** **The number theory is a one of the main branches of math. The number theory is pure mathematics. The number theory consists of numbers, that is, whole
numbers or rational numbers or fractions. **Prime number** and prime factorization are very important in the
number theory. The number theory proofs are given below.

1. If m and n are even integers, then mn is divisible by 4.

2. The sum of two odd integers is odd.

3. The sum of two odd integers is even.

4. If n is a positive integer, then n is even if 3n^{2}+ 8 is even.

5. n^{2} + n + 1 is a prime number whenever given n is a positive integer.

6. n^{2} + n + 1 is a prime number whenever n is a prime number.

Having problem with **Definition of Perfect Number** keep reading my upcoming
posts, i will try to help you.

**Proof 1:**

** ** If m and n are even integers, then mn is divisible 4

**Proof:**

m and n are even means that there real integer number a and b such that m =2p and n = 2q

Therefore mn = 4pq. Since ab is an integer, mn is 4 times an integer so it is divisible by 4.

**Proof 2:**

The sum of two odd integer numbers are even.

**Proof:**

If m and n are odd integers then there real integer number is a,b

such that m = 2a+1 and n = 2b+1. m + n = 2a+1+2b+1 = 2(a+b+1). Since (a+b+1) is an integer, m+n must be even.

**Proof 3:**

If n is a positive integer, then n is even if 3n^{2} + 8 is even.

**Proof:**

We must show that n is even 3n^{2}+ 8 is even, and that 3n^{2}+ 8 is even n is even.

First we will show if n is even, then 3n^{2}+ 8 is even. n even means there exists integer a such that n =
2a.

Then 3n^{2}+ 8 = 3(2a)^{2} + 8 = 12a^{2 }+ 8 = 2(6a^{2} + 4) which is even
since (6a^{2} + 4) is an integer.

**Proof 4: **

n^{2 }+ n + 1 is a prime number. Whenever n is a positive integer

Try some examples:

n = 1, 1+1+1 = 3 is prime

n = 2, 4+2+1 = 7 is prime

n = 3, 9+3+1 = 13 is prime

n = 4, 16+4+1 = 21 is not prime and is a counter example.

Not true.

Learn **What is a Square Root** online keep checking blogs for more help.

**Proof 1:**

n^{2 }+ n + 1 is prime number. Whenever given n is a prime number.

Try to some examples:

n = 1, 1+1+1 = 3 is prime

n = 2, 4+2+1 = 7 is prime

n = 3, 9+3+1 = 13 is prime

n = 5, 25+5+1 = 31 is prime

n = 7, 49+7+1 = 57 is not prime (19*3).

Not true.