E:16893: Difference between revisions

Latest revision as of 13:13, 20 September 2025

E:16893 (Traian Covaciu)

Arătați că numerele 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 7n-1} și 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 17n-1} sunt simultan prime doar dacă 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 n} este un multiplu natural al lui 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 6} .

Soluție

Pentru 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 n=6} se obțin numerele prime 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 42} și 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 101} .

Dacă 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 n} este impar, atunci numerele 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 7n-1} și 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 17n-1} sunt pare, deci nu pot fi prime, ceea ce implică faptul 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 2 |\, n} .

Dacă există 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 k\in \mathbb{N}} astfel încât 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 n = 3k+1} , atunci numărul 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 7n-1} nu este prim, căci 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 7n-1 = 7\left(3k+1\right)-1 = 3\left(7k+2\right) \vdots \, 3 }

Dacă există 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 k\in \mathbb{N}} astfel încât 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 n = 3k+2} , atunci numărul 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 17n-1} nu este prim, căci 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 17n-1 = 17\left(3k+2\right)-1 = 3\left(17k+11\right) \vdots \, 3. } Cum 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 n} trebuie să fie un număr par care este și multiplu al lui 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 3} , 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 6 |\, n} .

Observație. Perechi 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 \left(7n-1, 17n-1\right)}
 formate din numere prime se obține pentru 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 n=6}
, 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 n=60}
, 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 n=120}
, 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 n=300}
.

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 Dacă <math>n</math> este impar, atunci numerele <math>7n-1</math> și <math>17n-1</math> sunt pare, deci nu pot fi prime, ceea ce implică faptul că <math>2 |\, n</math>.
-Dacă există <math>k\in \mathbb{N}</math> astfel încât <math>n = 3k+1</math>, atunci numărul <math>7n-1</math> nu este prim, căci  <math>7n-1 = 7\left(3k+1\right)-1 = 3\left(7k+2\right) \vdots \, 3 </math>
+Dacă există <math>k\in \mathbb{N}</math> astfel încât <math>n = 3k+1</math>, atunci numărul <math>7n-1</math> nu este prim, căci  <math display="block">7n-1 = 7\left(3k+1\right)-1 = 3\left(7k+2\right) \vdots \, 3 </math>
-Dacă există <math>k\in \mathbb{N}</math> astfel încât <math>n = 3k+2</math>, atunci numărul <math>17n-1</math> nu este prim, căci  <math><math>17n-1 = 17\left(3k+2\right)-1 = 3\left(17k+11\right) \vdots \, 3. </math>
+Dacă există <math>k\in \mathbb{N}</math> astfel încât <math>n = 3k+2</math>, atunci numărul <math>17n-1</math> nu este prim, căci  <math display="block">17n-1 = 17\left(3k+2\right)-1 = 3\left(17k+11\right) \vdots \, 3. </math>Cum <math>n</math> trebuie să fie un număr par care este și multiplu al lui <math>3</math>, deducem că <math>6 |\, n</math>.
-Cum <math>n</math> trebuie să fie un număr par care este și multiplu al lui <math>3</math>, deducem că <math>6 |\, n</math>.
   Observație. Perechi <math>\left(7n-1, 17n-1\right)</math> formate din numere prime se obține pentru <math>n=6</math>, <math>n=60</math>, <math>n=120</math>, <math>n=300</math>.