In the quest to understand high-temperature superconductivity in copper oxides, a vigorous debate has been focused on the pseudogap - a partial gap that opens over portions of the Fermi surface in the normal state above the bulk critical temperature ($T_{c}$). The pseudogap has been attributed to precursor superconductivity, to the existence of preformed pairs, or to competing orders such as charge-density waves. A direct determination of the charge of carriers as a function of temperature and bias could help resolve among these alternatives. Here, we report measurements of the shot noise of tunneling current in high-quality La$_{2-x}$Sr$_{x}$CuO$_{4}$/La$_{2}$CuO$_{4}$/La$_{2-x}$Sr$_{x}$CuO$_{4}$ (LSCO/LCO/LSCO) heterostructures fabricated using atomic-layer-by-layer molecular beam epitaxy, for several doping levels. The data delineate three distinct regions in the bias voltage-temperature ($V-T$) space. Well outside the superconducting gap region, the shot noise agrees quantitatively with independent tunneling of charge-e carriers. Deep within the gap, shot noise is greatly enhanced, reminiscent of multiple Andreev reflections. Starting above $T_{c}$ and extending to biases much larger than the gap, there is a broad region in which the noise substantially exceeds the expectations of single-charge tunneling, indicating pairing of carriers. Pairs are detectable deep into the pseudogap region of temperature and bias.
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