IRRATIONALITY OF LAMBERT SERIES ASSOCIATED WITH A PERIODIC SEQUENCE

Let q be an integer with |q| > 1 and {an}n≥1 be an eventually periodic sequence of rational numbers, not identically zero from some point on. Then the number [Formula: see text] is irrational. In particular, if the periodic sequences [Formula: see text] of rational numbers are linearly independent over ℚ, then so are the following m + 1 numbers: [Formula: see text] This generalizes a result of Erdős who treated the case of m = 1 and [Formula: see text]. The method of proof is based on the original approaches of Chowla and Erdős, together with some results about primes in arithmetic progressions with large moduli of Ahlford, Granville and Pomerance.

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Product of primes in arithmetic progressions

International Journal of Number Theory ◽

10.1142/s1793042120500384 ◽

2019 ◽

Vol 16 (04) ◽

pp. 747-766

Author(s):

Olivier Ramaré ◽

Priyamvad Srivastav ◽

Oriol Serra

Keyword(s):

Natural Number ◽

Arithmetic Progressions ◽

Residue Classes ◽

Primes In Arithmetic Progressions ◽

Number Formula ◽

Special Case

We prove that, for all [Formula: see text] and for all invertible residue classes [Formula: see text] modulo [Formula: see text], there exists a natural number [Formula: see text] that is congruent to [Formula: see text] modulo [Formula: see text] and that is the product of exactly three primes, all of which are below [Formula: see text]. The proof is further supplemented with a self-contained proof of the special case of the Kneser Theorem we use.

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Divisibility of integers obtained from truncated periodic sequences

International Journal of Number Theory ◽

10.1142/s1793042122500129 ◽

2021 ◽

pp. 1-10

Author(s):

Artūras Dubickas ◽

Lukas Jonuška

Keyword(s):

Prime Number ◽

Prime Numbers ◽

Arithmetic Progressions ◽

Periodic Sequences ◽

Finite Set ◽

Number Formula

A finite set of prime numbers [Formula: see text] is called unavoidable with respect to [Formula: see text] if for each [Formula: see text] the sequence of integer parts [Formula: see text], [Formula: see text] contains infinitely many elements divisible by at least one prime number [Formula: see text] from the set [Formula: see text]. It is known that an unavoidable set exists with respect to [Formula: see text] and that it does not exist if [Formula: see text] is an integer such that [Formula: see text] is not square free. In this paper, we show that no finite unavoidable sets exist with respect to [Formula: see text] if [Formula: see text] is a prime number or [Formula: see text] belongs to some explicitly given arithmetic progressions, for instance, [Formula: see text] and [Formula: see text], [Formula: see text]

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Productly linearly independent sequences

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2015.52.3.1315 ◽

2015 ◽

Vol 52 (3) ◽

pp. 350-370

Author(s):

Jaroslav Hančl ◽

Katarína Korčeková ◽

Lukáš Novotný

Keyword(s):

Rational Numbers ◽

Infinite Products ◽

Linearly Independent ◽

New Concepts ◽

Infinite Sequences

We introduce the two new concepts, productly linearly independent sequences and productly irrational sequences. Then we prove a criterion for which certain infinite sequences of rational numbers are productly linearly independent. As a consequence we obtain a criterion for the irrationality of infinite products and a criterion for a sequence to be productly irrational.

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