representing all odd naturals as the sum of four squares, two of them equal can anyone prove that every natural odd number can be represented as the form $a^2+b^2+2c^2$ where $a,b$ and $c$ are nonnegative integers?
I've thinked on this problem for a long time, but I couldn't solve it. I'd really appreciate your help.
 A: Bhargava has solved the problem of which integral quadratic forms represent all odd naturals: it's sufficient that the form represent 1, 3, 5, 7, 11, 15, and 33. It's easy to check that your form represents these numbers, and hence all odd natural numbers.
A: An actual proof is in a 1939 book (Theorem 86, page 96) by Leonard Eugene Dickson called Modern Elementary Theory of Numbers. I give Dickson's list (pages 111-113) of "diagonal" regular ternary forms at DIAGONAL. Bhargava's work depends heavily on existing results such as these. Alright, I put Manjul's article, with a preface by Conway, at BHARGAVA. Finally, for a positive quadratic form to represent all numbers, it must have at least four variables. However, it is possible to represent all odd numbers with only three variables. Kaplansky identified all possible such ternary forms in KAPLANSKY. Kap gave 23 ternary forms that seemed to work, 19 he could prove (or had already been proved, such as your $x^2 + y^2 + 2 z^2$) along with four plausible candidates. I proved one of the four, but three are still uncertain. Henri Cohen likes to start his classes by mentioning one of those three forms, pointing out that we cannot prove what numbers it represents.
So, a few cautions. The 15 Theorem and the 33 Theorem are about "classically integral" forms, all mixed terms $x_i x_j$ have even coefficients. There is a 290 theorem by Bhargava and Hanke about positive forms that represent all numbers. In this case, analytic methods are able to finish the problem. It is not clear the 290 result will ever be published, it has been submitted and withdrawn once already. 
Finally, there is no theorem giving a bound that guarantees a positive form does indeed represent all odd positive integers, as soon as we allow forms that are not classically integral. There cannot be such a theorem as long as there is no proof that Kaplansky's three stubborn forms really work.
Plenty of other information is at TERNARY_SITE
EDIT: if one is willing to take as an axiom the three-squares theorem, Legendre 1798, Gauss 1801, it is an easy additional step. So, Legendre, a positive integer $r$ can represented as $r = x^2 + y^2 + z^2$ if and only if $r$ is not of type $4^k (8m+7),$ with integers $k,m \geq 0.$ So, take an odd positive number $n.$ We can see that $2n$ satisfies the three-square theorem, and we have
$$ 2 n = x^2 + y^2 + z^2.$$
The three numbers cannot all be odd, so, perhaps by renaming variables, demand that $z$ be even, or $z = 2 c.$ So far we have
$$ 2 n = x^2 + y^2 + 4 c^2.$$ Next, $x,y$ are either both odd or both even, anyway both $x+y$ and $x-y$ are both even. Take $$ a = \frac{x-y}{2}, \; \; b = \frac{x+y}{2}  $$ and we get
$$ n = a^2 + b^2 + 2 c^2  $$ in integers. 
