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Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.
In this paper, we study equations of the form , where is a binary form, homogeneous of degree , which is supposed to be primitive and irreducible, and is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result...
T. Cochrane and R. E. Dressler [CD] proved that the abc-conjecture implies that, for every > 0, the gap between two consecutive numbers A with two exceptions given in Table 2.
Let be an odd integer and be any given real number. We prove that if , , , , are nonzero real numbers, not all of the same sign, and is irrational, then for any real number with , the inequality
has infinitely many solutions in prime variables , where for and for odd integer with . This improves a recent result in W. Ge, T. Wang (2018).
Consider the linear congruence equation for , . Let denote the generalized gcd of and which is the largest with dividing and simultaneously. Let be all positive divisors of . For each , define . K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on . We generalize their result with generalized gcd restrictions on and prove that for the above linear congruence, the number of solutions...
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