Some extensions and refinements of a theorem of Sylvester

Abstract

This paper, we show Theorem 1. Let k 6. Then (4) !( 0 ) > 6 5 (k) + 1 except when (n; d; k) 2 f(1; 2; 6); (1; 3; 6); (1; 2; 7); (1; 3; 7); (1; 4; 7); (2; 3; 7); (2; 5; 7); (3; 2; 7); (1; 2; 8); (1; 2; 11); (1; 3; 11); (1; 2; 13); (3; 2; 13); (1; 2; 14)g: We see that Theorem 1 includes (1), (2), (3) with k 6 and the result of Saradha and Shorey stated above. In Theorem 1 and the subsequent results of this paper, we observe that the statements are not valid for the exceptions mentioned therein. As a consequence of Theorem 1, we derive Corollary 1. Let k 3 and (n; d; k) 62 f(1; 3; 3); (2; 3; 3); (2; 7; 3); (1; 5; 4); (3; 5; 4)g: Then Q( 0 ) ( Y p<k gcd (p;d)=1 p)k 2+[ 1 5 (k)] : The assumption on the triples (n; d; k) in Corollary 1 is necessary. We observe that the second factor on the right hand side of the above inequality is > e k=5 . Corollary 1 is related to a generalised version of a problem of Erd os and Woods, see [2]. The inequality (4) of Theorem 1 is a consequence of the following result with t = k. We dene d (k) to be the number of primes k and coprime to d: Theorem 2. Let k 6 and t k 6 5 (k) + d (k) 1. Suppose (5) !() d (k): Then (n; d; k; ) 2 V 2 : Further, we get from Theorem 2 the following result.