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(n^4) + 4 is composite ?

by Gokul43201
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Gokul43201
#1
Apr19-04, 10:03 PM
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How do you prove that (n^4) +4 is composite for all n>1 ?

I found this problem in a book I was reading. The even n part is obvious. The proof for odd n cannot be done by trying to factorize [(2k+1)^4]+4. I know this is true, because I've tried and moreover, the factors of the first few such numbers are : (2*2*5),(5*17),(2*2*5*13),(17*37),(2*2*5*5*13),(5*13*37),... The expected factors of 4 appear in the even numbers, but the factors for the odds seem patternless. So, I imagine that the proof must be done by some other means.

Any ideas ?
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Hurkyl
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Apr19-04, 10:06 PM
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Are there any factors that tend to occur frequently? Maybe you can solve the problem by cases.
cragwolf
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Apr20-04, 02:10 AM
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Quote Quote by Gokul43201
(2*2*5),(5*17),(2*2*5*13),(17*37),(2*2*5*5*13),(5*13*37),... The expected factors of 4 appear in the even numbers, but the factors for the odds seem patternless.
Are you sure they're patternless? Just look at the odd n for now:

n=3: n^4+4 = 5*17
n=5: n^4+4 = 17*37
n=7: n^4+4 = 5*13*37 = 37*65

And I'll add n=9 for you:

n=9: n^4+4 = 5*13*101 = 65*101

See any pattern yet? That should lead you to the general proof.

Gokul43201
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Apr20-04, 08:19 AM
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(n^4) + 4 is composite ?

thanks crag,
I must be blind !
cragwolf
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Apr20-04, 07:39 PM
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On no! The blind leading the blind!
yrch
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May27-04, 10:06 AM
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how about:
[tex]n^4 + 4 = n^4 + 4 - 4n^2 + 4n^2 = (n^2 + 2)^2 - (2n)^2 = (n^2 + 2n + 2)(n^2 - 2n + 2)[/tex]
:)

edit:
sorry, i pulled up an old post... just realized that it's from April not May 20th...
arildno
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May27-04, 10:41 AM
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Quote Quote by yrch
edit:
sorry, i pulled up an old post... just realized that it's from April not May 20th...
What a terrible misdemeanor on your part...
I am almost tempted not to welcome you
Gokul43201
#8
May27-04, 01:34 PM
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thanks, yrch...that's nice.

Now I really feel like $#!+.


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