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Question Number 191028 by Mingma last updated on 16/Apr/23

Answered by Frix last updated on 16/Apr/23

$$\mathrm{Not}\mathrm{a}\mathrm{proof}\mathrm{but}\mathrm{an}\mathrm{idea}\mathrm{for}\mathrm{a}\mathrm{proof}:$$$$(n-1)n(n+1)\times N=k^3;k\in \mathbb{N}$$$$\mathrm{We}\mathrm{know}\mathrm{that}$$$$1.\gcd (n-1,n)=1$$$$2.\gcd (n,n+1)=1$$$$3.\gcd (n-1,n+1)\in \{1,2\}$$$$\Rightarrow (n-1)n(n+1)\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{perfect}\mathrm{cube}$$$$\Rightarrow N\mathrm{must}\mathrm{have}\mathrm{enough}\mathrm{prime}\mathrm{factors}\mathrm{to}$$$$\mathrm{complete}\mathrm{the}\mathrm{perfect}\mathrm{cube}.\mathrm{Obviously}$$$$\mathrm{this}\mathrm{is}\mathrm{impossible}\mathrm{for}N=n\pm 2$$$$$$$$\mathrm{Let}N\in \mathbb{N}\mathrm{the}\mathrm{necessary}\mathrm{factor}\mathrm{to}\mathrm{get}{k}^3$$$$n=2N=36$$$$n=3N=9$$$$n=4N=450$$$$n=5N=225$$$$n=6N=44100$$$$n=7N=1764$$$$n=8N=147$$$$...$$

Answered by a.lgnaoui last updated on 16/Apr/23

$$\mathrm{soient}\mathbf{a}(\mathbf{a}+1)(\mathbf{a}+2)\mathrm{et}(\mathbf{a}+3)\mathrm{les}4\mathrm{nombres}$$$$\mathrm{entiers}\mathrm{successives}$$$$\mathrm{alirs}{\mathbf{a}}^4+6{\mathbf{a}}^3+11{\mathbf{a}}^2+6\mathbf{a}={\mathbf{a}}^3(\mathbf{a}+6+\frac{11}{\mathrm{a}}+\frac{6}{{\mathrm{a}}^3})$$$$\mathrm{on}\mathrm{remarque}\mathrm{que}\frac{11}{\mathrm{a}}+\frac{6}{{\mathrm{a}}^3}=\frac{1}{\mathrm{a}}(11+\frac{6}{{\mathrm{a}}^2})$$$$\mathrm{E}={\mathrm{a}}^3[\mathrm{a}+6+\frac{1}{\mathrm{a}}(11+\frac{6}{{\mathrm{a}}^2})]$$$$\max \mathrm{valdur}\mathrm{possible}(\mathrm{a}=1)$$$$(7+17){\mathrm{a}}^3\mathrm{n}\mathrm{est}[\mathrm{pas}\mathrm{cube}\mathrm{parfait}$$$${\left(2\mathrm{a}\right)}^3<{24\mathrm{a}}^3<{\left(3\mathrm{a}\right)}^3$$$$\mathrm{donc}\nexists \mathrm{a}/\mathrm{a}\in \mathbb{N}\mathrm{pour}\mathrm{lequel}$$$$\mathrm{a}(\mathrm{a}+1)(\mathrm{a}+2)(\mathrm{a}+3)\mathrm{est}\mathrm{cube}\mathrm{parfait}$$$$\bullet 2-\mathrm{suposons}\mathrm{que}\mathrm{a}+1={\mathit{\lambda }}^3$$$$\mathrm{alors}\mathrm{a}={\lambda }^3-1=(\lambda -1)({\lambda }^2+\lambda +1)$$$$\mathrm{n}\mathrm{est}\mathrm{pa}\mathrm{un}\mathrm{cube}\mathrm{parfait}$$$$\mathrm{meme}\mathrm{cas}\mathrm{pour}\mathrm{a}+2\mathrm{et}\mathrm{a}+3\mathrm{respectivement}$$$$\mathrm{par}\mathrm{consequent}$$$$\Rightarrow \mathrm{\exists }!\mathrm{un}\mathrm{terme}{\mathrm{a}}_{\mathrm{i}}/{\mathrm{a}}_{\mathrm{i}}^3\mathrm{non}\mathrm{cube}\mathrm{parfait}$$$$1<\mathrm{i}<4\prod _{\mathrm{i}=1}^{\mathrm{i}=4}{\mathrm{a}}_{\mathrm{i}}({\mathrm{a}}_{\mathrm{i}+1}={\mathrm{a}}_{\mathrm{i}}+1\mathrm{entier}):\mathbf{n}\mathbf{est}\mathbf{pas}\mathbf{cube}\mathbf{parfait}$$$$\mathbf{alors}$$$$\mathrm{le}\mathrm{priduit}\mathrm{de}4\mathrm{entiers}\mathrm{successives}\mathrm{ne}$$$$\mathrm{peut}\mathrm{pas}\mathrm{etre}\mathrm{un}\mathrm{cube}\mathrm{parfait}$$

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