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

Quick Info

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

Quick Info

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

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

Ernst G. Straus

Quick Info

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

Pick's Theorem

Georg Alexander Pick

Quick Info

Born: 10 August 1859, Vienna, Austria-Hungary.

Died:  26 July 1942 (aged 82), Czechoslovakia.

Known for: Pick’s Theorem

Introduction

Pick’s Theorem provides a simple formula for the area of any lattice polygon. A lattice polygon is a simple polygon embedded on a grid, or lattice, whose vertices have integer coordinates, otherwise known as grid or lattice points. Given a lattice polygon P, the formula involves simply adding the number of lattice points on the boundary, b, dividing b by 2, and adding the number of lattice points in the interior of the polygon, i, and subtracting 1 from i. Then the area of P is:

\begin{equation} {\color{Blue} Area}={\color{Red} i}+\frac{{\color{Green} b}}{2}-1 \end{equation}

The theorem was first stated by Georg Alexander Pick, an Austrian mathematician, in 1899. However, it was not popularized until Polish mathematician Hugo Steinhaus published it in 1969, citing  Pick. Georg Pick was born in Vienna in 1859 and attended the University of Vienna when he was just 16, publishing his first mathematical paper at only 17.

References

1.  Kiradjiev, Kristian (October 2018). "Connecting the dots with Pick's theorem" (PDF). Mathematics Today.       pp. 212–214.