Results 1 to 8 of 8

Math Help - Find all prime numbers

  1. #1
    Member
    Joined
    Nov 2006
    Posts
    152

    Find all prime numbers

    Find all prime numbers K

    and all positive integers N

    such that


    K  = N^4+4.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member JaneBennet's Avatar
    Joined
    Dec 2007
    Posts
    293
    The only solution is N=1,K=5.

    N clearly has to be odd. If N>1, then N must also be divisible by 5; if it isnít, N^4+4 would be divisible by 5 by Fermatís little theorem.

    Hence if N>1, let N=5(2k-1), k\ge1.

    Then N^4+4=625(2k-1)^4+4=\left[4(5k-2)^2+1\right]\left[4(5k-3)^2+1\right] is a product of two integers both greater than 1 and hence cannot be prime.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2008
    From
    Tezpur
    Posts
    5

    Wrong!!!

    We get only 1 as the solution!!!!!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    o_O
    o_O is offline
    Primero Espada
    o_O's Avatar
    Joined
    Mar 2008
    From
    Canada
    Posts
    1,408
    Yes, that's what JaneBennet has proven. N = 1 is the only solution yielding K = 5.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Aug 2008
    From
    Tezpur
    Posts
    5

    Another variant!

    Another variant of the problem is this : Find all n such that n^4+4^n is a prime!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by manjil View Post
    Another variant of the problem is this : Find all n such that n^4+4^n is a prime!
    See This
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by manjil View Post
    We get only 1 as the solution!!!!!
    Not even wrong, a solution consists of a pair a positive integer N and a prime K. As Miss Bennet demonstrates there is indeed only one solution and it is N=1, K=5

    RonL
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member JaneBennet's Avatar
    Joined
    Dec 2007
    Posts
    293
    Quote Originally Posted by perash View Post
    Find all prime numbers K

    and all positive integers N

    such that


    K  = N^4+4.
    I found a better solution.

    If N is even, then N^4+4 is an even number greater than 2, so it is not prime.

    If N is odd, let N=2m+1, m\ge0.

    Then N^4+4=(2m+1)^4+4=\left(4m^2+1\right)\left(4(m+1)^2  +1\right).

    If m>0, both the factors are greater than 1 and so N^4+4 is not prime. Hence the only solution is when m=0, i.e. N=1.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 22nd 2011, 01:37 PM
  2. find prime numbers
    Posted in the Math Forum
    Replies: 2
    Last Post: April 17th 2011, 10:44 AM
  3. find the prime factor of two numbers
    Posted in the Algebra Forum
    Replies: 3
    Last Post: March 5th 2010, 06:04 AM
  4. Replies: 1
    Last Post: September 24th 2009, 12:37 AM
  5. find all prime numbers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: September 16th 2009, 03:38 PM

Search Tags


/mathhelpforum @mathhelpforum