Largest Prime Number M44 ?

If you seen the Rajinikanth Robort(2010) movie. Then you might remember a seen of the film where Dr. Vasikaran introduce his robot (Chitti) in front of journalist. Where one of the journalist asked about the Largest Prime Number according to robot and here robot answers the "M44 is the largest prime number" Although this movie is fictional and not based on any actual mathematical concept or theory related to prime numbers. But lets see how this M44 comes from and what is the largest known prime number?

Firstly, Let's understand something

Euclid's Theorem, which states that there are infinitely many prime numbers. The proof of Euclid's theorem is a classic proof by contradiction that assumes there is a largest prime number, and then shows that this assumption leads to a contradiction. Therefore, there can be no largest prime number.

So, If there is no largest prime number exists, then what is M44?

Actually, M44 is shorthand for a Mersenne number (NOT a Mersenne Prime!)

Mersenne numbers are numbers of the form

Mn=2n−1${\mathrm{<br>}}_{}$

Not all Mersenne numbers are primes, although some are.
So, M44 is 17592186044415, which by the way, is not a prime number.

If M44 is not the largest prime then what it is?

The largest known prime number as of my knowledge cutoff date (September 2021) is 2^82,589,933 − 1, which is a Mersenne prime discovered by the Great Internet Mersenne Prime Search (GIMPS) project in December 2018.

The number has 24,862,048 digits when written in base 10. However, it's important to note that new prime numbers are constantly being discovered as computational power and algorithms improve. So, while this was the largest known prime number as of my knowledge cutoff, it's possible that an even larger one has been discovered since then.

If you want to know the mathematics behind season change then click here 

Stable Marriage Problem

The Stable Marriage problem, also known as the Stable Matching problem, is a classic problem in the field of computer science and mathematics that deals with finding a stable match between two sets of participants based on their preferences. The problem was first introduced by mathematician David Gale and economist Lloyd Shapley in 1962.

The problem involves a set of n men and n women who are trying to find a suitable partner for marriage. Each individual has a list of preferences, which ranks their potential partners in order of preference. The goal is to find a matching between the men and women such that there are no two individuals who would both prefer each other over their current partner.

The Gale-Shapley algorithm is a mathematical method for solving the Stable Marriage problem. The algorithm involves a series of proposals and rejections between the men and women, which ultimately leads to a stable match. The algorithm works as follows:

1. Each man proposes to his top-ranked woman.

2. Each woman who receives a proposal accepts the proposal from her most preferred man and rejects all other proposals.

3. Each rejected man proposes to his next-ranked woman, and the process continues until each man is either engaged or has proposed to all women.

4. Once all the men are engaged, the algorithm is complete.

This is the most useful video on the Stable Marriage Problem, If you want to know how this algorithm works with an example.

The key insight of the Gale-Shapley algorithm is that each individual acts in their own self-interest by proposing to their most preferred partner. This ensures that each individual is matched with the most preferred partner who is willing to accept them. The algorithm also ensures that once a match is made, it is stable and cannot be disrupted by any pair of individuals who would prefer to be with each other.

The Gale-Shapley algorithm has several important properties. First, it always produces a stable matching, which means that there are no two individuals who would both prefer each other over their current partner. Second, it is efficient and can be implemented in `n^2` time, where n is the number of individuals. This makes the algorithm practical for large-scale applications.

If you want to use this algorithm on a large scale then it not possible to do that using a copy and pen as seen on above video. So, R Software help you with these. You can do this in 2 way:

• Use of hri function of matchingMarkets package. For more information of this library click here
• You also use one2one function of matchingR package. For more information and example of this library click here

For download R software latest version go to https://www.r-project.org/ and click on download R

The Stable Marriage problem and the Gale-Shapley algorithm have many applications beyond the domain of marriage. The problem can be used to match medical students with residency programs, job seekers with employers, and even kidney donors with recipients. The algorithm has also been used in the design of computer networks and in the allocation of resources.

In conclusion, the Stable Marriage problem and the Gale-Shapley algorithm are important concepts in the fields of computer science and mathematics. The problem deals with finding a stable match between two sets of participants based on their preferences, and the algorithm provides a practical and efficient method for solving the problem. The algorithm has many applications beyond the domain of marriage and has the potential to revolutionize the way we match individuals with different resources and opportunities.

Why Should We Admire Aristotle

Who is Aristotle ?

Aristotle was a Greek philosopher who lived from 384-322 BCE. He was a student of Plato and tutored Alexander the Great. Aristotle is considered one of the greatest minds in Western philosophy and science. He made significant contributions to a wide range of fields, including metaphysics, ethics, politics, biology, and aesthetics. His works cover a vast range of subjects, including logic, rhetoric, and metaphysics. He believed in finding a mean between two extremes, known as the "Golden Mean," and this idea has influenced the development of ethics and moral philosophy. Aristotle's works have been highly influential in the development of Western thought and continue to be studied and discussed by philosophers, scholars, and students today.

