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Question Number 180877 by Sheshdevsahu last updated on 18/Nov/22

$$\mathbf{Q}.\mathbf{find}\mathbf{the}\mathbf{largest}\mathbf{value}\mathbf{of}\mathbf{such}\mathbf{that}\mathbf{the}$$$$\mathbf{positive}\mathbf{integers}\mathbf{a},\mathbf{b}>1\mathbf{satisfy}.$$$${\mathbf{a}}^{\mathbf{b}}.{\mathbf{b}}^{\mathbf{a}}+{\mathbf{a}}^{\mathbf{b}}+{\mathbf{b}}^{\mathbf{a}}=5329$$

Commented by MJS_new last updated on 18/Nov/22

$$\mathrm{the}\mathrm{largest}\mathrm{value}\mathrm{of}what?$$$$\mathrm{anyway}\mathrm{if}a,b\in {\mathbb{Z}}^{+}\mathrm{\setminus }\left\{1\right\}\mathrm{we}\mathrm{have}\mathrm{these}\mathrm{solutions}:$$$$a=3\wedge b=4\vee a=4\wedge b=3$$

Answered by Rasheed.Sindhi last updated on 19/Nov/22

$${\mathbf{a}}^{\mathbf{b}}.{\mathbf{b}}^{\mathbf{a}}+{\mathbf{a}}^{\mathbf{b}}+{\mathbf{b}}^{\mathbf{a}}=5329$$$${\mathrm{a}}^{\mathrm{b}}({\mathrm{b}}^{\mathrm{a}}+1)+{\mathrm{b}}^{\mathrm{a}}+1=5330$$$$({\mathrm{a}}^{\mathrm{b}}+1)({\mathrm{b}}^{\mathrm{a}}+1)=5330=2\cdot 5\cdot 13\cdot 41$$$$k\mid 5330:$$$${\mathrm{a}}^{\mathrm{b}}+1=k\wedge {\mathrm{b}}^{\mathrm{a}}+1=5330/k$$$$\Rightarrow k=65,82$$$$\{\begin{array}{l}{\mathrm{a}}^{\mathrm{b}}+1=65\wedge {\mathrm{b}}^{\mathrm{a}}+1=82\\ {\mathrm{a}}^{\mathrm{b}}+1=82\wedge {\mathrm{b}}^{\mathrm{a}}+1=65\end{array}$$$$\{\begin{array}{l}{\mathrm{a}}^{\mathrm{b}}=64\wedge {\mathrm{b}}^{\mathrm{a}}=81\Rightarrow 4^3=64\wedge 3^4=81\\ {\mathrm{a}}^{\mathrm{b}}=81\wedge {\mathrm{b}}^{\mathrm{a}}=64\Rightarrow 3^4=81\wedge 4^3=64\end{array}$$$$\{\mathrm{a},\mathrm{b}\}=\{3,4\}$$

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