Udowodnij indukcyjnie, że
Udowodnij indukcyjnie, że
\(\displaystyle{ 19}\) dzieli \(\displaystyle{ 2^{2^{6k+2}}+3}\) dla \(\displaystyle{ k=0,1,2,\dots}\).
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Re: Udowodnij indukcyjnie, że
\(\displaystyle{ 2 ^{2^2} +3=19=19 \cdot 1\\
2 ^{2^{6n+2}} +3=19 \cdot N\\
2 ^{2^{6(n+1)+2} }+3=19 \cdot K\\
L=2 ^{2^{6(n+1)+2}} +3=2 ^{2^{6n+6+2}} +3=2 ^{2^{6n+2} \cdot 2^6} +3=\left( 2 ^{2^{6n+2}} \right) ^{64}+3=\\=\left( 2 ^{2^{6n+2}} +3-3\right) ^{64}+3=\left( 19N-3\right) ^{64}+3=19M+3^{64}+3=\\=19M+3 \left( 3 ^{63} +1\right) =19M+3 \left( 3 ^{9} +1\right)\left( 3^{54}-3 ^{45}+3 ^{36}-3 ^{27}+3 ^{18}-3 ^{9} +1 \right) =\\= 19M+3 \left( 19 \cdot 1036\right)\left( 3^{54}-3 ^{45}+3 ^{36}-3 ^{27}+3 ^{18}-3 ^{9} +1 \right)=19K=P}\)
lub
\(\displaystyle{ L=2 ^{2^{6(n+1)+2}} +3=2 ^{2^{6n+6+2}} +3=2 ^{2^{6n+2} \cdot 2^6} +3=\left( 2 ^{2^{6n+2}} \right) ^{64}+3=\\=\left( 2 ^{2^{6n+2}} \right) ^{64}-3^{64}+3^{64}+3=
\left( 2 ^{2^{6n+2}}-3\right) \left( 2 ^{2^{6n+2}}+3\right)\left( \left( 2 ^{2^{6n+2}}\right) ^{2} +3 ^{2} \right) \cdot \\
\cdot \left( \left( 2 ^{2^{6n+2}}\right) ^{4} +3 ^{4} \right) \left( \left( 2 ^{2^{6n+2}}\right) ^{8} +3 ^{8} \right) \left( \left( 2 ^{2^{6n+2}}\right) ^{16} +3 ^{16} \right) \left( \left( 2 ^{2^{6n+2}}\right) ^{32} +3 ^{32} \right) + \\
+3(3^{63}+1)=
\left( 2 ^{2^{6n+2}}-3\right) \left( 19N\right)\left( \left( 2 ^{2^{6n+2}}\right) ^{2} +3 ^{2} \right) \left( \left( 2 ^{2^{6n+2}}\right) ^{4} +3 ^{4} \right) \cdot \\
\cdot \left( \left( 2 ^{2^{6n+2}}\right) ^{8} +3 ^{8} \right) \left( \left( 2 ^{2^{6n+2}}\right) ^{16} +3 ^{16} \right) \left( \left( 2 ^{2^{6n+2}}\right) ^{32} +3 ^{32} \right) +3(3^{63}+1)=\\=19M+3 \left( 3 ^{9} +1\right)\left( 3^{54}-3 ^{45}+3 ^{36}-3 ^{27}+3 ^{18}-3 ^{9} +1 \right) =\\= 19M+3 \left( 19 \cdot 1036\right)\left( 3^{54}-3 ^{45}+3 ^{36}-3 ^{27}+3 ^{18}-3 ^{9} +1 \right)=19K=P}\)
2 ^{2^{6n+2}} +3=19 \cdot N\\
2 ^{2^{6(n+1)+2} }+3=19 \cdot K\\
L=2 ^{2^{6(n+1)+2}} +3=2 ^{2^{6n+6+2}} +3=2 ^{2^{6n+2} \cdot 2^6} +3=\left( 2 ^{2^{6n+2}} \right) ^{64}+3=\\=\left( 2 ^{2^{6n+2}} +3-3\right) ^{64}+3=\left( 19N-3\right) ^{64}+3=19M+3^{64}+3=\\=19M+3 \left( 3 ^{63} +1\right) =19M+3 \left( 3 ^{9} +1\right)\left( 3^{54}-3 ^{45}+3 ^{36}-3 ^{27}+3 ^{18}-3 ^{9} +1 \right) =\\= 19M+3 \left( 19 \cdot 1036\right)\left( 3^{54}-3 ^{45}+3 ^{36}-3 ^{27}+3 ^{18}-3 ^{9} +1 \right)=19K=P}\)
lub
\(\displaystyle{ L=2 ^{2^{6(n+1)+2}} +3=2 ^{2^{6n+6+2}} +3=2 ^{2^{6n+2} \cdot 2^6} +3=\left( 2 ^{2^{6n+2}} \right) ^{64}+3=\\=\left( 2 ^{2^{6n+2}} \right) ^{64}-3^{64}+3^{64}+3=
\left( 2 ^{2^{6n+2}}-3\right) \left( 2 ^{2^{6n+2}}+3\right)\left( \left( 2 ^{2^{6n+2}}\right) ^{2} +3 ^{2} \right) \cdot \\
\cdot \left( \left( 2 ^{2^{6n+2}}\right) ^{4} +3 ^{4} \right) \left( \left( 2 ^{2^{6n+2}}\right) ^{8} +3 ^{8} \right) \left( \left( 2 ^{2^{6n+2}}\right) ^{16} +3 ^{16} \right) \left( \left( 2 ^{2^{6n+2}}\right) ^{32} +3 ^{32} \right) + \\
+3(3^{63}+1)=
\left( 2 ^{2^{6n+2}}-3\right) \left( 19N\right)\left( \left( 2 ^{2^{6n+2}}\right) ^{2} +3 ^{2} \right) \left( \left( 2 ^{2^{6n+2}}\right) ^{4} +3 ^{4} \right) \cdot \\
\cdot \left( \left( 2 ^{2^{6n+2}}\right) ^{8} +3 ^{8} \right) \left( \left( 2 ^{2^{6n+2}}\right) ^{16} +3 ^{16} \right) \left( \left( 2 ^{2^{6n+2}}\right) ^{32} +3 ^{32} \right) +3(3^{63}+1)=\\=19M+3 \left( 3 ^{9} +1\right)\left( 3^{54}-3 ^{45}+3 ^{36}-3 ^{27}+3 ^{18}-3 ^{9} +1 \right) =\\= 19M+3 \left( 19 \cdot 1036\right)\left( 3^{54}-3 ^{45}+3 ^{36}-3 ^{27}+3 ^{18}-3 ^{9} +1 \right)=19K=P}\)