No abstract

No abstract

It is proved that if [Formula: see text] are nonzero real numbers, not all of the same sign and [Formula: see text] is irrational, then for given real numbers [Formula: see text] and [Formula: see text], [Formula: see text], the inequality [Formula: see text] has infinitely many solutions in prime variables [Formula: see text]. This result constitutes an improvement upon that of Liu for the range [Formula: see text].

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Let [Formula: see text] be an integer with [Formula: see text] and [Formula: see text] be any real number. Suppose that [Formula: see text] are nonzero real numbers, not all of the same sign and [Formula: see text] is irrational. It is proved that the inequality [Formula: see text] has infinitely many solutions in prime variables [Formula: see text], where [Formula: see text] for [Formula: see text], and [Formula: see text] for [Formula: see text]. This gives an improvement of the recent result.