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Sur la Conjecture de Fermat

Authors:

Gaston Casanova

Institution:

(1) 6 Avenue Paul Apell, 75014 Paris, France

Abstract:

All the letters represent relative integers, except $\mathbb{Z},$ and $\mathbb{J} = \{ a + bj\}$ and i = e₁e₂ in R(2, 0) oder e₁ in R(1, 0). We study the Fermat’s equation

$a^{n} + b^{n} = c^{n}$

(1)

a, b, c being prime two and two and by utilizing an elementary method. We use the Gauss’ formula

$4(c^{n} - a^{n} ) = 4b^{n} = (c - a)(A^{2} \pm nB^{2} )$

where n = 5, 7, 11, 17.

If 2 is the p · g · c · d· of A and B we put

$A\,=\,2A^{\prime},\quad B\,=\,2B^{\prime}$

and A′ and B′ are prime between themselves.

If βⁿ = bⁿ / (c − a) is not divisible by n, we write the expansion

$(u + k\upsilon )^{n} = {A}\ifmmode{'}\else$'$\fi + k{B}\ifmmode{'}\else$'$\fi$

(2)

by puting $k = i{\sqrt n }$ oder $k = e_{1} {\sqrt n }$ whereas $\ifmmode\expandafter\bar\else\expandafter\=\fi{k} = - i{\sqrt n }$ oder $- e_{1} {\sqrt n }.$ It follows that B ′ is divisible by n.

If one a, b oder c is divisible by n we prove the impossibility

In the case n = 3 the ring {a + bj} is euclidian which permits to conclude in favour of the impossibility.

Keywords:

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