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PROPOSED PROBLEM (4) 


Let m be a fixed positive integer. Calculate: 


a m 
Lim n (P. ) / P, 


L —> Co 


where n(n) is Smarandache Function defined as the smallest integer 


m such that m! is divisible by n, and p. the prime series. 


k , 
Solution: 





We note by Ps a prime number greater than m. We show that 
m . ‘ 
1 {P. ) = mp., for any i > 3 
t by absurd q (p,” ) =ąa< mp, then 
a! =1°2- e.t DS ara (2p)... ° ((M=K) DO) * ya a, with k > 0, will 
(5 & 
eee A mak m 
divisible by P; Dut not by Pe > 


Then this limit is equal tom. 


Pedro Melendez 

Av. Cristovao Colombo 336 
30.000 Belo Horizonte, MG 
BRAZIL 


40 


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