Glasnik matematički, Vol. 54 No. 1, 2019.
Izvorni znanstveni članak
https://doi.org/10.3336/gm.54.1.05
Diophantine m-tuples with the property D(n)
Riley Becker
; Department of Mathematics, Queen's University, Kingston, Ontario, K7L 3N6, Canada
M. Ram Murty
; Department of Mathematics, Queen's University, Kingston, Ontario, K7L 3N6, Canada
Sažetak
Let n be a non-zero integer. A set of m positive integers { a1,a2,⋯ ,am} such that aiaj+n is a perfect square for all 1≤ i < j≤ m is called a Diophantine m-tuple with the property D(n). In a series of papers, Dujella studied the quantity Mn= sup {|????|: ???? has the property D(n)} and showed for |n|≥ 400 that Mn ≤ 15.476 log |n| and if |n| >10100, then Mn < 9.078 log |n|. We refine his argument to show that Cn≤ 2log |n|+ O(log |n|/(log log |n|)2), where the implied constant is effectively computable and Cn = sup {|???? ∩ [1,n2]|:???? has the property D(n)}. Together with earlier work of Dujella, this implies Mn≤ 2.6071 log |n|+ O(log |n|/ (log log |n|)2), where the implied constant is effectively computable.
Ključne riječi
Diophantine m-tuples; Gallagher's sieve; Vinogradov's inequality
Hrčak ID:
220843
URI
Datum izdavanja:
7.6.2019.
Posjeta: 1.365 *