58 research outputs found

    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

    Arithmetic progressions in binary quadratic forms and norm forms

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    We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms, this improves significantly upon an earlier result of Dey and Thangadurai.Comment: 7 pages; minor revision; to appear in BLM
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