Thread: Sums of Squares problem.

1. Sums of Squares problem.

Let $\displaystyle n = 17, 51, 37, 407, 629, 40885.$

(a) Decide whether $\displaystyle n \in S_2$.

(b) If $\displaystyle n \in S_2$ find $\displaystyle a,b \in \mathbb{Z}$ with $\displaystyle n=a^2+b^2.$

(c) If $\displaystyle n \notin S_2$ find $\displaystyle a,b,c,d \in \mathbb{Z}$ with $\displaystyle n=a^2+b^2+c^2+d^2.$

Thank you all for checking this out, and any ideas/suggestions are very welcome. Thanks!

2. Originally Posted by Proof_of_life
Let $\displaystyle n = 17, 51, 37, 407, 629, 40885.$

(a) Decide whether $\displaystyle n \in S_2$.

(b) If $\displaystyle n \in S_2$ find $\displaystyle a,b \in \mathbb{Z}$ with $\displaystyle n=a^2+b^2.$

(c) If $\displaystyle n \notin S_2$ find $\displaystyle a,b,c,d \in \mathbb{Z}$ with $\displaystyle n=a^2+b^2+c^2+d^2.$

Thank you all for checking this out, and any ideas/suggestions are very welcome. Thanks!
$\displaystyle S_2$ is the set of all numbers that are the sum of the squares of two integers?

3. $\displaystyle S_2$ is the set of all sums of 2 squares.

Definition:
For each integer $\displaystyle k$, greater than or equal to 1, let $\displaystyle S_k$ = {$\displaystyle n$ | $\displaystyle n$ = $\displaystyle (x_1)^2$ + . . . + $\displaystyle (x_k)^2$ for some $\displaystyle x_1$ ,..., $\displaystyle x_k \in \mathbb{Z}$ },the set of all sums of $\displaystyle k$ squares.

Sorry, and thanks!

4. Hint (just to get you started): a square has to be congruent to 0 or 1 mod 4.

5. So, let me say for the first one that n=17. Since 17 is congruent to 1 mod 4, that would satisfy part (a)? Since part (a) is satisfied, I would take part (b)'s approach and ignore part (c). The answer to part (b) would be $\displaystyle 4^2 + 1^2 = 17$, where a=4 and b=1.

I think I will get confused when n does not belong to $\displaystyle S_2$, and I will not know how to approach part (c).

6. Sum of squares

I should have just googled this earlier! Thanks to the last replier, it got me on the right track.