SOLUTION: Hi guys, this is a proofing type question. There are three parts - I don't know how to answer the last one. Part 1 and 2 should help with the final part. They are summarised bel

Question 1172305: Hi guys, this is a proofing type question.
There are three parts - I don't know how to answer the last one. Part 1 and 2 should help with the final part. They are summarised below.
Every odd number is one more or one less than a multiple of 4
and
The product of any two positive integers of the form , n is positive integer, is also of the form .
The actual question is:
Hence, prove by contradiction that any composite number of the form must have at least one prime factor of the form .
Answer by ikleyn(52802) (Show Source): You can put this solution on YOUR website!
.


So, the statement, which you want to prove, is THIS


   +------------------------------------------------------------------------+
   |  prove by contradiction that any composite number of the form (4n-1)   |
   |  must have at least one prime factor of the form (4n-1).               |
   +------------------------------------------------------------------------+



Let N be a composite number of the form (4n-1).


Then it is a product of the odd prime numbers; the prime number 2 is not its divisor.


Let assume that all its prime divisors are of the form (4n+1).


Notice that the product of any two odd numbers of the form (4n+1) is the number of the form (4n+1).
 


    It can be proven by direct multiplication of the numbers of this form.



It implies that a product of ANY number of the primes of the form (4n+1) has the form (4n+1).


But our number N has the form (4n-1).


So, we got a CONTRADICTION, which proves the statement.

The proof is completed.

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