Using high-resolution echelle spectra obtained with Magellan/MIKE, we present a chemical abundance analysis of both stars in the planet-hosting wide binary system HD20782 + HD20781. Both stars are G dwarfs, and presumably coeval, forming in the same molecular cloud. Therefore we expect that they should possess the same bulk metallicities. Furthermore, both stars also host giant planets on eccentric orbits with pericenters $lesssim 0.2,$ AU. We investigate if planets with such orbits could lead to the host stars ingesting material, which in turn may leave similar chemical imprints in their atmospheric abundances. We derived abundances of 15 elements spanning a range of condensation temperatures ($T_{C}approx 40-1660,$ K). The two stars are found to have a mean element-to-element abundance difference of $0.04pm0.07,$ dex, which is consistent with both stars having identical bulk metallicities. In addition, for both stars, the refractory elements ($T_{C} > 900,$ K) exhibit a positive correlation between abundance (relative to solar) and $T_{C}$, with similar slopes of $approx$ $1times10^{-4},$ dex K$^{-1}$. The measured positive correlations are not perfect; both stars exhibit a scatter of $approx$ $5times10^{-5},$ dex K$^{-1}$ about the mean trend, and certain elements (Na, Al, Sc) are similarly deviant in both stars. These findings are discussed in the context of models for giant planet migration that predict the accretion of H-depleted rocky material by the host star. We show that a simple simulation of a solar-type star accreting material with Earth-like composition predicts a positive---but imperfect---correlation between refractory elemental abundances and $T_{C}$. Our measured slopes for HD 20782/81 are consistent with what is predicted for the ingestion of 10--20 Earths by both stars.
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