# A New Look at the Single Ladder Problem (SLP) via Integer Parametric Solutions to the Corresponding Quartic Equation

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- Find complete integer solutions by parametric representation for the quartic in question, both for the two solutions in the first quadrant and for all four solutions.
- Use this information to find new information concerning SLP.

## 2. Complete Parametric Representation for Integer Solutions in the First Quadrant

**Theorem**

**1.**

**Proof.**

**Example**

**1**

**.**If $r=3,\phantom{\rule{4pt}{0ex}}s=2,\phantom{\rule{4pt}{0ex}}t=2,\phantom{\rule{4pt}{0ex}}w=1$, then we get the following integer solution of SLP, after scaling for common factors:

**Remark**

**1.**

**Corollary**

**1.**

**Remark**

**2.**

**Example**

**2**

**.**If $t=2$ and $w=1$, then we have the following integer solution of SLP in this minimum case:

## 3. Parametric Representation for Integer Solutions to All Four Real Solutions to the Quartic (1) When $\mathit{L}>{\mathit{L}}_{\mathbf{min}}$

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

**Example**

**3**

**.**If $t=2$ and $w=1,$ then we obtain the following integer solution $(a,b,L)$ to the SLP:

**Remark**

**4.**

**Example**

**4.**

**Proof.**

**Example**

**5**

**.**If $t=2$ and $w=1$, then we obtain the following integer solution $(a,b,L)$ to the SLP:

**Example**

**6.**

**Example**

**7**

**.**If $m=2$ and $n=1$, then we obtain the following solution $(a,b,L)$ to the SLP:

## 4. Final Examples and Remarks

**Remark**

**5.**

**Example**

**8.**

**Example**

**9**

**.**We see that when $a=b$, then ${x}_{1}={y}_{2}$ and ${x}_{2}={y}_{1}$. In particular, if $t=2$ and $w=1$, then

**Example**

**10.**

**Example**

**11**

**.**We see that when ${z}_{1}=L-{z}_{1},$ then ${x}_{1}=a$ and ${y}_{1}=b$. In particular, if $t=2$ and $w=1$, we obtain that

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The crossed ladder problem (CLP): find a complete integer parametric representation of the CLP variables.

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**MDPI and ACS Style**

Høibakk, R.; Lukkassen, D.; Meidell, A.; Persson, L.-E.
A New Look at the Single Ladder Problem (SLP) via Integer Parametric Solutions to the Corresponding Quartic Equation. *Mathematics* **2020**, *8*, 267.
https://doi.org/10.3390/math8020267

**AMA Style**

Høibakk R, Lukkassen D, Meidell A, Persson L-E.
A New Look at the Single Ladder Problem (SLP) via Integer Parametric Solutions to the Corresponding Quartic Equation. *Mathematics*. 2020; 8(2):267.
https://doi.org/10.3390/math8020267

**Chicago/Turabian Style**

Høibakk, Ralph, Dag Lukkassen, Annette Meidell, and Lars-Erik Persson.
2020. "A New Look at the Single Ladder Problem (SLP) via Integer Parametric Solutions to the Corresponding Quartic Equation" *Mathematics* 8, no. 2: 267.
https://doi.org/10.3390/math8020267