Pythagorean Triples from Harmonic Sequences

2001 ◽  
Vol 94 (3) ◽  
pp. 218-222
Author(s):  
Angelo S. DiDomenico ◽  
Randy J. Tanner
Keyword(s):  
Mathematics Teacher ◽  
Fibonacci Numbers ◽  
Whole Numbers ◽  
Multiplication Tables ◽  
Pythagorean Triples ◽  
Mathematical Experience ◽  
Ancient Times

Pythagorean triples have intrigued generations of mathematics explorers, including students, since ancient times. One of their most charming features is their connection with various other areas of mathematics. In the Mathematics Teacher, for example, authors have shown that Pythagorean triples can be generated from the Fibonacci numbers (Bertucci 1991), from geometric sequences (Carbeau 1993), and from both the addition and multiplication tables of whole numbers (DiDomenico 1993, 1995). These findings are indeed fascinating; when shared with students, they spark interest and curiosity and lead to a truly enriching mathematical experience. Students, in fact, independently found that Pythagorean triples could be generated from Fibonacci numbers and geometric sequences. This article reveals another surprising connection that shows how all primitive Pythagorean triples can be generated from harmonic sequences.

How to give your class the business (math)

1975 ◽  
Vol 22 (4) ◽  
pp. 313-319
Author(s):  
Joseph B. Shapiro
Keyword(s):  
Mathematics Teacher ◽  
Basic Number ◽  
Whole Numbers ◽  
Good Morning ◽  
Number Facts

Good morning, students. My name is Mr. Shapiro. I will be your mathematics teacher this year. During the course we will review the basic number facts of addition, subtraction, multiplication, and division of whole numbers. Then we will work on fractions, decimals, and percents. We will also try to ….”

Teaching concepts incorrectly

1972 ◽  
Vol 19 (2) ◽  
pp. 137-140
Author(s):  
Bill Bompart
Keyword(s):  
Mathematics Teachers ◽  
Mathematics Teacher ◽  
Whole Numbers

A very brief but provocative article by Julia Adkins appeared in the Mathematics Teacher several years ago (1964). It was entitled “Leave the Door Open!” and emphasized the importance of “providing an atmosphere conducive to the development of creativity,” (p. 486) by keeping questions open for later extensions. For instance, to tell students that it is impossible to subtract seven from three is to deny them the opportunity to think beyond the set of whole numbers. It is to be hoped that all mathematics teachers endorse this philosophy and try to “leave the door open” at every opportunity.

Delving Deeper: Fibonacci Identities, Matrices, and Graphs

2005 ◽  
Vol 98 (6) ◽  
pp. 400-403 ◽  
Author(s):  
Danrun Huang
Keyword(s):  
Mathematics Teacher ◽  
Fibonacci Numbers

In “Fibonacci and Related Sequences” published in the “Delving Deeper” department of the Mathematics Teacher February 2004, Richard Askey presents “general strategies used to help discover, prove, and generalize” identities for Fibonacci numbers. He also suggests some problems for further delving, including proving one key identity by using matrices. That proof is the focus of this article.

scholarly journals A SHORT NOTE ON FIBONACCI NUMBERS IN A GENERALIZED PYTHAGOREAN TRIPLES

2021 ◽  
Vol 4 (1) ◽  
pp. 90-95
Author(s):  
Emily L Casinillo ◽  
Leomarich F Casinillo
Keyword(s):  
Fibonacci Numbers ◽  
Short Note ◽  
Pythagorean Triples ◽  
New Formula

This paper aims to construct a new formula that generates a Fibonacci numbers in a generalized Pythagorean triples. In addition, the paper formulates some Fibonacci identities and discuss some important findings.

scholarly journals Triangle of Triangular Numbers

2021 ◽  
Vol 09 (10) ◽  
Author(s):  
Dr. R. Sivaraman ◽  
Keyword(s):  
Whole Numbers ◽  
Pythagorean Triples ◽  
Pascal’S Triangle ◽  
Mathematical Properties ◽  
Triangular Numbers ◽  
Pascal's Triangle

Among several interesting number triangles that exist in mathematics, Pascal’s triangle is one of the best triangle possessing rich mathematical properties. In this paper, I will introduce a number triangle containing triangular numbers arranged in particular fashion. Using this number triangle, I had proved five interesting theorems which help us to generate Pythagorean triples as well as establish bijection between whole numbers and set of all integers.

scholarly journals WUHAN COVID-19 SYNTHETIC ORIGINS AND EVOLUTION

2020 ◽  
Vol 8 (2) ◽  
pp. 285-324 ◽  
Author(s):  
Jean-Claude PEREZ
Keyword(s):  
Fibonacci Numbers ◽  
Small Region ◽  
Adaptation Strategy ◽  
Formal Proof ◽  
Global Strategy ◽  
Whole Numbers ◽  
Synthetic Genome ◽  
Entire Sequence ◽  
Different Strains ◽  
Artificial Origin

