For an integer and for , define , where denotes the fractional part of the real number . A number of properties of are derived, and then a connection between and the rumor conjecture is established. To form a rumor sequence , first select integers and . Then select an integer , and for let , where the right side is the least non-negative residue of modulo . The rumor sequence conjecture asserts that all such rumor sequences are eventually 0. A condition on is shown to be equivalent to the rumor conjecture.
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