An awful problem about integers in base four

Skillful writing of an awful research paper

A Editor I have from time to time in this column offered advice to authors on the desirable elements of a good research report. Like contrary children, for some authors such advice seems to vanish like smoke in a wind. So I take here a different approach, based on the idea that some folks have a knack for doing the opposite of what is recommended to them (like contrary children). I present some...

A "Four Integers" Theorem and a "Five Integers" Theorem

The recent exciting results by Bhargava, Conway, Hanke, Kaplansky, Rouse, and Schneeberger concerning the representabiltity of integers by positive integral quadratic forms in any number of variables are presented. These results build on the earlier work of Dickson, Halmos, Ramanujan, and Willerding on quadratic forms. Two results of this type for positive diagonal ternary forms are proved. The...

Some Conjectures About Cyclotomic Integers

Skillful writing of an awful research paper.

A s Editor I have from time to time in this column offered advice to authors on the desirable elements of a good research report. Like contrary children, for some authors such advice seems to vanish like smoke in a wind. So I take here a different approach, based on the idea that some folks have a knack for doing the opposite of what is recommended to them (like contrary children). I present so...

Frobenius Problem and Dead Ends in Integers

Let a and b be positive, relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In...