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Question Number 31109 by naka3546 last updated on 02/Mar/18

$${\mathit{a}}^4+{\mathit{b}}^4+13isapossiblelargestprimenumber.$$$$\mathit{a}and\mathit{b}areprimenumbers.$$$$Find\mathit{a}and\mathit{b}.$$

Commented by rahul 19 last updated on 02/Mar/18

$$i{don}^{\prime }tthinksuchques.haveproper$$$${sol}^n.theycanbedoneonlybyhitnd$$$$trialmethod?$$

Commented by Tinkutara last updated on 02/Mar/18

Many of such questions indeed have a good thinking and proper logic. Many such questions can be solved by AOPS Community. I think posting questions there can help him.

Commented by MJS last updated on 03/Mar/18

$$\mathrm{what}\mathrm{does}‘‘\mathrm{possible}\mathrm{largest}\mathrm{prime}$$$${\mathrm{number}}^{″}\mathrm{mean}?$$

Commented by naka3546 last updated on 04/Mar/18

$$possibleprimenumberof\mathit{a}and\mathit{b}primenumbers.$$

Commented by MJS last updated on 04/Mar/18

$$a=3,b=5,a^4+b^4+13=719$$$$\mathrm{what}\mathrm{I}\mathrm{found}\mathrm{but}\mathrm{cannot}\mathrm{explain}:$$$$a,b\mathrm{are}\mathrm{prime}\mathrm{numbers},$$$$c=a^4+b^4+13$$$$\mathrm{for}a=3,b>3:5\mid c$$$$\mathrm{for}a=5,b>5:3\mid c$$$$\mathrm{for}a>5,b>a:(3\cdot 5)\mid c$$$$\mathrm{can}\mathrm{somebody}\mathrm{explain}\mathrm{why}\mathrm{this}\mathrm{is}?$$

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