's theorem on primes in arithmetic progressions, which is way beyond the scope of this course, states that for all coprime positive integers a and b, there are infinitely many primes p such that p = a mod b; in particular, there are in fact infinitely many primes p such that p = 1 mod 6. Why, in the statement of this theorem, is it necessary to assume that a and b are coprime?

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1 Let p be a prime number which is neither 2 nor 3 Prove that either p 1 mod 6 or p 1 mod 6 Proof Since p is a prime number it is divisible by a singl... View the full answer