Showing papers on "Coprime integers published in 1969"
01 Jan 1969
TL;DR: In this article, a finite continued fraction (FFLF) representation of a natural integer with respect to a given natural integer N and a non-natural integer N can be found.
Abstract: Many years ago Dr. J. Gillis asked me the following question: Let N and a be coprime natural integers, 1 ≦ a < N, so that a/N can be represented by a finite continued fraction
$$ a/N = 1/c_1 + 1/c_2 + \cdots + 1/c_{n(a)} $$
where the c i are natural integers depending on N and a, and where c n(a) > 1 (to make the representation unique).
91 citations
45 citations
TL;DR: In this paper, it was shown that whenever m, n are coprime, each subvariety of the abelian-by-nilpotent variety has a finite basis for its laws.
Abstract: We show that, whenever m , n are coprime, each subvariety of the abelian-by-nilpotent variety has a finite basis for its laws. We further Show that the just non-Cross subvarieties of are precisely those already known.
22 citations
TL;DR: In this paper, it was shown that d(A, f) is relatively prime to f(b1, b2,…, bn) when f is primitive and when b1 is a polynomial with rational integral coefficients.
6 citations
01 Feb 1969
TL;DR: The result on locally finite groups depends mainly on a result of Amitsur (see [I]). A complete discussion of the following notation can be found in his paper as discussed by the authors, where a group G has property E if it can be embedded in a sfield D and property EE if every automorphism of G can be extended to be an automorphsim of D.
Abstract: NOTATION AND DEFINITIONS. A group G has property E if it can be embedded in a sfield D and property EE if every automorphism of the group G can be extended to be an automorphsim of D. For a more complete discussion see [5 ]. The result on locally finite groups depends mainly on a result of Amitsur (see [I]). A complete discussion of the following notation can be found in his paper. 7r will denote the set of all primes and ri the set of all odd primes p such that 2 has odd order mod p. Let m and r be relatively prime integers. Put s = (r -1, m), t = m/s and n = minimal integer satisfying rn= 1 mod m. Denote by Gm,r a group generated by two elements A and B satisfying Am=1, Bn=At and BAB-'= A. Denote by G., a group G which has a countable ascending tower of subgroups Hi: O < i< oo } such that G = U1 Hi and each Hi is isomorphic to Gmi,ri. T*, 0*, 1* will denote the binary tetrahedral, octahedral and icosahedral groups. Let p be a fixed prime dividing m. a= aP is the highest power of p dividing m. 7, is the minimal integer satisfying rap= 1 mod (Mp-a). AuP is the minimal integer satisfying r/,P=p'2 mod (Mp-a) for some integer ,u. bp is the minimal integer such that ppr_ 1 mod (mp-a).
4 citations
TL;DR: In this article, it was proved that if the rank of the equation over the field does not exceed unity, and if is not divisible by any fourth power and is relatively prime to the discriminant, then, provided that is sufficiently large relative to, the equation does not have more than three positive integer solutions.
Abstract: In this paper it is proved that if the rank of the equation over the field does not exceed unity, and if is not divisible by any fourth power and is relatively prime to the discriminant, then, provided that is sufficiently large relative to , the equation does not have more than three positive integer solutions.Bibliography 10 items.