Number Theory Proofs

 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 3n2+ 8 is even.

      5. n2 + n + 1 is a prime number whenever given n is a positive integer.

      6. n2 + 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.

 

Number theory proofs:

 

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 3n2 + 8 is even.

 Proof:

           We must show that n is even  3n2+ 8 is even, and that 3n2+ 8 is even  n is even.

           First we will show if n is even, then 3n2+ 8 is even. n even means there exists integer a such that n = 2a. 

           Then  3n2+ 8 = 3(2a)2 + 8 = 12a+ 8 = 2(6a2 + 4) which is even since (6a2 + 4)  is an integer.

Proof 4: 

             n+ 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.

 

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Additional number theory proofs:

 

Proof 1:

        n+ 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.