8,400 research outputs found
On sets of integers whose shifted products are powers
AbstractLet N be a positive integer and let A be a subset of {1,β¦,N} with the property that aaβ²+1 is a pure power whenever a and aβ² are distinct elements of A. We prove that |A|, the cardinality of A, is not large. In particular, we show that |A|βͺ(logN)2/3(loglogN)1/3
Shifted distinct-part partition identities in arithmetic progressions
The partition function , which counts the number of partitions of a
positive integer , is widely studied. Here, we study partition functions
that count partitions of into distinct parts satisfying certain
congruence conditions. A shifted partition identity is an identity of the form
for all in some arithmetic progression. Several
identities of this type have been discovered, including two infinite families
found by Alladi. In this paper, we use the theory of modular functions to
determine the necessary and sufficient conditions for such an identity to
exist. In addition, for two specific cases, we extend Alladi's theorem to other
arithmetic progressions
Stochastic Models for the 3x+1 and 5x+1 Problems
This paper discusses stochastic models for predicting the long-time behavior
of the trajectories of orbits of the 3x+1 problem and, for comparison, the 5x+1
problem. The stochastic models are rigorously analyzable, and yield heuristic
predictions (conjectures) for the behavior of 3x+1 orbits and 5x+1 orbits.Comment: 68 pages, 9 figures, 4 table
Almost all primes have a multiple of small Hamming weight
Recent results of Bourgain and Shparlinski imply that for almost all primes
there is a multiple that can be written in binary as with or ,
respectively. We show that (corresponding to Hamming weight )
suffices.
We also prove there are infinitely many primes with a multiplicative
subgroup , for some
, of size , where the sum-product set
does not cover completely
- β¦