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Teorema de Mertens, 1874). 3n2 + O(n log n): (n) = 2 Demostracion: n ( )= = = n X d=1 n X i=1 i ( )= n X X i i=1 d i j d d ( ) = X dd n d (d) 0 0 ! n 2 X 1 n n (d) d =2 (d) + d d d 1 d 1 0 n 1 ! 3. Card(Fn) = 1 + (n) n 1: Demostracion: por induccion. 4. Card(Fn) = 3n2 + n + 1): ( O(n log n): Demostracion: es consecuencia directa de los dos teoremas anteriores. 5. Cada tres terminos consecutivos de la sucesion de Fibonacci un son primos dos a dos. Demostracion: supongamos que existen dos terminos consecutivos un y un 1 tales que mcd(un un 1 ) = d > 1.

Sylvester establecio en 1881 las siguientes cotas 0 96695 < A < 1 y 1 < B < 1 04423: x las cotas 0 949x < (x) < 1 052 E. Aparicio 1, p. 390] ha obtenido para ( ) para x > 501000. 8. Sea M(Fn) el m nimo comun multiplo de los n primeros numeros enteros positivos. Existen dos constantes A1 A2 mayores que 1 tales que A2 en M(Fn) A1 en para valores de n su cientemente grandes. Demostracion: como log(M(Fn)) = (n), tomando A1 = eC1 y A2 = eC2 se tiene que C2 n log(M(Fn)) C1 n eC2 n M(Fn) eC1 n A2 en M(Fn) A1 en para valores de n su cientemente grandes.

6. 12, en Bn aparecen los denominadores + + - + - - + - + un 1 - un un 1 + que, por el lema anterior, son primos dos a dos. 7. (Teorema de Chebyshev). Existen dos constantes positivas C1 y C2 tales que para cada numero real x 2, se veri ca C2 x (x) C1 x: La demostracion puede verse en 14, p. 132]. No obstante, queremos se~nalar que 1 X log x log p = (x) log x (x) (x) p x log p 1 Designamos por (x) , como es costumbre, el numero de primos no superiores a x. 1 Resultados previos. Teorema de Chebyshev 25 y, por tanto, este resultado esta estrechamente relacionado con el clasico teorema que en 1850 estableciera Chebyshev sobre la distribucion de los numeros primos.

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