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I wish to know the numbers which can be written in the form of sum of squares of two numbers and cube of two numbers and is there any pattern in it?

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What do you mean by "numbers"? Integers? Rational numbers? – Álvaro Lozano-Robledo Feb 21 '13 at 15:28

A number $n$ is the sum of two squares (i.e., $n=a^2+b^2$ for some $a,b\in\mathbb{Z}$) if and only if every prime divisor $p$ of $n$ of the form $p\equiv 3\bmod 4$ occurs in the factorization of $n$ to an even power.

The classification of numbers that are a sum of two cubes is not known. A number $n$ is a sum of two cubes (i.e., $n=a^3+b^3$ for some $a,b\in\mathbb{Z}$) if and only if the curve $n=x^3+y^3$ has an integral point, but at the moment we do not know how to check this condition. If $n\geq 1$, the curve $E_n: n=x^3+y^3$ is an elliptic curve, and there is no proven algorithm to determine whether an elliptic curve has a rational point (nevermind an integral point). If the Birch and Swinnerton-Dyer conjecture is proven to be true, then we can check whether $E_n$ has a rational point by means of studying the vanishing of the Hasse-Weil $L$-series of $E_n$ at $s=1$.

Here is a very interesting paper on sums of two cubes, by J.H. Silverman.

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A number n is expressible as a sum of 2 squares if and only if in the prime factorization of n, every prime of the form (4k+3) occurs an even number of times. See also Fermat's theorem on sums of two squares.

Also well known is the result that every natural number can be written using at most $4$ squares: Lagrange four square theorem

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Thanks for the edit, I need to learn that markup... – Landei Feb 21 '13 at 13:52
Sorry, I corrected it. – Landei Feb 21 '13 at 15:23

The question whether a positive integer $n$ can be expressed as a sum of two cubes of positive integers is addressed in this paper by K Broughan. His Theorem 2.1 states a rather complex necessary and sufficient condition on $n$ for this to be possible.

He also lists 38 integers $k$ $(0 < k < 63)$ such that if $n \equiv k$ mod 63 then $n$ cannot be a sum of two cubes of positive integers. However, it is not the case that if a positive integer does not meet this condition then it can be expressed as such a sum: for example 7 is not on the list but cannot be expressed as a sum of two cubes of positive integers.

The paper also states, in its introduction, the well-known result for a sum of two squares.

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The other possible interpretation of the question: every integer $n$ (positive or negative) can be expressed in integers as $$ n = w^2 + x^2 + y^3 + z^3, $$ as long as we allow $y,z$ to be positive or negative or $0$ as needed. In fact, every integer $n$ can be expressed in integers as $$ n = x^2 + y^2 + z^3, $$ as long as we do not restrict $z.$ If we do demand $z \geq 0,$ then computer search suggests that every $n > 10,000,000$ can be so expressed, as well as the majority of smaller numbers.

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