Dihedral Quintic Fields with a Power Basis

It is shown that there exist infinitely many dihedral quintic fields with a power basis.</p

The Prime Ideal Factorization of 2 in Pure Quartic Fields with Index 2

The prime ideal decomposition of 2 in a pure quartic field with field index 2 is determined explicitly.</p

Class Numbers and Biquadratic Reciprocity

The research of the first author was supported by Natural Sciences and Engineering Research Council Canada Grant No. A-7233, while that of the second was supported by a Natural Sciences and Engineering Research Council Canada Undergraduate Summer Research Award.https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/class-numbers-and-biquadratic-reciprocity/00FECEE10C8CB10943F2ED8E7AAE2B8

A divisibility property of binomial coefficients viewed as an elementary sieve

The triangular array of binomial coefficients 012301111212131331… is said to have undergone a j-shift if the r-th row of the triangle is shifted rj units to the right (r=0,1,2,…). Mann and Shanks have proved that in a 2-shifted array a column number c>1 is prime if and only if every entry in the c-th column is divisible by its row number. Extensions of this result to j-shifted arrays where j>2 are considered in this paper. Moreover, an analog of the criterion of Mann and Shanks [2] is given which is valid for arbitrary arithmetic progressions

Arithmetic Properties of the Ternary Quadratic Form 3x^2+6y^2+14z^2+4yz+2zx+2xy

In this paper we study the arithmetic properties of the ternary quadratic form 3x^2+6y^2+14z^2+4yz+2zx+2xy. This is the ternary quadratic form of least discriminant, which is conjectured, but has not been proven to be regular

Infinite product representations of some q-series

For integers $a$ and $b$ (not both $0$) we define the integers $c(a,b;n)\ \ (n=0,1,2,\ldots)$ by \sum_{n=0}^{infty} c(a,b,;n)q^n = \prod_{n=1}^\infty \left(1-q^n\right)^a (1-q^{2n})^b \quad (|q|<1). These integers include the numbers $t_k(n) = c(-k,2k;n)$, which count the number of representations of $n$ as a sum of $k$ triangular numbers, and the numbers $(-1)^n r_k(n) = c(2k,-k;n)$, where $r_k(n)$ counts the number of representations of $n$ as a sum of $k$ squares. A computer search was carried out for integers $a$ and $b$, satisfying $-24\leq a,b\leq 24$, such that at least one of the sums \begin{align} \sum_{n=0}^{infty} c(a,b;3n+j)q^n, \quad j=0,1,2, \end{align} (0.1) is either zero or can be expressed as a nonzero constant multiple of the product of a power of $q$ and a single infinite product of factors involving powers of $1-q^{rn}$ with $r\in\{1,2,3,4,6,8,12,24\}$ for all powers of $q$ up to $q^{1000}$. A total of 84 such candidate identities involving 56 pairs of integers $(a,b)$ all satisfying a\equiv b\pmd3 were found and proved in a uniform manner. The proof of these identities is extended to establish general formulas for the sums (0.1). These formulas are used to determine formulas for the sums \[\sum_{n=0}^{infty} t_k(3n+j)q^n, \quad \sum_{n=0}^{infty} r_k(3n+j)q^n, \quad j=0,1,2. \

Leg Injuries To Coyotes Captured In Standard And Modified Soft Catch® Traps

Leg injuries of coyotes (Canis latrans) captured in standard No. 3 Soft Catch traps were compared with those captured in the same trap type modified with two additional coil springs. One hundred thirteen coyotes were trapped in southern California in conjunction with livestock predator control operations, 53 in standard traps, and 60 in modified traps. Observed injuries were similar in both trap types. The most frequent injuries were edematous hemorrhages and small cutaneous lacerations. Injuries, such as joint luxations and bone fractures, were noted more frequently for coyotes trapped in standard Soft Catch traps

On the Common Index Divisors of a Dihedral Field of Prime Degree

A criterion for a prime to be a common index divisor of a dihedral field of prime degree is given. This criterion is used to determine the index of families of dihedral fields of degrees 5 and 7