On the family of $D(4)$-triples {k-2, k+2, 4k^3-4k}

On the family of Diophantine triples { k − 1 , k + 1 , 16 k 3 − 4 k }

It is proven that if k ≥ 2 is an integer and d is a positive integer such that the product of any two distinct elements of the set {k − 1, k + 1, 16k − 4k, d} increased by 1 is a perfect square, then d = 4k or d = 64k−48k+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k − 1, k + 1, c, d} are regular.

On the Sums ∞ K=−∞ (4k + 1)

The sum in the title is a rational multiple of π for all integers n = 2, 3, 4, . . . for which the sum converges absolutely. This is equivalent to a celebrated theorem of Euler. Of the many proofs that have appeared since Euler, a simple one was discovered only recently by Calabi [4]: the sum is written as a definite integral over the unit n-cube, then transformed into the volume of a polytope ...

A 4K × 2 K-Pixel Color Image Pickup System

We have developed an experimental 4K × 2K pixel progressive scan color camera system. This new camera system has a data rate of 297MHz pixel/sec and 60 frame/sec and we are sure that horizontal and vertical limiting resolution of 1500TVL (TV lines) can be achieved on a color monitor. Instead of the previous approach of improving resolution simply by increasing the pixel count in a imager, a nov...

On the sums P∞ k=−∞ (4k + 1)−n

On the irreducible characters of Camina triples

The Camina triple condition is a generalization of the Camina condition in the theory of finite groups. The irreducible characters of Camina triples have been verified in the some special cases. In this paper, we consider a Camina triple (G,M,N)  and determine the irreducible characters of G in terms of the irreducible characters of M and G/N.