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The harmonic numbers [Formula: see text] play important roles in mathematics. Let [Formula: see text] be a prime. With the help of some combinatorial identities, we establish the following two new congruences: [Formula: see text] and [Formula: see text] where [Formula: see text] denotes the Bernoulli polynomial of degree [Formula: see text]. As an application, we determine [Formula: see text] and [Formula: see text] modulo [Formula: see text], where [Formula: see text] with [Formula: see text].
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Mao et al. (2015) studied this question.
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