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» identité du mois de janvier 2022 »
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Soit ![\phi\](http://s0.wp.com/latex.php?latex=%5Cphi%5C&bg=T&fg=000000&s=0 "\phi\")
( x + 1 )² – ( x – 1 )² = x² – 1 lorsque
x = ![\frac{3+\sqrt5}{-1+\sqrt5}\\\](http://s0.wp.com/latex.php?latex=%5Cfrac%7B3%2B%5Csqrt5%7D%7B-1%2B%5Csqrt5%7D%5C%5C%5C&bg=T&fg=000000&s=0 "\frac{3+\sqrt5}{-1+\sqrt5}\\\")
![\frac{\phi+1}{\phi-1}\\\](http://s0.wp.com/latex.php?latex=%5Cfrac%7B%5Cphi%2B1%7D%7B%5Cphi-1%7D%5C%5C%5C&bg=T&fg=000000&s=0 "\frac{\phi+1}{\phi-1}\\\")
![\sqrt(x+1)^2-\sqrt(x-1)^2=\sqrt(x^2-1)\\\](http://s0.wp.com/latex.php?latex=%5Csqrt%28x%2B1%29%5E2-%5Csqrt%28x-1%29%5E2%3D%5Csqrt%28x%5E2-1%29%5C%5C%5C&bg=T&fg=000000&s=0 "\sqrt(x+1)^2-\sqrt(x-1)^2=\sqrt(x^2-1)\\\")
x =
![\frac{(1+\sqrt5)^2+4}{(1+\sqrt5)^2-4}\\\](http://s0.wp.com/latex.php?latex=%5Cfrac%7B%281%2B%5Csqrt5%29%5E2%2B4%7D%7B%281%2B%5Csqrt5%29%5E2-4%7D%5C%5C%5C&bg=T&fg=000000&s=0 "\frac{(1+\sqrt5)^2+4}{(1+\sqrt5)^2-4}\\\")
^2-\sqrt[3](x-1)^2=\sqrt[3](x^2-1)\\\](http://s0.wp.com/latex.php?latex=%5Csqrt%5B3%5D%28x%2B1%29%5E2-%5Csqrt%5B3%5D%28x-1%29%5E2%3D%5Csqrt%5B3%5D%28x%5E2-1%29%5C%5C%5C&bg=T&fg=000000&s=0 "\sqrt[3](x+1)^2-\sqrt[3](x-1)^2=\sqrt[3](x^2-1)\\\")
x =
![\frac{(1+\sqrt5)^3+8}{(1+\sqrt5)^3-8}\\\](http://s0.wp.com/latex.php?latex=%5Cfrac%7B%281%2B%5Csqrt5%29%5E3%2B8%7D%7B%281%2B%5Csqrt5%29%5E3-8%7D%5C%5C%5C&bg=T&fg=000000&s=0 "\frac{(1+\sqrt5)^3+8}{(1+\sqrt5)^3-8}\\\")
^2-\sqrt[4](x-1)^2=\sqrt[4](x^2-1)\\\](http://s0.wp.com/latex.php?latex=%5Csqrt%5B4%5D%28x%2B1%29%5E2-%5Csqrt%5B4%5D%28x-1%29%5E2%3D%5Csqrt%5B4%5D%28x%5E2-1%29%5C%5C%5C&bg=T&fg=000000&s=0 "\sqrt[4](x+1)^2-\sqrt[4](x-1)^2=\sqrt[4](x^2-1)\\\")
x=
![\frac{(1+\sqrt5)^4+16}{(1+\sqrt5)^4-16}\\\](http://s0.wp.com/latex.php?latex=%5Cfrac%7B%281%2B%5Csqrt5%29%5E4%2B16%7D%7B%281%2B%5Csqrt5%29%5E4-16%7D%5C%5C%5C&bg=T&fg=000000&s=0 "\frac{(1+\sqrt5)^4+16}{(1+\sqrt5)^4-16}\\\")
^2-\sqrt[5](x-1)^2=\sqrt[5](x^2-1)\\\](http://s0.wp.com/latex.php?latex=%5Csqrt%5B5%5D%28x%2B1%29%5E2-%5Csqrt%5B5%5D%28x-1%29%5E2%3D%5Csqrt%5B5%5D%28x%5E2-1%29%5C%5C%5C&bg=T&fg=000000&s=0 "\sqrt[5](x+1)^2-\sqrt[5](x-1)^2=\sqrt[5](x^2-1)\\\")
x =
![\frac{(1+\sqrt5)^5+32}{(1+\sqrt5)^5-32}\\\](http://s0.wp.com/latex.php?latex=%5Cfrac%7B%281%2B%5Csqrt5%29%5E5%2B32%7D%7B%281%2B%5Csqrt5%29%5E5-32%7D%5C%5C%5C&bg=T&fg=000000&s=0 "\frac{(1+\sqrt5)^5+32}{(1+\sqrt5)^5-32}\\\")
^2-\sqrt[6](x-1)^2=\sqrt[6](x^2-1)\\\](http://s0.wp.com/latex.php?latex=%5Csqrt%5B6%5D%28x%2B1%29%5E2-%5Csqrt%5B6%5D%28x-1%29%5E2%3D%5Csqrt%5B6%5D%28x%5E2-1%29%5C%5C%5C&bg=T&fg=000000&s=0 "\sqrt[6](x+1)^2-\sqrt[6](x-1)^2=\sqrt[6](x^2-1)\\\")
x =
![\frac{(1+\sqrt5)^6+64}{(1+\sqrt5)^6-64}\\\](http://s0.wp.com/latex.php?latex=%5Cfrac%7B%281%2B%5Csqrt5%29%5E6%2B64%7D%7B%281%2B%5Csqrt5%29%5E6-64%7D%5C%5C%5C&bg=T&fg=000000&s=0 "\frac{(1+\sqrt5)^6+64}{(1+\sqrt5)^6-64}\\\")
…
généralisation
^2-\sqrt[n](x-1)^2=\sqrt[n](x^2-1)\\\](http://s0.wp.com/latex.php?latex=%5Csqrt%5Bn%5D%28x%2B1%29%5E2-%5Csqrt%5Bn%5D%28x-1%29%5E2%3D%5Csqrt%5Bn%5D%28x%5E2-1%29%5C%5C%5C&bg=T&fg=000000&s=0 "\sqrt[n](x+1)^2-\sqrt[n](x-1)^2=\sqrt[n](x^2-1)\\\")
x = ![\frac{(1+\sqrt5)^n+2^n}{(1+\sqrt5)^n-2^n}\\\](http://s0.wp.com/latex.php?latex=%5Cfrac%7B%281%2B%5Csqrt5%29%5En%2B2%5En%7D%7B%281%2B%5Csqrt5%29%5En-2%5En%7D%5C%5C%5C&bg=T&fg=000000&s=0 "\frac{(1+\sqrt5)^n+2^n}{(1+\sqrt5)^n-2^n}\\\")
