Results 1 to 6 of 6

Math Help - Prime Numbers!

  1. #1
    Member
    Joined
    Mar 2009
    Posts
    182
    Thanks
    1

    Prime Numbers!

    In this question arithmetic in a restricted subset of Z, similar to arithmetic with E-numbers,
    is investigated. Let D = {4a + 1 | a ∈ Z} and call the elements of D the D-numbers. The
    first few positive D-numbers are

    1
    , 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57.


    Put another way, the D-numbers are the integers which are congruent to 1 modulo 4. Use
    of arithmetic modulo 4 makes answers to some of the the following questions very simple
    (but is not mandatory).

    (a) Show that if two D-numbers are multiplied together the result is a D-number. Give an
    example to show that the same is not true when two D-numbers are added together.

    (b) If a and b are D-numbers we say that a D-divides b if b = ac, where c is a D-number.
    Show that 5 D-divides 25 and 45. Show that 1 D-divides every D-number.

    (c) Now show that if a and b are D-numbers such that a|b (in the usual sense of Definition
    1.5) then a D-divides b. Give an example to show that there are integers n and m,
    which are not both D-numbers but are such that mn is a D-number.

    (d) If a is a positive D-number greater than 1 and the only positive D-divisors of a are
    1
    and a , then we say that a is D-prime. List the first 10 D-primes and the first two
    positive D-numbers (> 1) which are D-composite (i.e. not D-prime).

    (e) All (ordinary) odd primes are either of the form 4m+1 or 4m+3: that is are congruent
    to 1 or 3 modulo 4. Show that a D-number is D-prime if and only if it is prime (in Z)
    or its prime factorisation is pq where p and q are congruent to 3 mod 4 (and may be
    equal).

    (f) Find a D-number which has two distinct D-prime factorisations.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Which question do you need help with?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2009
    Posts
    182
    Thanks
    1
    Well for part (a), am i just to do (4a+1)*(4a+1) and then somehow show that this always = a multiple of (4a+1)?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,415
    Thanks
    1853
    Quote Originally Posted by sirellwood View Post
    Well for part (a), am i just to do (4a+1)*(4a+1) and then somehow show that this always = a multiple of (4a+1)?
    Not (4a+1)(4a+1) but two numbers of that form- the numbers multipying "4" may be different. (4a+1)(4b+1)= 16ab+ 4(a+b)+ 1= 4( ? )+ 1. That's pretty easy, isn't it?

    What is (4a+ 1)+ (4b+1)? In particular, what is 5+ 9? Is it a "D number"?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Mar 2009
    Posts
    182
    Thanks
    1
    Haha, yes it is very easy! yes i was able to do the 2nd part of part (a) :-)

    For part (b) i have started off by saying that 5 = (4a+1) and 25 = (4b+1) and then hopefully going to say that (4b+1)/(4a+1) = a D-number?

    Is this the right route?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Mar 2009
    Posts
    182
    Thanks
    1
    I have also just tried this for (b) but not sure it is sufficient?

    5=(4a+1)
    25 = 5(4a+1)

    \frac{25}{5} = \frac{5(4a+1)}{(4a+1)}

    = 5

    = D-Number

    Then a similar idea for 45
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Prime Numbers
    Posted in the Algebra Forum
    Replies: 5
    Last Post: November 17th 2011, 07:39 AM
  2. Replies: 1
    Last Post: October 22nd 2011, 01:37 PM
  3. Help with prime numbers
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: August 4th 2011, 08:42 PM
  4. Prime numbers
    Posted in the Algebra Forum
    Replies: 5
    Last Post: December 29th 2010, 05:54 AM
  5. Prime numbers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: April 26th 2010, 07:35 PM

Search Tags


/mathhelpforum @mathhelpforum