On powers in shifted products
نویسندگان
چکیده
منابع مشابه
On shifted products which are powers
Fermat gave the first example of a set of four positive integers {a1, a2, a3, a4} with the property that aiaj + 1 is a square for 1 ≤ i < j ≤ 4. His example was {1, 3, 8, 120}. Baker and Davenport [1] proved that the example could not be extended to a set of 5 positive integers such that the product of any two of them plus one is a square. Kangasabapathy and Ponnudurai [6], Sansone [9] and Grin...
متن کاملShifted products that are coprime pure powers
A set A of positive integers is called a coprime Diophantine powerset if the shifted product ab + 1 of two different elements a and b of A is always a pure power, and the occurring pure powers are all coprime. We prove that each coprime Diophantine powerset A ⊂ {1, . . . , N} has |A| 8000 log N/ log log N for sufficiently large N. The proof combines results from extremal graph theory with numbe...
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Let N be a positive integer and let A be a subset of {1, . . . , N} with the property that aa′ + 1 is a pure power whenever a and a′ are distinct elements of A. We prove that |A|, the cardinality of A, is not large. In particular, we show that |A| ≪ (logN)2/3(log logN)1/3.
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We call shifted power a polynomial of the form (x− a)e. The main goal of this paper is to obtain broadly applicable criteria ensuring that the elements of a finite family F of shifted powers are linearly independent or, failing that, to give a lower bound on the dimension of the space of polynomials spanned by F . In particular, we give simple criteria ensuring that the dimension of the span of...
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Let {uk} be a Lucas sequence. A standard technique for determining the perfect powers in the sequence {uk} combines bounds coming from linear forms in logarithms with local information obtained via Frey curves and modularity. The key to this approach is the fact that the equation uk = xn can be translated into a ternary equation of the form ay2 = bx2n + c (with a, b, c ∈ Z) for which Frey curve...
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ژورنال
عنوان ژورنال: Glasnik Matematicki
سال: 2007
ISSN: 0017-095X
DOI: 10.3336/gm.42.2.02