Integral distances from (two) lattice points

DOI10.4171/LEM/1026arXiv2103.14932MaRDI QIDQ6363891

Publication date: 27 March 2021

Abstract: {it .}We completely characterize pairs of lattice points

P_{1} e q P_{2}

in the plane with the property that there are infinitely many lattice points

Q

whose distance from both

P_{1}

and

P_{2}

is integral. In particular we show that it suffices that

P_{2} - P_{1} e q (p m 1, p m 2), (p m 2, p m 1)

, and we show that

| P_{1} - P_{2} | > s q r t 20

suffices for having infinitely many such

Q

outside any finite union of lines. We use only elementary arguments, the crucial ingredient being a theorem of Gauss which does not appear to be often applied. We further include related remarks (and open questions), also for distances from an arbitrary prescribed finite set of lattice points %

P_{1}, l d o t s, P_{r}

. }

Mathematics Subject Classification ID

Quadratic and bilinear Diophantine equations (11D09)

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