Prove: There are inﬁnitely many primes congruent to 3 modulo 8. Hint: Consider N = (p1p2···pr)2...

Prove: There are inﬁnitely many primes congruent to 3 modulo 8. Hint: Consider N = (p₁p₂···p_r)² + 2.

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Colby Messinger answered 2 years ago

Prove: There are infinitely many primes of the form 6n − 1 (n is an integer).

a) Suppose that a ∈ Z is a unit modulo n. Prove that its inverse modulo...

a) Suppose that a ∈ Z is a unit modulo n. Prove that its inverse modulo n is well defined as a residue class in Zn, and depends only on the residue class a in Zn. b) Let Z × n ⊆ Zn be the set of invertible residue classes modulo n. Prove that Z × n forms a group under multiplication. Is this group a subgroup of Zn? c) List the elements of Z × 9 . How many...

Show that is [(n^2) + 2] and [(n^2) - 2] are both primes, then 3 divides...

Show that is [(n^2) + 2] and [(n^2) - 2] are both primes, then 3 divides n

Discrete math problem: Prove that there are infinitely many primes of form 4n+3.

Let “ ·n” be multiplication modulo n, and consider the set Un = { [a] ∈...

Let “ ·n” be multiplication modulo n, and consider the set Un = { [a] ∈ Zn | there is a [b] ∈ Zn with [a] ·n [b] = [1]} (a) Show that (Un, ·n ) is a group. (b) Write down the Cayley table for U5. Hint: |U5| = 4. (c) Write down the Cayley table for U12. Hint: |U12| = 4.

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 +...

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for every n ∈ N. That is, the sum of the first n perfect cubes is the square of the sum of the first n natural numbers. (As a student, I found it very surprising that the sum of the first n perfect cubes was always a perfect square at all.)

a) Prove that n 3 − 91n 2 − 7n − 14 = Ω(n 3 )....

a) Prove that n 3 − 91n 2 − 7n − 14 = Ω(n 3 ). Your answer must clearly specify the constants c and n0. b) Let g(n) = 27n 2 + 18n and let f(n) = 0.5n 2 − 100. Find positive constants n0, c1 and c2 such that c1f(n) ≤ g(n) ≤ c2f(n) for all n ≥ n0. Be sure to explain how you arrived at the constants.

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Prove: There are inﬁnitely many primes congruent to 3 modulo 8. Hint: Consider N = (p1p2···pr)2...

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