Number Conjecture

Seth D. Humphries

18 April, 2005

Let $x_{k}\in\mathbb{N}$ and given:

$\displaystyle z^{n}=\sum\limits_{k=1}^M x^{n}_{k}$     (1)
$\displaystyle n,M \in \mathbb{N}$      

Then $\forall\ M$ and $\forall\ n \leq M$ there exist solutions such that $z \in\mathbb{N}$. Furthermore, $\forall\ n\ >\ M$ there exist only solutions such that $z \notin{\mathbb{N}}$

To give examples, I have found solutions for $n = M$ for various $M$. They are given below. Note that Eq. 2 and Eq. 7 are examples demonstrating Fermat's Last Theorem.


$\displaystyle 5^2$ $\textstyle =$ $\displaystyle 3^2 + 4^2$ (2)
$\displaystyle 12^{3}$ $\textstyle =$ $\displaystyle 6^{3} + 8^{3} + 10^{3}$ (3)
$\displaystyle 353^{4}$ $\textstyle =$ $\displaystyle 30^{4} + 120^{4} + 272^{4} + 315^{4}$ (4)
$\displaystyle 144^{5}$ $\textstyle =$ $\displaystyle 38^{5} + 86^{5} + 92^{5} + 94^{5} + 134^{5}$ (5)

For $n < M$, an example is:

$\displaystyle 14^{2}$ $\textstyle =$ $\displaystyle 4^{2} + 6^{2} + 12^{2}$ (6)

For solutions where $n > M$ an example is:

$\displaystyle 6.54^{3} \approx 4^{3} + 6^{3}$     (7)