A note on the least prime in an arithmetic progression
نویسندگان
چکیده
منابع مشابه
On the Least Prime in Certain Arithmetic Progressions
We nd innnitely many pairs of coprime integers, a and q, such that the least prime a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators.
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Let nr(a,q) be the least /--free number in the arithmetic progession a modulo q. Several results are proved that give lower bounds for n,.(a, q), improving on previous results due to Erdös and Warlimont. In addition, a heuristic argument is given, leading to two conjectures that would imply that the results of the paper are close to best possible.
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Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of G if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....
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The rings considered in this article are commutative with identity $1neq 0$. By a proper ideal of a ring $R$, we mean an ideal $I$ of $R$ such that $Ineq R$. We say that a proper ideal $I$ of a ring $R$ is a maximal non-prime ideal if $I$ is not a prime ideal of $R$ but any proper ideal $A$ of $R$ with $ Isubseteq A$ and $Ineq A$ is a prime ideal. That is, among all the proper ideals of $R$,...
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We show that the abc conjecture implies that the number of terms of any arithmetic progression consisting of almost perfect ”inhomogeneous” powers is bounded, moreover, if the exponents of the powers are all ≥ 4, then the number of such progressions is finite. We derive a similar statement unconditionally, provided that the exponents of the terms in the progression are bounded from above.
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1980
ISSN: 0022-314X
DOI: 10.1016/0022-314x(80)90056-6