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        riddles >> easy >> Splitting 2005
        
(Message started by: JocK on Jan 4^th, 2005, 12:21pm)

Title: Splitting 2005
Post by JocK on Jan 4^th, 2005, 12:21pm

Can you write 2005 as the sum of:
a prime squared, a prime cubed and a prime to the fourth power?

Title: Re: Splitting 2005
Post by Sir Col on Jan 4^th, 2005, 1:15pm

Yes...

::[hide]
2005^1/4~=6.69, so the largest prime 4th power we can use is 5⁴: 2005-5⁴=1380.
1380^1/3~=11.13, and as 11 is prime: 1380-11³=49, which happens to be square.

Therefore, 2005=7²+11³+5⁴
[/hide]::

Title: Re: Splitting 2005
Post by Trebla on Jan 4^th, 2005, 1:16pm

One such answer, though I'd wager there are more...
<<<
7²+11³+5⁴=2005
>>>

Title: Re: Splitting 2005
Post by Trebla on Jan 4^th, 2005, 1:17pm

Bah, I'm too slow...

Title: Re: Splitting 2005
Post by JocK on Jan 4^th, 2005, 1:29pm

on 01/04/05 at 13:17:56, Trebla wrote:

Bah, I'm too slow...

Well... grab your chance: no one has posted alternative solutions (yet)...! ;D

Title: Re: Splitting 2005
Post by towr on Jan 4^th, 2005, 1:49pm

What alternatives are there? Or would you allow subtraction as well as addition in a sum?
Or Gaussian primes?

Title: Re: Splitting 2005
Post by JocK on Jan 4^th, 2005, 2:35pm

No gaussian primes please... I have taken Trebla's wager and won't allow cheating... ;)

Title: Re: Splitting 2005
Post by Sir Col on Jan 4^th, 2005, 4:09pm

If you're working in base 10 then there is only one solution.

However, if 2005 is not necessarily given in base 10, then there certainly exists solutions in base 12 and base 16.
2005₁₂ = 3461 = 53²+3³+5⁴
2005₁₆ = 8197 = 79²+11³+5⁴

But you did say...

on 01/04/05 at 14:35:17, JocK wrote:

I ... won't allow cheating

:-/