Intereting Posts

covariance of increasing functions
Is $p(p + 1)$ always a friendly number for $p$ a prime number?
What is a conormal vector to a domain intuitively?
For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
Why is the minimum size of a generating set for a finite group at most $\log_2 n$?
Rigorous Proof: Circle cannot be embedded into the the real line!
Characteristic of a Non-unital Integral Ring
Is my understanding of quotient rings correct?
Finding a primitive element of a finite field
How to solve an nth degree polynomial equation
How to prove that the Fibonacci sequence is periodic mod 5 without using induction?
Suppose f is differentiable on an interval I. Prove that f' is bounded on I if and only if exists a constant M such that $|f(x) – f(y)| \le M|x – y|$
Does an uncountable discrete subspace of the reals exist?
Poles on the curve
Dimension analysis of an integral

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 $0^2+1^2+2^2+3^2$).

- On products of ternary quadratic forms $\prod_{i=1}^3 (ax_i^2+by_i^2+cz_i^2) = ax_0^2+by_0^2+cz_0^2$
- Non-squarefree version of Pell's equation?
- Existence of rational points on ellipses equivalent to existence of integral points?
- Proof that $26$ is the one and only number between square and cube
- Representation of a number as a sum of squares.
- Find all integer solutions of $1+x+x^2+x^3=y^2$

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 $\mathcal R (U_p)$?**

- What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?
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- Sum of digits of repunits
- Proof that $(3^n - 2^n) / n$ is not an integer, $n \geq 2$
- Proof that Fibonacci Sequence modulo m is periodic?
- Super Perfect numbers
- Solve the equation $2φ(x)=x $ for $x\in\mathbb N^+.$
- Show that $2^n$ is not a sum of consecutive positive integers
- If $a|bc$ and $\gcd(a,b) = 1$, then $a|c$

- Graph of continuous function has measure zero by Fubini
- Spectrum of Indefinite Integral Operators
- $ \cos(\hat{A})BC+ A\cos(\hat{B})C+ AB\cos(\hat{C})=\frac {A^2 + B^2 + C^2}{2} $
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- $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
- What are the 2125922464947725402112000 symmetries of a Rubik's Cube?
- fundamental group of $U(n)$
- Square covered with tiles
- Connectedness of the boundary
- continuous onto function from irrationals in onto rationals in
- Show that the group is cyclic.
- How to show pre-compactness in Holder space?
- How to geometrically prove the focal property of ellipse?