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This article presents a vertical multiplication formula for calculating the multiplication of any two multi-digit integers, which may be not only used to design the multiplier but also to the mental multiplication. Our algorithm is a generalization of that of Karatsuba and Offman, but it is superior to the latter because the latter is not suitable for oral calculation of multiplication of large integers. These two algorithms are both the vertical multiplication method, but in our algorithm the horizontal subtraction in stead of the horizontal addition used in the Kartsuba algorithm makes the involved numbers are much smaller and their algebraic sum are much smaller too because the positive and negative numbers probably cancel each other out. Because of the above advantages, our vertical multiplication formula allows more practical and efficient multipliers to be designed., Comment: 15 pages, 4 tables, and 23 references

This paper gives an heuristic lower bound for the number of integers connected to 1 and less than $x$, $\theta(x) > 0.9x,$ in the context of the $3n+1$ problem.

The paper found a geometric and algebraic interpretation of the parameters m and n from the formulas for obtaining primitive Pythagorean triples, which are solutions of the equation ${x^2+y^2=z^2}$, namely: ${x=m^2-n^2}$, ${y=2mn}$, ${z=m^2+n^2}$. The study was based on the process of building figurate numbers using gnomons. The paper discusses the process of building squares. The addition of the gnomon U to the original square leads to a larger square: ${x^2+U=z^2}$. The first stage of the investigation was the construction of the gnomon U, which is equal to the area of a square ${y^2}$. The construction is based on a generating square with a side equal to an even number. The area of the generating square is represented as the sum of the areas of two equal rectangles in all possible ways. At the same time, using the generating square, the gnomon U and the sides of all squares are also constructed: x, y, z. The second stage of the investigation was to obtain a formula for the parameters m and n through the partition elements t and l of the side of the generating square., Comment: 10 pages, 3 figures

This paper gives a simple proof of the Wirsching-Goodwin representation of integers connected to 1 in the $3x+1$ problem (see \cite{Wirsching} and \cite{Goodwin}). This representation permits to compute all the ascending Collatz sequences $(f^{(i)}(n),\: i=1,b-1)$ with a last value $f^{(b)}(n)=1.$ Other periodic sequences connected to $1$ are also identified.

In this work, we define a triangle area number to be the area number of a triangle whose sides have integer lengths, and whose area is a rational number. In Result 3, on page 17, we prove that every triangle area number is in fact an integer which is a multiple of 6. Certain divisibility and other conditions and formulas are also derived, which the three integer sidelengths must satisfy. On pages 20 and 21, we list all the triangle area numbers not exceeding 999., Comment: 22 pages

There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is Pythagorean, the four diameters are integers. Consider a Pythagorean triangle with the property that one leglength is a perfect(or integer)square, and with one of the four diameters also a integer square.Of the eight resulting combinations, we prove that only six are possible or can occur. We then completely parametrically describe the six families; each corresponding to one of the six combinations., Comment: 27 pages