DIOPHANTINE APPROXIMATION BY PRIMES
نویسندگان
چکیده
منابع مشابه
Diophantine Approximation by Primes
We show that whenever δ > 0 and constants λi satisfy some necessary conditions, there are infinitely many prime triples p1, p2, p3 satisfying the inequality |λ0 + λ1p1 + λ2p2 + λ3p3| < (max pj)−2/9+δ. The proof uses Davenport–Heilbronn adaption of the circle method together with a vector sieve method. 2000 Mathematics Subject Classification. 11D75, 11N36, 11P32.
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Let λ1, . . . , λs be non-zero with λ1/λ2 irrational and let S be the set of values attained by the form λ1x 3 1 + · · ·+ λsxs when x1 has at most 6 prime divisors and the remaining variables are prime. In the case s = 4, we establish that most real numbers are “close” to an element of S. We then prove that if s = 8, S is dense on the real line.
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Let λ1, . . . , λ4 be non-zero with λ1/λ2 irrational and negative, and let S be the set of values attained by the form λ1x 3 1 + · · · + λ4x4 when x1 has at most 3 prime divisors and the remaining variables are prime. We prove that most real numbers are close to an element of S.
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In this paper, we show that if p is a prime and ifA = {a1, a2, . . . , am} is a set of positive integers with the property that aiaj +p is a perfect square for all 1 ≤ i < j ≤ m, then m < 3 · 2168. More generally, when p is replaced by a squarefree integer n, the inequality m ≤ f(ω(n)) holds with some function f , where ω(n) is the number of prime divisors of n. We also give upper bounds for m ...
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ژورنال
عنوان ژورنال: Glasgow Mathematical Journal
سال: 2009
ISSN: 0017-0895,1469-509X
DOI: 10.1017/s0017089509990176