Products of binomial coefficients and unreduced Farey fractions

Products of Farey Fractions

PRODUCTS OF BINOMIAL COEFFICIENTS MODULO p 2

As usual Z, Q, R and C denote the ring of integers, the rational field, the real field and the complex field respectively. We also let Z = {1, 2, 3, · · · } and C∗ = C \ {0}. For a ∈ Z and n ∈ Z, by (a, n) we mean th greatest common divisor of a and n, if n is odd then the Jacobi symbol ( a n ) is defined in terms of Legendre symbols (see, e.g. [IR]). For x ∈ R, [x] and {x} stand for the integr...

The Correlations of Farey Fractions

We prove that all correlations of the sequence of Farey fractions exist and provide formulas for the correlation measures.

Neighboring Fractions in Farey Subsequences

We present explicit formulas for computation of the neighbors of several elements in Farey subsequences.

Products and Sums Divisible by Central Binomial Coefficients

In this paper we study products and sums divisible by central binomial coefficients. We show that 2(2n+ 1) ( 2n n ) ∣∣∣∣ (6n 3n )( 3n n ) for all n = 1, 2, 3, . . . . Also, for any nonnegative integers k and n we have ( 2k k ) ∣∣∣∣ (4n+ 2k + 2 2n+ k + 1 )( 2n+ k + 1 2k )( 2n− k + 1 n ) and ( 2k k ) ∣∣∣∣ (2n+ 1)(2n n ) Cn+k ( n+ k + 1 2k ) , where Cm denotes the Catalan number 1 m+1 ( 2m m ) = (...