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Prove that the equation $x^2+y^2=5^k$ has $4k+4$ integral solution.

Any ideas would be appreciated.

Thanks

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    @AndréNicolas no, I haven't done. The wikipedia page doesn't give a lot of information IMO, any suggestions??2012-10-13

2 Answers 2

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Lets use Strong form of mathematical induction.

Base Case: $k=0$ and $k=1$ are obvious.

Assume $x^2+y^2=5^k$ has exactly $8$ solutions $(x,y)$ such that $x$ and $y$ are not divisible by $5$ along with $4k-4$ solutions of the form $(5a,5b)$ where $(a,b)$ are solution of $a^2+b^2=5^{k-2}$. We need to show similar for $k+1$.

Note that $(x,y)$ are all obtained from each other by permutations of $x$ and $y$ and changes of signs,lets call them new solutions.

If $x^2 + y^2$ be divisible by $5$. Then $(x + 2y)(x − 2y) = x^2 + y^2 − 5y^2$ is also divisible by $5$. Hence, one of the numbers $x + 2y$ and $x − 2y$ is divisible by $5$. Note that if both are divisible by $5$, then both $x$ and $y$ are divisible by $5$.

If $(x, y)$ is a new solution of equation $x^2 + y^2 = 5^k$ , then $(x + 2y, 2x − y)$ and $(x − 2y, 2x + y)$ are solutions of equation $m^2+n^2=5^{k+1}$ and precisely one of them is new. Hence each new solution pair $(x,y)$ of $x^2+y^2=5^k$ yields $1$ new solution and $1$ not new solution for $x^2+y^2=5^{k+1}$. Hence we are done as there are exactly $8$ new solutions for $x^2+y^2=5^k$ yielding $8$ new and $8$ not new solutions for $x^2+y^2=5^{k+1}$ in the form of $(\pm (2x \pm y),\pm(x \pm 2y)) \text{ and }(\pm (x \pm 2y), \pm(2x \pm y)).$

Note that the $8$ not new solutions will be contained in case of $x^2+y^2=5^{k-1}$.

Thanks

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Not a full solution, but enough to get you started:

I assume that you are counting solutions such which differ only by changing sign or by swapping $x$ and $y$ as being distinct?

If you have two numbers $a$ and $b$ which can be written as a sum of two squares in one (essentially distinct) way, then $ab$ can be written as a sum of two squares in two essentially distinct ways - this is covered here: Brahmagupta–Fibonacci identity.

Now we know that $5=2^2+1^2$, and from this we can use the above identity to find solutions of $25 = x^2+y^2$, (i.e. (5,0) and (3,4)) and so on ...