Hi,
here's a "little" problem i found in a book about induction :
Let's f a function from the naturals integers N to N that satisfy:
* f(1) > 0
* f(m² + n²) = (f(m))² + (f(n))²
I've calculate f(m) for m = 0 to 20 and find f(m) = m for all these values, but how can we give a good induction proof of this fact, i don't know ? ...
Could someone help me, please ! ...
Apologize for my poor english . Thanks
Best regards

Try simple cases. What do you get plugging in m = n = 0? What do you get if only one of them is 0?

>Try simple cases. What do you get plugging in m = n =
>0? What do you get if only one of them is 0?
Of course it'easy to prove that f(0)=0 and f(n²)=(f(n))², even it'rather easy to calculate f(1),f(2), then f(4),f(8) , then f(5) (plug m=2,n=1), then f(3) (plug m=3,n=4) ,etc...
In fact i have calculate f(m) for m=0 to 20 and i stop to 20 because I wanted a general proof of the "obvious" fact that f(m) = m by induction, but it does'nt seem to be quite so easy
Thanks a lot if you could help me
sincerely

Induction generally can only be used on integer
arguments because the proof depends on a process
that hits every number. Keeping this in mind, here
is a general guideline that should assist you in
solving the induction problem above:

Firstly, make sure you know where you are
starting from. It is usually a good idea to
include both 0 and 1 in the base cases. (They
can be calculated by hand and then assumed in the
rest of the induction.)

f(0) = 0
f(1) = 1

In induction, we assume that we have already proved
that f(m) = m for all values of m <= k, and then
we prove that f(k+1) = k+1. This should help you
get started on your inductive proof.

Sanjay
Sanjay

One thing that might be useful to know is that the summation property of this function does not have to be restricted to two terms. For example, f(a^2 + b^2 + c^2) = (f(a))^2 + (f(b))^2 + (f(c))^2, which is easily proven through iteration of the sum property.

The reason why this property is useful is that, although not every number is a sum of two perfect squares, every number must be a sum of some finite number of perfect squares (at worst, n is the sum of n ones).