# On the Nature of Some Euler’s Double Equations Equivalent to Fermat’s Last Theorem

## Abstract

**:**

## 1. **Introduction**

## 2. **Indeterminate Analysis of Second Degree**

**Theorem**

**1.**

## 3. On Homogeneous Ternary Quadratic Diophantine Equations ${\mathit{aX}}^{\mathbf{2}}+{\mathit{bY}}^{\mathbf{2}}-{\mathit{cZ}}^{\mathit{2}}=\mathbf{0}$

**Theorem**

**2.**

**Proof.**

## 4. From the Concordant Forms of Euler to Fermat’s Last Theorem

**Conjecture**

**1.**

**Theorem**

**3**

## 5. The Nature of Euler’s Double Equations through the Algebraic Geometry

## 6. The Determination of the Parameter $\mathit{Q}$ in Euler’s Double Equations

## 7. Conclusions

## Additional Remarks

**Remark**

**1.**

**Remark**

**2.**

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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Ossicini, A.
On the Nature of Some Euler’s Double Equations Equivalent to Fermat’s Last Theorem. *Mathematics* **2022**, *10*, 4471.
https://doi.org/10.3390/math10234471

**AMA Style**

Ossicini A.
On the Nature of Some Euler’s Double Equations Equivalent to Fermat’s Last Theorem. *Mathematics*. 2022; 10(23):4471.
https://doi.org/10.3390/math10234471

**Chicago/Turabian Style**

Ossicini, Andrea.
2022. "On the Nature of Some Euler’s Double Equations Equivalent to Fermat’s Last Theorem" *Mathematics* 10, no. 23: 4471.
https://doi.org/10.3390/math10234471