An additive problem over Piatetski–Shapiro primes and almost-primes

Abstract Let $F=x^2+y^2-z^2$, and let $x_0 \in \mathbb{Z}^3$ be a $primitive$ solution to $F(x_0)=0$, e.g., so that its coordinates share no nontrivial divisor. Let $\Gamma \leq \mathrm{SO_F(\mathbb{Z})}$ be a thin subgroup. We consider the resulting thin orbits of Pythagorean triples $x_0 \cdot \Gamma$—specifically which hypotenuses, areas, and products of all three coordinates arise. We produce infinitely many $R$-almost primes in these three cases whenever $\Gamma$ has exponent $\delta_\Gamma>\delta_0(R)$ for explicit $R$, $\delta_0$.

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On almost-primes in arithmetic progressions, III

Proceedings of the Japan Academy ◽

10.3792/pja/1195518371 ◽

1976 ◽

Vol 52 (3) ◽

pp. 116-118 ◽

Cited By ~ 2

Author(s):

Yoichi Motohashi

Keyword(s):

Arithmetic Progressions ◽

Almost Primes ◽

Primes In Arithmetic Progressions

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On the distribution in short intervals of products of a prime and integers from a given set

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004197002284 ◽

1998 ◽

Vol 124 (1) ◽

pp. 1-14

Author(s):

J. KACZOROWSKI ◽

A. PERELLI

Keyword(s):

Number Theory ◽

Classical Problem ◽

Analytic Number Theory ◽

Multiplicative Structure ◽

Short Intervals ◽

Analytic Number ◽

Almost Primes ◽

Prime Factors

A classical problem in analytic number theory is the distribution in short intervals of integers with a prescribed multiplicative structure, such as primes, almost-primes, k-free numbers and others. Recently, partly due to applications to cryptology, much attention has been received by the problem of the distribution in short intervals of integers without large prime factors, see Lenstra–Pila–Pomerance [3] and section 5 of the excellent survey by Hildebrand–Tenenbaum [1].In this paper we deal with the distribution in short intervals of numbers representable as a product of a prime and integers from a given set [Sscr ], defined in terms of cardinality properties. Our results can be regarded as an extension of the above quoted results, and we will provide a comparison with such results by a specialization of the set [Sscr ].

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