Fie \( (l_{n})_{n\geq 1} \) sirul numerelor libere de patrate. Sa se arate ca
\( \limsup_{n\to\infty}(l_{n+1}-l_{n})=\infty. \)
Diferenta dintre numere libere de patrate consecutive
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Cezar Lupu
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Diferenta dintre numere libere de patrate consecutive
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.
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Dragos Fratila
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Fie \( p_1,p_2,...,p_k,.. \) sirul numerelor prime
Conform teoremei chineze a resturilor exista \( n \) astfel incat:
\( n\equiv -i (mod p_i^2) \) \( i=1,2,...,m \)
Avem atunci \( n, n+1,...,n+m \) nu sunt libere de patrate.
Asta demonstreaza ca exista oricat de multe numere consecutive care nu sunt libere de patrate, de unde rezulta concluzia problemei.
Conform teoremei chineze a resturilor exista \( n \) astfel incat:
\( n\equiv -i (mod p_i^2) \) \( i=1,2,...,m \)
Avem atunci \( n, n+1,...,n+m \) nu sunt libere de patrate.
Asta demonstreaza ca exista oricat de multe numere consecutive care nu sunt libere de patrate, de unde rezulta concluzia problemei.