How many solution in integers have to the following equation:

( )

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- July 17th 2010, 12:03 PMAlso sprach Zarathustrax^n+y^n=z^(n+1)
How many solution in integers have to the following equation:

( )

- July 17th 2010, 12:44 PMOpalg
Obviously loads of solutions if n=1. Also if n=2, for example . In fact, let z be any number with the property that each of its prime factors of the form 4k+3 occurs to an even power. Then z^3 is the sum of two squares.

Another solution, valid for all n, is that . - July 17th 2010, 12:48 PMAlso sprach Zarathustra
And what about n>3?

- July 17th 2010, 12:59 PMOpalg
- July 17th 2010, 02:36 PMundefined
For n=3, we have some solutions x=y=2^1, 2^5, 2^9, 2^13, ...

For n=4, we have some solutions x=y=2^1, 2^6, 2^11, 2^16, ...

etc.

Edit: More solutions for n=3 given by (x,y)

9,18

28,84

65,260

70,105

126,630

144,288

162,162

266,665

273,364

456,760

469,603

756,945

793,854

n = 4

17,34

82,246

194,291

486,486

PARI/GP

Code:`f(x,y,z)=for(i=1,x,for(j=i,y,a=i^z+j^z;b=round(a^(1/(z+1)));if(b^(z+1)==a,print(i," ",j))))`

Code:`f(1000,1000,3)`

f(1000,1000,4)

- July 18th 2010, 09:00 PMsimplependulum
- July 19th 2010, 12:41 AMAlso sprach Zarathustra