Leonardo P. Fibonacci

 

Quick Info

Born: 1170, Pisa, Republic of Pisa.

Died:  1250 (aged 79-80), Pisa, Republic of Pisa.

Introduction

Leonardo Fibonacci, also known as Leonardo of Pisa, was an Italian mathematician born around 1170 and died 1250. He is best known for introducing the Hindu-Arabic numeral system to Europe through his book “Liber Abaci“ (The book of calculation), which was published in 1202. Here are some key points about Leonardo Fibonacci and his contributions:

Introduction of Hindu-Arabic Numerals :

Fibonacci most significant contribution was the introduction of the Hindu-Arabic numeral system with ten digits including a zero and positional notation to the Western world. Before this, Roman numerals were predominatly used in Europe. Fibonacci’s book “Liber Abaci“ explained the advantages of the decimal system with its place value notation, which greatly simplified arithmetic operations.

Fibonacci Sequence:

Fibonacci is also associated with the Fibonacci sequence is the sum of the previous two numbers. Fibonacci omitted the “0“ and the first “1“ include todayand began the sequence with 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,...

The Fibonacci numbers may be defined by the recurrence relation:

\begin{equation} F_{n}=F_{n-1}+F_{n-2} \end{equation}

\begin{equation} with\ F_{0}=0\ and\ F_{1}=F_{2}=1. \end{equation}

Fibonacci Numbers in Nature:

The Fibonacci sequence and the related golden ratio (the limit oft he ratios of consecutive Fibonacci numbers). This ratio is found in various aspects of art, architecture and nature, such as in the arrangement of leaves on a stem, the spirals of a pinecone, or the seeds in a sunflower.

The value of the golden ratio is

\begin{equation} \varphi =\frac{1+\sqrt{5}}{2}=1.61803398874... \end{equation}

Binet's Formula for the Fibonacci numbers

\begin{equation} F_{n}=\frac{\left ( \frac{1+\sqrt{5}}{2} \right )^{n}-\left ( \frac{1-\sqrt{5}}{2} \right )^{n}}{\sqrt{5}} \end{equation}

Generating Function

\begin{equation} \sum_{n=0}^{\infty }F_{n}x^{n}=\frac{x}{1-x-x^{2}} \end{equation}

Leonhard Euler

Quick Info

Born: 15 April 1707, Basel, Switzerland.

Died:  18 September 1783, St Petersburg, Russia.

Introduction

Leonhard Euler (1707-1783) was a Swiss mathematician and physicist who made significant contributions to various branches of mathematics and physics. Euler is widely regarder as one of the greatest mathematicians of all time.

Here are some key aspects of Euler’s work:

• Inventing functions for sine, cosine and tangent.

• Introducing and conceptualizing functions for mathematical and complex analysis.

• Discovering the relationships  between trigonometric functions and exponential functions.

• Creating an entirely new branch of mathematics, analytic Number Theory.

• Establishing and proving a product (known as Euler's product) that is the Riemann Zeta Function.

• First formulating the law of quadratic reciprocity in number theory.

• And more...

Constants and Theorems named after Euler

Constants

• Euler’s Number: 

\begin{equation}  e=2.718281828459... \end{equation}

• Euler-Mascheroni Constant:

\begin{equation}  \gamma =0.577215664901... \end{equation}

• Euler-Gompertz constant:

 \begin{equation} \delta =0.596347362323... \end{equation}

Theorems

• Euler’s theorem : If a and n are coprime positive integers, then

\begin{equation} a^{\phi \left ( n \right )}\equiv 1 \left ( modn \right ) \end{equation}

• Euler's partition theorem, used in combinatorics and numbers theory, states that the amount of partitions of an integer n into odd parts is equal to the numbers  of partitions of n  into distinct parts.

• The Euclid-Euler theorem, which connects Mersenne primes to perfect numbers, states that an even number is perfect if can be written as:

\begin{equation} 2^{p-1}\left ( 2^{p}-1 \right ) \end{equation}

Formulas and identities

• Euler's Formula:

\begin{equation} e^{i\theta }=cos\left ( \theta  \right )+isin\left ( \theta  \right ) \end{equation}

With a special case known as Euler's Identity:

\begin{equation} e^{i\pi  }+1=0 \end{equation}

• Euler’s reflection formula:

 \begin{equation} \Gamma \left ( z \right )\Gamma \left ( 1-z \right )=\frac{\pi }{sin\left ( \pi z \right )} \end{equation}

• Euler’s polyhedron formula:

\begin{equation} V-E+F=2 \end{equation}

• Euler's Totient functions Phi is a method for determining the amount of relatively prime numbers.

Wolstenholme's theorem

 Joseph Wolstenholme

Quick Info

Born: 30 September 1829, Eccles near salford, Lancashire, England.

Died: 18 November 1891.

Known for: Wolstenholme primes

                       Wolstenholme's theorem

                       Wolstenholme numbers

Introduction

 In 1862, J. Wolstenholme proved that for any prime p>3 the numerator of the fraction

\begin{equation} \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1} \end{equation}

written in reduced form is divisible by  p^{2}  and that the numerator of the fraction

\begin{equation} \frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+....+\frac{1}{\left ( p-1 \right )^{2}} \end{equation}

written in reduced form is divisible by p.

The first of the above congruences, the so-called Wolstenholme’s theorem, is a fundamental congruence in Combinatorial Number Theory.

