S:E15.316

S:E15316 (Cristina Vijdeluc și Mihai Vijdeluc)

Determinați numărul prim $p$ și numărul natural $q$ astfel încât

$$p^{2}+5^{p}+31=3181^{q}$$

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Soluție:

Putem scrie $p^{2}+5^{p}=3181^{q}-31$. Cum ultima cifră a lui $3181^{q}-31$ este $0$ și $5^{p}$ este divizibil cu Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5} , deducem că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=5} . Atunci relația devine Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5^2 + 5^5 + 31 = 3181^{q}} sau Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3181 = 3181^{q}} , de unde Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q = 1} .