Curious congruences for cyclotomic polynomials

Curious Congruences for Fibonacci Numbers

In this paper we establish some sophisticated congruences involving central binomial coefficients and Fibonacci numbers. For example, we show that if p = 2, 5 is a prime then p−1

Lucas-type Congruences for Cyclotomic Ψ-coefficients

Let p be any prime and a be a positive integer. For l, n ∈ {0, 1, . . . } and r ∈ Z, the normalized cyclotomic ψ-coefficient {n r } l,pa := p − ⌊ n−pa−1−lpa pa−1(p−1) ⌋ ∑ k≡r (mod pa) (−1) (n k )( k−r pa l ) is known to be an integer. In this paper, we show that this coefficient behaves like binomial coefficients and satisfies some Lucas-type congruences. This implies that a congruence of Wan i...

Cyclotomic Polynomials

If n is a positive integer, then the n cyclotomic polynomial is defined as the unique monic polynomial having exactly the primitive n roots of unity as its zeros. In this paper we start off by examining some of the properties of cyclotomic polynomials; specifically focusing on their irreducibility and how they relate to primes. After that we explore some applications of these polynomials, inclu...

Some Curious Congruences modulo Primes

Let n be a positive odd integer and let p > n + 1 be a prime. We mainly derive the following congruence: ∑ 0<i1<···<in<p ( i1 3 ) (−1)i1 i1 · · · in ≡ 0 (mod p).

Cyclotomic Polynomials