Binomial Coefficients, Roots of Unity and Powers of Prime Numbers

Let t ∈ N + be given. In this article, we are interested in characterizing those d ∈ N + such that the congruence 1 t ∑ s = 0 t - 1 n + d ζ t s d - 1 ≡ n d - 1 ( mod d ) is true for each n ∈ Z . In particular, assuming that d has a prime divisor greater than t , we show that the above congruence hol...

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Bibliographic Details
Published inBulletin of the Malaysian Mathematical Sciences Society Vol. 45; no. 4; pp. 1489 - 1506
Main Author Miska, Piotr
Format Journal Article
LanguageEnglish
Published Singapore Springer Nature Singapore 01.07.2022
Springer Nature B.V
Subjects
Applications of Mathematics
Binomial coefficients
Mathematics
Mathematics and Statistics
Prime numbers
Root of unity
11A07
11A41
adic valuation
11B65
Binomial coefficient
Prime number
Power
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Summary:Let t ∈ N + be given. In this article, we are interested in characterizing those d ∈ N + such that the congruence 1 t ∑ s = 0 t - 1 n + d ζ t s d - 1 ≡ n d - 1 ( mod d ) is true for each n ∈ Z . In particular, assuming that d has a prime divisor greater than t , we show that the above congruence holds for each n ∈ Z if and only if d = p r , where p is a prime number greater than t and r ∈ { 1 , … , t } .
ISSN:0126-6705
2180-4206
DOI:10.1007/s40840-022-01266-4