Abstract. Our main theorem shows that the regularity of non-degenerate homogeneous prime ideals is not bounded by any polynomial function of the degree; this holds over any field k. In particular, we provide counterexamples to the longstanding Regularity Conjecture, also known as the Eisenbud-Goto Conjecture (1984). We introduce a method which, starting from a homogeneous ideal I, produces a pr...

has non-negative power series coefficients. The purpose of this note is prove the Conjecture. The conjecture has been established for prime values of n by Andrews [1], and for n ≤ 99, using a computer proof (see [2], [4]). The proof given here relies upon an identity for the rational function of the conjecture, which is our Lemma. A similar identity was found by Andrews [1] to establish the cas...

The Alon-Tarsi conjecture states that for even n, the number of even latin squares of order n diiers from the number of odd latin squares of order n. Zappa 6] found a generalization of this conjecture which makes sense for odd orders. In this note we prove this extended Alon-Tarsi conjecture for prime orders p. By results of Drisko 2] and Zappa 6], this implies that both conjectures are true fo...

Let V (ZG) be the normalized unit group of the integral group ring ZG of a finite group G. One of most interesting conjectures in the theory of integral group ring is the conjecture (ZC) of H. Zassenhaus [25], saying that every torsion unit u ∈ V (ZG) is conjugate to an element in G within the rational group algebra QG. For finite simple groups, the main tool of the investigation of the Zassenh...

We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson-Lichtenbaum conjecture, we apply our results to show the vanishing of SK2(A) for a central simple algebra A of square-free index (prime to the characteristic). This pro...

Schmutz Schaller’s conjecture regarding the lengths of the hexagonal versus the lengths of the square lattice is shown to be true. The proof makes use of results from (computational) prime number theory. Using an identity due to Selberg, it is shown that, in principle, the conjecture can be resolved without using computational prime number theory. By our approach, however, this would require a ...

We prove the following conjecture of Narayana: there are no nontrivial dominance refinements of the Smirnov two-sample test if and only if the two sample sizes are relatively prime. We also count the number of natural significance levels of the Smirnov two-sample test in terms of the sample sizes and relate this to the Narayana conjecture. In particular, Smirnov tests with relatively prime samp...

Let E be an elliptic curve over Q. In 1988, N. Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy–Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s Conjecture is still widely open. In this paper we prove that Koblitz’s Conjecture is true on average...

In 1980, Carl Pomerance and J. L. Selfridge proved D. J. Newman’s coprime mapping conjecture: If n is a positive integer and I is a set of n consecutive integers, then there is a bijection f :{1, 2, . . . , n}→ I such that gcd(i, f(i)) = 1 for 1 ≤ i ≤ n. The function f described in their theorem is called a coprime mapping. Around the same time, Roger Entringer conjectured that all trees are pr...

ardy and Littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. They believed that the number $mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically given by begin{equation*} mathcal{R}(n) sim frac{sqrt{n}}{log n}prod_{p=3}^{infty}left(1-frac{1}{p-1}left(frac{n}{p}right)right), end{equation*} where $p$ is a prime, $m$ is a...