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Question Number 136516 by metamorfose last updated on 22/Mar/21

$$findallintegers(x,y):5^x=3^x+2021y$$

Answered by MJS_new last updated on 23/Mar/21

$$\mathrm{solution}:x=322n\wedge y=\frac{5^{322n}-3^{322n}}{2021}\mathrm{\forall }n\in \mathbb{N}$$

Commented by MJS_new last updated on 23/Mar/21

$$\mathrm{yes}$$$$3^x=2021n_3+r_3$$$$5^x=2021n_5+r_5$$$$y=\frac{5^x-3^3}{2021}\Rightarrow r_3=r_5$$$$\mathrm{obviously}\mathrm{this}\mathrm{is}\mathrm{given}\mathrm{for}x=0\Rightarrow r_3=r_5=1$$$$\mathrm{we}\mathrm{have}\mathrm{no}\mathrm{other}\mathrm{chance}\mathrm{but}\mathrm{to}\mathrm{find}\mathrm{the}\mathrm{next}$$$$x\mathrm{by}\mathrm{trying}...$$

Commented by Rasheed.Sindhi last updated on 23/Mar/21

$$\mathcal{S}ir,didyoumakeaprogramforthis?$$

Commented by Rasheed.Sindhi last updated on 23/Mar/21

$$\mathcal{T}h\alpha n\mathbb{k}s\mathcal{S}ir!$$

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