First problems from text on number theory 
Express $635,318,657$ as sum of two fourth powers in two different ways. It is the smallest number with this property?

and 

$1105$ can be expressed as the sum of two squares in 4 different ways. Find them. 

The solution to these problems wouldn't be of any help but how to think about such problems and what tools to use?
Source: "Elementary Numebr Theory with applications" by Thomas Koshy 
 A: Question $3$ is the only one that is connected to long-established theory.
Note that
$1105=5 \times 13\times 17$, and that each of $5$, $13$, and $17$ is easily seen to be a sum of two squares.
By an identity that is a special case of Brahmagupta's Identity, the product of two sums of two squares is itself a sum of two squares. The identity is
$$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2=(ad-bc)^2 +(ac+bd)^2.$$
You can use the above identity to generate the representations of $1105$ as a sum of squares.  Do it!
The identity is interesting in many ways, and is less "magic" than it looks. 
Let $p+qi$ be a complex number.  Then $|p+qi|^2$, the square of the norm of $a+bi$, is precisely $p^2+q^2$.
The Brahmagupta Identity then can be interpreted as saying that
$$ |(a+bi)(c+di)|^2 =|a+bi|^2|c+di|^2$$
(the square of the norm of a product is the product of the squares of the norms, or, taking square roots, the norm of a product is the product of the norms.) 
Expressing  $1105$ as a sum of two squares can be thought of as finding all the ways to express
$$(2-i)(2+i)(3-2i)(3+2i)(4-i)(4+i)$$
in the form $(a-bi)(a+bi)$, where $a$ and $b$ are integers.
There is a quite thoroughly worked out theory of sums of two squares, that you can find in most books on Elementary Number Theory. For example, a prime of the form $4k+1$ can be expressed in essentially one way as a sum of two squares.  This is a result of Fermat.  The proof, though "elementary", is not all that easy.
A: Nowadays, the best (anyway, most efficient) way to solve the first problem is to type the number into Google to see what turns up. Alternatively, type it into the search window at the Online Encyclopedia of Integer Sequences, and it will lead you to http://oeis.org/A018786 which has some useful citations. 
