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Question Number 19329 by Tinkutara last updated on 09/Aug/17

$$\mathrm{Solve}\mathrm{the}\mathrm{equation}{y}^3=x^3+8x^2-6x+8$$$$\mathrm{for}\mathrm{positive}\mathrm{integers}x\mathrm{and}y.$$

Commented by Tinkutara last updated on 12/Aug/17

$$\mathrm{RMO}2000$$

Commented by Tinkutara last updated on 12/Aug/17

$$\mathrm{help}\mathrm{pls}$$

Commented by Tinkutara last updated on 12/Aug/17

$$\mathrm{How}\mathrm{is}{(x+3)}^3-y^3=x^3+9x^2+27x$$$$+27-(x^3+8x^2-6x+8)=x^2+33x$$$$+19>0?$$

Commented by ajfour last updated on 12/Aug/17

$$\mathrm{given}:\mathrm{y}={\mathrm{x}}^3+{8\mathrm{x}}^2-6\mathrm{x}+8$$$${(\mathrm{x}+3)}^3={\mathrm{x}}^3+{9\mathrm{x}}^2+27\mathrm{x}+27$$$$\mathrm{So}{(\mathrm{x}+3)}^3-{\mathrm{y}}^3={\mathrm{x}}^2+33\mathrm{x}+19$$$$\mathrm{for}\mathrm{positive}\mathrm{integer}\mathrm{x},\mathrm{R}.\mathrm{H}.\mathrm{S}.⩾53$$

Commented by Tinkutara last updated on 12/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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