Wolstenholme numbers is the numerator of sum

\begin{equation} \sum_{k=1}^{n}\frac{1}{k^{2}} \end{equation}

The first few terms of the Wolstenholme's numbers: 1, 5, 49, 205, 5269, 5369, 266681, 1077749, 9778141, 1968329,... (See A007406).

References

1   Granville, Andrew (1997), "Binomial coefficients modulo prime powers" , Canadian Mathematical Society Conference Proceedings, 20: 253–275

2   McIntosh, R. J.; Roettger, E. L. (2007), "A search for Fibonacci−Wieferich and Wolstenholme primes", Mathematics of Computation, 76 (260)

 

Wilson Quotient

 

John Wilson

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Born: 6 August 1741, Applethwaite, Westmorland, England.

Died: 18 October 1793 (aged 52) Kendal, Westmorland, England.

Known for: Wilson Quotient

                        Wilson primes

                      Wilson's theorem

Introduction

The Wilson Quotient

\begin{equation}  W\left ( {\color{Red} p} \right )=\frac{\left ({\color{Red} p\textbf{}}-1  \right )!+1}{{\color{Red} p}}\end{equation}

If p is a prime number, the quotient is an integer.

if p is composite, the quotient is not an integer.

A Wilson prime is a prime p that divides its Wilson quotient w(p) (see A007619). The known Wilson primes are 5, 13, 563 (see A007540).

References

• 1 R. Crandall & C. Pomerance, Prime Numbers: A Computational Perspective. New York: Springer (2001): 29.

Pascal's Triangle

Blaise Pascal

Quick Info

Born: 19 June 1623, Clermont-Ferrand, Auvergne, France.

Died:  19 August 1662 (aged 39), Paris, France.

Known for:  Pascal’s triangle

                     Pascal’s law

                     Pascal’s rule

                     Pascal’s theorem

                     Pascal’s calculator

                     Pascale’s wager

                     Probability theory

                     Pascal distribution

Introduction

One of the most interesting Numbers patterns is Pascal's triangle (named After Blaise Pascal, a famous Mathematician).
A Formula for any entry in the triangle.

In fact there is a Formula from combinations for working out the value at any place in Pascal's triangle:

It is commonly called "n choose k'' and written like this:

References

1.  Coolidge, J. L. (1949), "The story of the binomial theorem", The American Mathematical Monthly, 56 (3): 147–157, doi:10.2307/2305028, JSTOR 2305028, MR 0028222.

Erdos-Straus Conjecture

Paul Erdős

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Born: 26 March 1913, Budapest, Austria-Hungary.

Died:  20 September 1996 (aged 83), Warsaw, Poland.

Ernst G. Straus

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Born: 25 February 1922 , Munich, Germany.

Died:  12 July 1983 (aged 61), Los Angeles, California, U.S.

Introduction: Unsolved problem in mathematics

The Erdös-Straus conjecture states that the equation 4n=1x+1y+1z has positive integer solutions x,y,z for every postive integers n that is 2 or more.n≥

One important topic in number theory is the study of Diophantine equations, equations in which only integer solutions are permitted.

Example.

For n=2, \begin{equation} \frac{4}{2}=\frac{1}{2}+\frac{1}{2}+\frac{1}{1} \end{equation}

For n=3, \begin{equation} \frac{4}{3}=\frac{1}{1}+\frac{1}{4}+\frac{1}{12} \end{equation}

For n=4, \begin{equation} \frac{4}{4}=\frac{1}{2}+\frac{1}{3}+\frac{1}{6} \end{equation}

For n=5, \begin{equation} \frac{4}{5}=\frac{1}{2}+\frac{1}{4}+\frac{1}{20} \end{equation}

References

1  Bernstein, Leon (1962), "Zur Lösung der diophantischen Gleichung ${\displaystyle �/�=1/�+1/�+1/�}$, insbesondere im Fall ${\displaystyle �=4}$", Journal für die Reine und Angewandte Mathematik (in German), 211: 1–10, doi:10.1515/crll.1962.211.1, MR 0142508, S2CID 118098315

2  Kotsireas, Ilias (1999), "The Erdős-Straus conjecture on Egyptian fractions", Paul Erdős and his mathematics (Budapest, 1999), Budapest: János Bolyai Math. Soc., pp. 140–144, MR 1901903

Frullani's integral

Giuliano Frullani

Quick Info

Born: 1795, Italian.

Died:  1834

Introduction

The Italian mathematician G. Frullani, 1795-1834, reported to G. A. Plana, 1781-1864, the formula

\begin{equation}  \int_{0}^{\infty }\frac{f\left ( ax \right )-f\left ( bx \right )}{x}dx=f\left ( 0 \right )log\left ( \frac{a}{b} \right )\end{equation}

In 1823 and 1827, Cauchy gave a satisfactory proof of the formula 

\begin{equation} \int_{0}^{\infty }\frac{f\left ( ax \right )-f\left ( bx \right )}{x}dx=\left [ f\left ( \infty  \right )-f\left ( 0 \right ) \right ]log\left ( \frac{a}{b} \right ) \end{equation}

We look at a nice Integral identity that alows us to evaluate an entire family of integrals.

which is called Frullani’s integral, after the Italian mathematician Giuliano Frullani (1795–1834).

For example:

Solution:

References

1  Bravo, Sergio; Gonzalez, Ivan; Kohl, Karen; Moll, Victor Hugo (21 January 2017). "Integrals of Frullani type and the method of brackets". Open Mathematics. 15 (1). doi:10.1515/math-2017-0001