Intereting Posts

Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$
definit integral of Airy function
Finite Element Method for a Two-Point Problem
Show $f(x) = x^3 – \sin^2{x} \tan{x} < 0$ on $(0, \frac{\pi}{2})$
Proving $\binom{2n}{n}\ge\frac{2^{2n-1}}{\sqrt{n}}$
Prove that there is no element of order $8$ in $SL(2,3)$
What are some examples of coolrings that cannot be expressed in the form $R$?
Is this proof for Theorem 16.4 Munkers Topology correct?
Projectivity of $B$ over $C$, given $A \subset C \subset B$
Prove $p^2=p$ and $qp=0$
Slick proofs that if $\sum\limits_{k=1}^\infty \frac{a_k}{k}$ converges then $\lim\limits_{n\to\infty} \frac{1}{n}\sum\limits_{k=1}^n a_k=0$
Developing Mathematic Intuition
Let $R=M_n(D)$, $D$ is a division ring. Prove that every $R-$simple module is isomorphic to each other.
Easy way to show that $\mathbb{Z}{2}]$ is the ring of integers of $\mathbb{Q}{2}]$
Calculate exponential limit involving trigonometric functions

Edward Waring, asks whether for every natural number $n$ there exists an associated positive

integer s such that every natural number is the sum of at most $s$ $k$th powers of natural

numbers (for example, every number is the sum of at most 4 squares, or 9 cubes, or 19 fourth

powers, etc.) from here.

I ask: **Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$, with $a_u\neq a_v \geq0 \; , \; \forall u\neq v$?**

$g(k)$ being the minimum number $s$ of $k$th powers needed to represent all integers.

$14$ would be an example of a unique decomposition (into $0^2+1^2+2^2+3^2$).

- What are $x$, $y$ and $z$ if $\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y} = 4$ and $x$, $y$ and $z$ are whole numbers?
- On the equation $(a^2+1)(b^2+1)=c^2+1$
- $x^2+y^2=z^n$: Find solutions without Pythagoras!
- Is Legendre’s solution of the general quadratic equation the only one?
- Solutions to Diophantine Equation $x^2 - D y^2 = m^2$
- Can n! be a perfect square when n is an integer greater than 1?

Non-unique decompositions for $k=2$ can be constructed by letting $ n=(a_0+x)^2+\sum_{j=1}^3 a_j^2$ and $a_0=x+\sum_{j=1}^3 a_j$. Robert gave a formula for cubes below.

So another question is: How can one test that a certain $n$ has a unique solution, or even better how can I calculate the number of respresentations? As Gerry points out in his comment, a prime $p=4k+1\;$ **has** a unique representation $p=a^2+b^2$ with $0<a<b\;$ (Thue’s Lemma).

**Partial answer for $k=2$** (from here)

The sequence of positive integers whose representation as a sum of four squares is unique is:

1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32, 56, 96, 128, 224… (sequence A006431 in OEIS).

These integers consist of the seven odd numbers $1, 3, 5, 7, 11, 15, 23$ and all numbers of the form $2 × 4^k, 6 × 4^k$ or $14 × 4^k$.

**Partial**, therefore, because $23=1^2+2^2+2\times 3^2$ (**answer**, because $14\times 4^k$ is unique).

Another way to look at it, uses ${g(k)}$-dimensional vector spaces $V$ over $\mathbb N_0^+$ that are provided with an $k$-norm $||a||_k=\left( \sum_{j=1}^{g(k)} a_j^k \right)^{1/k}$ and the additional conditions that $a_u\neq a_v \geq0 \; , \; \forall u\neq v$.

Now there is a set of length-preserving operations $U_p$, that transforms vectors, from a defined range $\mathcal R (U_p)$. An example operation was shown above.

In this framework the question sounds like:

**Which vectors of $V$ lie outside the union of all $\mathcal R (U_p)$?**

- Proving $\sqrt2$ is irrational
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- How to get to the formula for the sum of squares of first n numbers?
- Is my proof that $U_{pq}$ is not cyclic if $p$ and $q$ are distinct odd primes correct?
- Proof: if $p$ is prime, and $0<k<p$ then $p$ divides $\binom pk$

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- Analytic continuation for $\zeta(s)$ using finite sums?