SOLUTION: Prove that every prime of the form 3m + 1 with m (in) N is also of the form 6n + 1
with n (in) N.
N for natural numbers
Algebra.Com
Question 849360: Prove that every prime of the form 3m + 1 with m (in) N is also of the form 6n + 1
with n (in) N.
N for natural numbers
Answer by LinnW(1048) (Show Source): You can put this solution on YOUR website!
We need to show that if p is a prime number of the form
6n + 1 , then there exists an m such that 3m + 1 = p.
We also need to show that if p is a prime number of the form
3m + 1, then there exists an n such that 6n + 1 = p.
-------
Taking the first case, showing that if p is a prime number of the form
6n + 1 , then there exists an m such that 3m + 1 = p.
Let p be such a prime number and n is a number such that 6n + 1 = p .
Certainly, 6n + 1 = 3(2n) + 1. So setting m = 2n , 3m + 1 = p.
-------
For the second case, we need to show that if p is a prime number of the form
3m + 1, then there exists an n such that 6n + 1 = p.
Let p be a prime number such that 3m + 1 = p.
-------
Suppose m is odd and greater than 2.
This means that 3m is odd, since an
odd number times an odd number is odd.
This means that 3m + 1 is even. But if
3m + 1 is even, 3m + 1 is not prime.
-------
Suppose m is even. Being even, there exists a number n
such that 2n = m. So 3m + 1 = 3(2n) + 1 = 6n + 1.
RELATED QUESTIONS
PLEASE HELP!!!
Prove that U
(n choose k-1) + (n choose k)=(n+1 choose k)
for all... (answered by KMST)
prove that:
Sum of n numbers in a sequence is... (answered by stanbon,TP)
Prove that the sum s of the first n natural numbers is given by the
equation s = n/2... (answered by AnlytcPhil)
If n is the set of natural numbers that are factors of 16, what is the set in roster... (answered by Edwin McCravy)
Some prime numbers are of the form {{{2^n-1}}}. For example, the prime number 7 is equal... (answered by ikleyn)
Some prime numbers are of the form {{{2^n-1}}}. For example, the prime number 7 is equal... (answered by ikleyn)
what is the simplest form of equation 1/(m+n)^3 - 1/m^3 /... (answered by Alan3354)
Solvetheequation(n+1)! 30(n−1)!forn∈N.
(Remember: N is the set of natural... (answered by Fombitz)
Prove that (5n)!/ 40^n * n! is natural number for every natural number... (answered by greenestamps)