A diophantine problem on groups IV

Let P(z1,…, zn) be a polynomial with positive coefficients. For positive x define A classical diophantine problem is to describe the asymptotic behavior of N1(x) as x → ∞. More generally, one can introduce a second polynomial φ satisfying the condition (0.1) Sign φ (m) is constant for all m outside at most a finite subset of ℕn.

Download Full-text

On Dividing Coconuts: A Linear Diophantine Problem

College Mathematics Journal ◽

10.2307/2687525 ◽

1997 ◽

Vol 28 (3) ◽

pp. 203 ◽

Cited By ~ 2

Author(s):

Sahib Singh ◽

Dip Bhattacharya

Keyword(s):

Diophantine Problem ◽

Linear Diophantine Problem

Download Full-text

A remark on spherical equations in free metabelian groups

Groups – Complexity – Cryptology ◽

10.1515/gcc-2017-0012 ◽

2017 ◽

Vol 0 (0) ◽

Author(s):

Evgeny I. Timoshenko

Keyword(s):

Metabelian Groups ◽

Diophantine Problem ◽

Magnus Embedding

AbstractI. Lysenok and A. Ushakov proved that the Diophantine problem for spherical quadric equations in free metabelian groups is solvable. The present paper proves this result by using the Magnus embedding.

Download Full-text

A Diophantine Problem with Unlike Powers of Primes

Journal of Mathematics ◽

10.1155/2021/5528753 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Quanwu Mu ◽

Liyan Xi

Keyword(s):

Real Number ◽

Infinitely Many Solutions ◽

Real Numbers ◽

Diophantine Problem

Let k be an integer with 4 ≤ k ≤ 6 and η be any real number. Suppose that λ 1 , λ 2 , … , λ 5 are nonzero real numbers, not all of them have the same sign, and λ 1 / λ 2 is irrational. It is proved that the inequality λ 1 p 1 + λ 2 p 2 2 + λ 3 p 3 3 + λ 4 p 4 4 + λ 5 p 5 k + η < max 1 ≤ j ≤ 5 p j − σ k has infinitely many solutions in prime variables p 1 , p 2 , p 3 , p 4 , and p 5 , where 0 < σ 4 < 1 / 36 , 0 < σ 5 < 4 / 189 , and 0 < σ 6 < 1 / 54 . This gives an improvement of the recent results.

Download Full-text