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A note on the primes in Smarandache power product sequences

By Maohua Le


Abstract. In this paper we IS not a power of 2, then product sequence of the first prime 2. prove that if r> 1 and r the Smarandache r-power kind contains onlv one J Key words. Smarandache power product sequence, fIrst kind, prune. For any positive integers n, r with r> 1, let P(n,r) be the n-th power of degree r. Fwther, let n (1) V(n,r) = n P(k,r)+ 1. k=1 Then the sequence V(r)= { V(n,r)}oon=l is called the Smarandache r-power product sequence of the In [2], Russo proposed the following question. fIrst kind. Question. How many tenns m V(2) and primes? V(3) In fact, Le and Wu [1] showed that if r is then V(r) contains only one prime 2. It implies V(3) contains only one pnme 2. In this paper prove a general result as follows. Theorem. If r is not a power of 2, then contains only one prime 2. are odd, that we V(r) Proof. Since r> 1, if r is not a power of 2, then r has an odd prime divisor p, By (1), we get V(n,r)=(n!Y+ 1 =«n!)'Ip+ 1)«n!y(P-l)4J_(n!),(P-2)1p+... _(n!YP+ 1), 230 Where rip is a poSItIve integer. Notice that if n> 1, then (n!)r/P+l> 1 and (n!)rUJ-1)-P _... +l>1. Therefore, we see from (2) that if n> 1, then V (n, r) IS not a prime. Thus, the sequence V(r) cotains only one pnme V(1,r) 2. The theorem is proved

Year: 1998
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