Infinitely many three-term arithmetic progressions of powerful numbers exist with d = 2√N + 1, with a conjecture that infinitely many are consecutive in the sequence of all powerful numbers.
Distribution modulo one of linear recurrent sequences
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abstract
We study the distribution modulo one of linear recurrent sequences of real numbers. We prove criteria for the finiteness of the set of limit values of the fractional parts of such a sequence and give lower bounds for the maximal distance between two limit values. Our results generalize theorems of Flatto, Lagarias, Pollington, and Dubickas.
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2026 1verdicts
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Three-term arithmetic progressions of consecutive powerful numbers
Infinitely many three-term arithmetic progressions of powerful numbers exist with d = 2√N + 1, with a conjecture that infinitely many are consecutive in the sequence of all powerful numbers.