The main result of this updated release is the formal proof that 2019-nCoV coronavirus is partially a SYNTHETIC genome. We proof the CONCENTRATION in a small région of wuhan New genome (300bp) of 3 different régions from HIV1 ENVELOPPE gene and 3 others from HIV2 and SIV (ENV and POL RT). All this is remarkable and bears the mark of a desire for organization of a human nature: LOGIC, SYMETRIES. In this article, we demonstrate also that there is a kind of global human hosts adaptation strategy of SARS viruses as well as a strategy of global evolution of the genomes of the different strains of SARS which have emerged, mainly in China, between years 2003 first SARS genomes and the last 2019 COVID-19 Wuhan seafood market pneumonia virus isolate Wuhan-Hu-1, complete genome. This global strategy, this temporal link, is materialized in our demonstration by highlighting stationary numerical waves controlling the entire sequence of their genomes. Curiously, these digital waves characterizing the 9 SARS genomes studied here are characteristic whole numbers: the "Fibonacci numbers", omnipresent in the forms of Nature, and which our research for several decades has shown strong links with the proportions of nucleotides in DNA. Here we demonstrate that the complexity and fractal multiplicity of these Fibonacci numerical waves increases over the years of the emergence of new SARS strains. We suggest that this increase in the overall organization of the SARS genomes over the years reflects a better adaptation of SARS genomes to the human host. The question of a link with pathogenicity remains open. However, we believe that this overall strategy for the evolution of the SARS genomes ensures greater unity, consistency and integrity of the genome. Finally, we ask ourselves the question of a possible artificial origin of this genome, in particular because of the presence of fragments of HIV1, HIV2 and SIV retroviruses.

Wuhan nCoV-2019 SARS Coronaviruses Genomics Fractal Metastructures Evolution and Origins

2020 ◽  
Author(s):  
Jean-Claude Perez
Keyword(s):  
Fibonacci Numbers ◽  
Adaptation Strategy ◽  
Formal Proof ◽  
Global Strategy ◽  
Whole Numbers ◽  
Johannes Kepler ◽  
Synthetic Genome ◽  
Entire Sequence ◽  
Different Strains ◽  
Artificial Origin

Wuhan nCoV-2019 SARS Coronaviruses Genomics Fractal Metastructures Evolution and Origins “Where there is matter, there is geometry.” Johannes Kepler Jean-claude PEREZ, PhD Maths § Computer Science Bordeaux University, RETIRED interdisciplinary researcher (IBM Emeritus, IBM European Research Center on Artificial Intelligence), 7 avenue de terre-rouge F33127 Martignas Bordeaux metropole France, phone 33 0781181112 [email protected] ABSTRACT : The main result of this updated release is the formal proof that 2019-nCoV coronavirus is partially a SYNTHETIC genome. We proof the CONCENTRATION in a small région of wuhan New genome of 3 different régions from HIV1 ENVELOPPE GENE. In this article, we demonstrate that there is a kind of global human hosts adaptation strategy of SARS viruses as well as a strategy of global evolution of the genomes of the different strains of SARS which have emerged, mainly in China, between years 2003 first SARS genomes and the last 2020 nCoV-2019 Wuhan seafood market pneumonia virus isolate Wuhan-Hu-1, complete genome. This global strategy, this temporal link, is materialized in our demonstration by highlighting stationary numerical waves controlling the entire sequence of their genomes. Curiously, these digital waves characterizing the 9 SARS genomes studied here are characteristic whole numbers: the "Fibonacci numbers", omnipresent in the forms of Nature, and which our research for several decades has shown strong links with the proportions of nucleotides in DNA. Here we demonstrate that the complexity and fractal multiplicity of these Fibonacci numerical waves increases over the years of the emergence of new sArs strains. We suggest that this increase in the overall organization of the SARS genomes over the years reflects a better adaptation of SARS genomes to the human host. The question of a link with pathogenicity remains open. However, we believe that this overall strategy for the evolution of the SARS genomes ensures greater unity, consistency and integrity of the genome. Finally, we ask ourselves the question of a possible artificial origin of this genome, in particular because of the presence of fragments of HIV1 retrovirus. KEYWORDS : SARS, Wuhan nCoV-2019, Fibonacci numbers, Fractal genome, Numerical stationary periodic waves, HIV1, synthetic genomes.

When Does A Subtraction Algorithm Involve Borrowing?

1982 ◽  
Vol 75 (9) ◽  
pp. 771-775
Author(s):  
John R. Kolb
Keyword(s):  
Mathematics Teacher ◽  
Whole Numbers ◽  
Interesting Article

In the February 1982 issue of The Mathematics Teacher, Irvin Vance wrote an interesting article describing two algorithms for the subtraction of whole numbers. He describes an algorithm presented by Colton (1980) and concludes that it involves borrowing. Vance calls the second algorithm he discusses the residue method, and he claims that it does not involve borrowing. One of Vance's colleagues claims that both algorithms do involve borrowing. Vance ends the article by asking "What do you think—is borrowing involved?”

A group structure on the golden triples

2021 ◽  
Vol 105 (562) ◽  
pp. 87-97
Author(s):  
Howard Sporn
Keyword(s):  
Mathematics Teachers ◽  
Group Structure ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Mathematics Students ◽  
Pythagorean Triples ◽  
And Mathematics

Some favourite topics of both mathematics teachers and mathematics students are the golden ratio, Fibonacci numbers, groups, and Pythagorean triples. The material of this paper involves the first three, and there are some analogies with the fourth.

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