Author(s)

Prosper Kouadio Kimou¹, François Emmanuel Tanoé², Kouassi Vincent Kouakou³

¹Laboratoire d’Informatique et de Mathématiques appliquées, Institut Polytechnique Félix Houphouët BOIGNY, Yamoussoukro, Cote d’Ivoire.
²UFR Mathématiques et Informatique, Université Félix Houphouet BOIGNY, Abidjan, Cote d’Ivoire.
³UFR Sciences Fondamentales Appliquées, Université NANGUI ABROGOUA, Abidjan, Cote d’Ivoire.

In this paper we prove in a new way, the well known result, that Fermat’s equation a⁴ + b⁴ = c⁴, is not solvable in $\mathbb{N}$ , when $abc\ne 0$ . To show this result, it suffices to prove that: $\left(F_0\right):a_1^4+{\left(2^s{b}_1\right)}^4=c_1^4$ , is not solvable in $\mathbb{N}$ , (where $a_1\mathrm{,}{b}_1\mathrm{,}{c}_1\in 2\mathbb{N}+1$ , pairwise primes, with necessarly $2\le s\in \mathbb{N}$ ). The key idea of our proof is to show that if (F₀) holds, then there exist ${\alpha }_2\mathrm{,}{\beta }_2\mathrm{,}{\gamma }_2\in 2\mathbb{N}+1$ , such that $\left(F_1\right):{\alpha }_2^4+{\left(2^{s-1}{\beta }_2\right)}^4={\gamma }_2^4$ , holds too. From where, one conclude that it is not possible, because if we choose the quantity 2 ≤ s, as minimal in value among all the solutions of $\left(F_0\right)$ , then $\left({\alpha }_2{\mathrm{,2}}^{s-1}{\beta }_2\mathrm{,}{\gamma }_2\right)$ is also a solution of Fermat’s type, but with $2\le s-1<s$ , witch is absurd. To reach such a result, we suppose first that (F₀) is solvable in $\left(a_1{\mathrm{,2}}^s{b}_1\mathrm{,}{c}_1\right)$ , s ≥ 2 like above; afterwards, proceeding with “Pythagorician divisors”, we creat the notions of “Fermat’s b-absolute divisors”: $\left(d_b\mathrm{,}{d^{\prime }}_b\right)$ which it uses hereafter. Then to conclude our proof, we establish the following main theorem: there is an equivalence between (i) and (ii): (i) (F₀): $a_1^4+{\left(2^s{b}_1\right)}^4=c_1^4$ , is solvable in $\mathbb{N}$ , with $2\le s\in \mathbb{N}$ , $\left(a_1\mathrm{,}{b}_1\mathrm{,}{c}_1\right)\in {\left(2\mathbb{N}+1\right)}^3$ , coprime in pairs. (ii) $\exists \left(a_1\mathrm{,}{b}_1\mathrm{,}{c}_1\right)\in {\left(2\mathbb{N}+1\right)}^3$ , coprime in pairs, for wich: $\exists \left({b^{\prime }}_2\mathrm{,}{b}_2\mathrm{,}{b^{″}}_2\right)\in {\left(2\mathbb{N}+1\right)}^3$ coprime in pairs, and $2\le s\in \mathbb{N}$ , checking $b_1={b^{\prime }}_2{b}_2{b^{″}}_2$ , and such that for notations: $S=s-\lambda \left(s-1\right)$ , with $\lambda \in \left\{\mathrm{0,1}\right\}$ defined by $\frac{c_1-a_1}{2}\equiv \lambda \left(\mathrm{mod}2\right)$ , $d_b=gcd\left(2^s{b}_1,c_1-a_1\right)=2^S{b}_2$ and ${d^{\prime }}_b=2^{s-S}{b^{\prime }}_2=\frac{2^s{B}_2}{d_b}$ , where ${\left(2^s{B}_2\right)}^2=gcd\left(b_1^2,c_1^2-a_1^2\right)$ , the following system is checked: $\{\begin{array}{l}{c}_1-a_1=\frac{d_b^4}{2^{2+\lambda }}=2^{2-\lambda }{\left(2^{S-1}{b}_2\right)}^4\\ c_1+a_1=2^{1+\lambda }{d^{\prime }}_b^4=2^{1+\lambda }{\left(2^{s-S}{b^{\prime }}_2\right)}^4\\ c_1^2+a_1^2=2{b^{″}}_2^4\end{array}$ ; and this system implies: ${\left(b_{1-\lambda \mathrm{,2}}^4\right)}^2+{\left(2^{4s-3}{b}_{\lambda \mathrm{,2}}^4\right)}^2={\left({b^{″}}_2^2\right)}^2$ ; where: $\left(b_{1-\lambda \mathrm{,2}}\mathrm{,}{b}_{\lambda \mathrm{,2}}\mathrm{,}{b^{″}}_2\right)=\{\begin{array}{l}\left({b^{\prime }}_2\mathrm{,}{b}_2\mathrm{,}{b^{″}}_2\right)\text{if}\lambda =0\hfill \\ \left(b_2\mathrm{,}{b^{\prime }}_2\mathrm{,}{b^{″}}_2\right)\text{if}\lambda =1\hfill \end{array}$ ; From where, it is quite easy to conclude, following the method explained above, and which thus closes, part I, of this article.

Factorisation in ℤ, Greatest Common Divisor, Pythagoras Equation, Pythagorician Triplets, Fermat's Equations, Pythagorician Divisors, Fermat's Divisors, Diophantine Equations of Degree 2, 4-Integral Closure of ℤ in ℚ

Kimou, P. , Tanoé, F. and Kouakou, K. (2024) Fermat and Pythagoras Divisors for a New Explicit Proof of Fermat’s Theorem:a⁴ + b⁴ = c⁴. Part I. Advances in Pure Mathematics, 14, 303-319. doi: 10.4236/apm.2024.144017.