Question

1. Prove that {2k+1: k ∈ Z}={2k+3 : k ∈ Z}

2. Prove/disprove: if p and q are prime numbers and p < q, then 2p + q^2 is odd (Hint: all prime numbers greater than 2 are odd)

Answer #1

If p = 2k − 1 is prime, show that k is an odd integer or k =
2.
Hint: Use the difference of squares 22m − 1 = (2m − 1)(2m +
1).

8. Prove or disprove the following statements about
primes:
(a) (3 Pts.) The sum of two primes is a prime number.
(b) (3 Pts.) If p and q are prime numbers both greater than 2,
then pq + 17 is a composite number.
(c) (3 Pts.) For every n, the number n2 ? n + 17 is always
prime.

Problem 2: (i) Let a be an integer. Prove that 2|a if and only
if 2|a3.
(ii) Prove that 3√2 (cube root) is irrational.
Problem 3: Let p and q be prime numbers.
(i) Prove by contradiction that if p+q is prime, then p = 2 or q
= 2
(ii) Prove using the method of subsection 2.2.3 in our book that
if
p+q is prime, then p = 2 or q = 2
Proposition 2.2.3. For all n ∈...

Prove or disprove (a) Z[x]/(x^2 + 1), (b) Z[x]/(x^2 - 1) is an
Integral domain.
By showing (a) x^2+1 is a prime ideal or showing x^2 + 1 is not
prime ideal.
By showing (b) x^2-1 is a prime ideal or showing x^2 - 1 is not
prime ideal.
(Hint: R/I is an integral domain if and only if I is a prime
ideal.)

(§2.1) Let a,b,p,n ∈Z with n > 1.
(a) Prove or disprove: If ab ≡ 0 (mod n), then a ≡ 0 (mod n) or
b ≡ 0 (mod n).
(b) Prove or disprove: Suppose p is a positive prime. If ab ≡ 0
(mod p), then a ≡ 0 (mod p) or b ≡ 0 (mod p).

4. Prove that if p is a prime number greater than 3, then p is
of the form 3k + 1 or 3k + 2.
5. Prove that if p is a prime number, then n √p is irrational
for every integer n ≥ 2.
6. Prove or disprove that 3 is the only prime number of the form
n2 −1.
7. Prove that if a is a positive integer of the form 3n+2, then
at least one prime divisor...

Let m = 2k + 1 be an odd integer. Prove that k + 1 is the
multiplicative inverse of 2, mod m.

Prove directly that the group 2Z = {2k | k ∈ Z} and
the group 5Z = {5k | k ∈ Z} are isomorphic.

(a) Prove that if y = 4k for k ≥ 1, then there exists a
primitive Pythagorean triple (x, y, z) containing y.
(b) Prove that if x = 2k+1 is any odd positive integer greater
than 1, then there exists a primitive Pythagorean triple (x, y, z)
containing x.
(c) Find primitive Pythagorean triples (x, y, z) for each of z =
25, 65, 85. Then show that there is no primitive Pythagorean triple
(x, y, z) with z...

Define the set E to be the set of even integers; that is,
E={x∈Z:x=2k, where k∈Z}. Define the set F to be the set of integers
that can be expressed as the sum of two odd numbers; that is,
F={y∈Z:y=a+b, where a=2k1+1 and
b=2k2+1}.Please prove E=F.

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