scholarly journals Arithmetic Table as an Integral Part of all Computational Mathematics

2020 ◽  
Vol 6 (6) ◽  
pp. 31-41
Author(s):  
V. Shcherban
Keyword(s):  
Mathematical Problem ◽  
Prime Numbers ◽  
Symmetric Polynomial ◽  
Computational Mathematics ◽  
Arithmetic Properties ◽  
Pascal’S Triangle ◽  
Numerical Table ◽  
Cryptographic System ◽  
Pascal's Triangle ◽  
First Time

The paper is devoted to studying the following issue as a statement. What do we know and what we don’t know about arithmetic tables. Perhaps there is no mathematical problem as naive or simple as finding a method for creating arithmetic tables. We confirm that the general method has not been found yet. This study provides nonterminal solution to this problem. Why? The presentation of arithmetic material in essence, plus some accompanying ideas, makes it possible to develop them further in the system. Materials and methods. The system looks like this: a numerical table as a Pascal's triangle and a symmetric polynomial in two or three variables. Some arithmetic properties of such tables will be found, studied and proved. All this was made possible only after successful decryption of the entire class of numeric tables of truncated triangles in the cryptographic system. Results. For example, the arithmetic properties of truncated Pascal’s triangle for finding all prime numbers have been found and presented, and then their formulas have been placed. In addition to elementary addition and subtraction tables, unlimited “comparison” tables of numbers are given and presented for the first time. Conclusions. For computer implementation of the objectives set, the rules of real actions that should exist for tables have been laid down. Only recurrent numeric series should be used for this purpose.

A Pattern: Pascal's Triangle and Prime Numbers

1975 ◽  
Vol 68 (1) ◽  
pp. 23-26
Author(s):  
David R. Duncan ◽  
Bonnie H. Litwiller
Keyword(s):  
Prime Numbers ◽  
Rich Source ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

Pascal's triangle (fig. 1) has been a rich source of patterns in mathematics. New patterns may be found if time is taken to search for them. All that is necessary is to ask the question, “What happens if …?” We asked ourselves the question, “What happens if Pascal's triangle is rewritten mod n? Are there any interesting patterns that may be observed?” The purpose of this article is to report our discussion.

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

Neural Correlates of Mathematical Problem Solving

2015 ◽  
Vol 25 (02) ◽  
pp. 1550004 ◽  
Author(s):  
Chun-Ling Lin ◽  
Melody Jung ◽  
Ying Choon Wu ◽  
Hsiao-Ching She ◽  
Tzyy-Ping Jung
Keyword(s):  
Problem Solving ◽  
Spectral Power ◽  
Mathematical Problem ◽  
Neural Correlates ◽  
Mathematical Problem Solving ◽  
Single Digit ◽  
Beta Power ◽  
First Time ◽  
Anterior Posterior ◽  
Arithmetical Problem Solving

This study explores electroencephalography (EEG) brain dynamics associated with mathematical problem solving. EEG and solution latencies (SLs) were recorded as 11 neurologically healthy volunteers worked on intellectually challenging math puzzles that involved combining four single-digit numbers through basic arithmetic operators (addition, subtraction, division, multiplication) to create an arithmetic expression equaling 24. Estimates of EEG spectral power were computed in three frequency bands — θ (4–7 Hz), α (8–13 Hz) and β (14–30 Hz) — over a widely distributed montage of scalp electrode sites. The magnitude of power estimates was found to change in a linear fashion with SLs — that is, relative to a base of power spectrum, theta power increased with longer SLs, while alpha and beta power tended to decrease. Further, the topographic distribution of spectral fluctuations was characterized by more pronounced asymmetries along the left–right and anterior–posterior axes for solutions that involved a longer search phase. These findings reveal for the first time the topography and dynamics of EEG spectral activities important for sustained solution search during arithmetical problem solving.

3057. A Note on Pascal's Triangle

1963 ◽  
Vol 47 (359) ◽  
pp. 57
Author(s):  
Robert Croasdale
Keyword(s):  
Pascal’S Triangle ◽  
Pascal's Triangle

scholarly journals The Fibonacci p-numbers and Pascal’s triangle

2016 ◽  
Vol 3 (1) ◽  
pp. 1264176 ◽  
Author(s):  
Kantaphon Kuhapatanakul ◽  
Lishan Liu
Keyword(s):  
Pascal’S Triangle ◽  
Pascal's Triangle

scholarly journals Arithmetic Triangle

2017 ◽  
Vol 9 (2) ◽  
pp. 100
Author(s):  
Luis Dias Ferreira
Keyword(s):  
Arithmetic Progression ◽  
Modus Operandi ◽  
Interesting Point ◽  
Pascal’S Triangle ◽  
Initial Term ◽  
The Common ◽  
Pascal's Triangle

The product of the first $n$ terms of an arithmetic progression may be developed in a polynomial of $n$ terms. Each one of them presents a coefficient $C_{nk}$ that is independent from the initial term and the common difference of the progression. The most interesting point is that one may construct an "Arithmetic Triangle'', displaying these coefficients, in a similar way one does with Pascal's Triangle. Moreover, some remarkable properties, mainly concerning factorials, characterize the Triangle. Other related `triangles' -- eventually treated as matrices -- also display curious facts, in their linear \emph{modus operandi}, such as successive "descendances''.

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