In 1974, Harry Ruderman proposed the following problem in the problem section of this MONTHLY: if m > n ≥ 0 are integers such that 2m − 2n divides 3m − 3n , then show that 2m − 2n divides xm − xn for all natural numbers x . This problem is still open. We prove that there are only finitely many pairs of natural numbers m, n such that 2m − 2n divides 3m − 3n . Since the proof involves the Schmidt...

androgenesis via anther culture or microspore culture is one of the current methods for producing haploid and double haploid plants in maize. to use of this method in maize breeding program it should be able to regenerate enough plantlets. anthers culture usually is carried out indirectly via callus induction and regeneration on at least two different media. in this study a responsive genotype,...

In [7, Proposition 2.9] we further proved that H1(p − 1) ≡ 0 (mod p ) if and only if p divides the numerator of Bp(p−1)−2, which never happens for primes less than 12 million. There is another important equivalent statement of Wolstenholme’s Theorem by using combinatorics. D.F. Bailey [1] generalizes it to the following form. Theorem 1.1. ([1, Theorem 4]) Let n and r be non-negative integers an...

The tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. We determine t2(K(n1, n2, . . . , nk)) of the complete k-partite graph K(n1, n2, . . . , nk). In particular, we prove that t2(K(n,m)) = (m − 2)/2n + 2, whe...

Denote by B the n-dimensional unit ball centred at o. It is known that in every lattice packing of B there is a cylindrical hole of infinite length whenever n ≥ 3. As a counterpart, this note mainly proves the following result: For any fixed , > 0, there exist a periodic point set P (n, ) and a constant c(n, ) such that B + P (n, ) is a packing in R, and the length of the longest segment contai...

Leighton’s columnsort algorithm sorts on an r × s mesh, subject to the restrictions that s is a divisor of r and that r ≥ 2s2 (so that the mesh is tall and thin). We show how to mitigate both of these restrictions. One result is that the requirement that s is a divisor of r is unnecessary; columnsort sorts correctly whether or not s divides r . We present two algorithms that, as long as s is a ...

In [7] we further proved that H1(p−1) ≡ 0 (mod p ) if and only if p divides the numerator of Bp−3, which never happens for primes less than 12 million [2]. There is another important equivalent statement of Wolstenholme’s Theorem by using combinatorics. D.F. Bailey [1] generalizes it to the following form. Theorem 1.1. ([1, Theorem 4]) Let n and r be non-negative integers and p ≥ 5 be a prime. ...

We consider an old problem of Linial and Wilf to determine the structure of graphs which allow the maximum number of q-colorings among graphs with n vertices and m edges. We show that if r divides q then for all sufficiently large n the Turán graph Tr(n) has more q-colorings than any other graph with the same number of vertices and edges. This partially confirms a conjecture of Lazebnik. Our pr...