8,400 research outputs found

    On sets of integers whose shifted products are powers

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    AbstractLet N be a positive integer and let A be a subset of {1,…,N} with the property that aaβ€²+1 is a pure power whenever a and aβ€² are distinct elements of A. We prove that |A|, the cardinality of A, is not large. In particular, we show that |A|β‰ͺ(logN)2/3(loglogN)1/3

    Shifted distinct-part partition identities in arithmetic progressions

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    The partition function p(n)p(n), which counts the number of partitions of a positive integer nn, is widely studied. Here, we study partition functions pS(n)p_S(n) that count partitions of nn into distinct parts satisfying certain congruence conditions. A shifted partition identity is an identity of the form pS1(nβˆ’H)=pS2(n)p_{S_1}(n-H) = p_{S_2}(n) for all nn 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

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    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

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    Recent results of Bourgain and Shparlinski imply that for almost all primes pp there is a multiple mpmp that can be written in binary as mp=1+2m1+β‹―+2mk,1≀m1<β‹―<mk,mp= 1+2^{m_1}+ \cdots +2^{m_k}, \quad 1\leq m_1 < \cdots < m_k, with k=66k=66 or k=16k=16, respectively. We show that k=6k=6 (corresponding to Hamming weight 77) suffices. We also prove there are infinitely many primes pp with a multiplicative subgroup A=βŠ‚Fpβˆ—A=\subset \mathbb{F}_p^*, for some g∈{2,3,5}g \in \{2,3,5\}, of size ∣Aβˆ£β‰«p/(log⁑p)3|A|\gg p/(\log p)^3, where the sum-product set Aβ‹…A+Aβ‹…AA\cdot A+ A\cdot A does not cover Fp\mathbb{F}_p completely
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