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Fractii de forma 1/p, p prim, p>5

 
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Cristi Popa
Euclid


Joined: 10 Nov 2007
Posts: 38
Location: Paris

PostPosted: Sun Jul 20, 2008 12:30 am    Post subject: Fractii de forma 1/p, p prim, p>5 Reply with quote

Fie p un numar prim, p\geq7 si \frac{1}{p}=0,(a_1...a_n). Atunci n=p-1 sau n\ |\ p-1 (n divizor propriu al lui p-1).
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Marius Mainea
Gauss


Joined: 26 May 2008
Posts: 1099
Location: Gaesti (Dambovita)

PostPosted: Sun Jul 20, 2008 9:57 pm    Post subject: Reply with quote

Se stie ca daca perioada lui \frac{1}{p} are n cifre , n este cel mai mic exponent pentru care 10^n-1=\mathcal{M}p (gaussianul lui 10 modulo p)

Conform teoremei lui Fermat (10^{p-1}-1=\mathcal{M}p), rezulta ca p-1=n sau p-1 divide n.


Vezi aici si alte probleme de acest fel.
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