Aristotle is Wrong

Aristotle made many significant contributions to a wide range of fields, but have you know many theories of Aristotle is proven incorrect. Some of the famous theories are:-

1. The geocentric universe: Aristotle believed that the Earth was the center of the universe and that all celestial bodies revolved around it. However, this view was later challenged and disproven by astronomers such as Copernicus and Galileo, who proposed a heliocentric model in which the sun was at the center of the universe.

2. The four elements: Aristotle believed that all matter was made up of four basic elements: earth, air, fire, and water. This theory was later replaced by the atomic theory, which states that matter is made up of atoms.

3. Spontaneous generation: Aristotle believed that certain types of animals and plants could arise spontaneously from non-living matter. This theory was later disproven by the experiments of Francesco Redi and Louis Pasteur, who demonstrated that living organisms only come from pre-existing living organisms.

4. Law of Natural Motion: Aristotle believed that objects have a natural tendency to move toward their natural place. He thought that heavy objects, such as rocks and metal, naturally move downward toward the center of the Earth, while light objects, such as air and fire, naturally move upward.

5. Law of Impressed Force: Aristotle believed that the force that is necessary to keep an object in motion is proportional to the speed of the object. This means that the faster an object is moving, the more force is required to keep it in motion. This idea was later challenged by Galileo and later by Newton, who showed that the laws of motion do not depend on the speed of the object.

Still...

Yes, Aristotle is wrong about many things. Still, he is one of the greatest thinkers and greatest scientist in the history of world. Because he laid the foundation for future scientific discoveries and also you may not find any other person in human history who work on more fields then the Aristotle.

It is important to do more research on the existing theories. But actually start the conversation on a new theory is more difficult. That's why, I admire Aristotle more than anyone.

Comment below your favorite scientist or mathematician.

Two Summer Vacations in a Year ?

Situation:

We might have saw this image in our 3rd or 4th grade science book and with this image a line is mentioned "earth revolve in an elliptical orbit around the sun and sun is at the center of that ellipse." with that you also read "the summer and winter on earth is depend on the distance between the earth and sun"

But if you see the image and the line that written on the book are completely contradictory. As the image says that at two times in a year earth closest and farthest from the sun. That simply means, we experience two times every season in a same year. Which is obviously wrong!

So, if all the things in our book is write then why we have only one summer vacation in a year?

Reality:

In reality, the sun is not at the center of the ellipse. The sun is at the focus of the ellipse and a ellipse has two focus point and a center point. which you see in the image.

So the image on our book is also not correct. The original orbit of earth is like:-

This is one thing about the season change. Along with that earth is tilted on its axis at an angle of about 23.5 degrees, which means that different parts of the planet receive different amounts of sunlight throughout the year. When the Northern Hemisphere is tilted towards the sun, it is summer in the Northern Hemisphere and winter in the Southern Hemisphere. Similarly, when the Southern Hemisphere is tilted towards the sun, it is summer in the Southern Hemisphere and winter in the Northern Hemisphere.

Therefore, because of the way the Earth is tilted and orbit path around the sun, it is not possible for there to be two summer vacations in a year.

If you want to know more about mathematics click here

Hierarchy of Numbers

In mathematics, the hierarchy of numbers refers to the ordering and classification of different types of numbers. There are several different levels in the hierarchy of numbers, including:

Natural Numbers:

The natural numbers are a set of positive integers used for counting and enumeration. They are the numbers 1, 2, 3, 4, 5, and so on. The natural numbers are also known as the counting numbers or positive integers. They play a crucial role in mathematics and are used in a variety of mathematical applications, including number theory, combinatorics, and set theory.

In mathematics, the set of natural numbers is often denoted by the symbol "N". The natural numbers have many properties that make them unique among other number sets. For example, they have a property known as the "well-ordering principle", which states that any non-empty set of natural numbers has a smallest element. Additionally, the natural numbers are the building blocks for the other number sets, such as the integers, rational numbers, and real numbers.

Whole Numbers:

Whole numbers are a subset of the set of integers and include the positive integers (1, 2, 3, 4, etc.) as well as zero. In other words, whole numbers are the numbers that you would use to count a set of objects and include 0.

The set of whole numbers is often denoted by the symbol "W". Whole numbers play a significant role in many mathematical applications, including arithmetic, algebra, and number theory. Additionally, whole numbers are used in computer programming and other technological applications, as they are often used to represent discrete values in a digital system.

Integers:

The integers are the set of whole numbers and their negatives. The set of integers includes positive whole numbers (1, 2, 3, 4, etc.), negative whole numbers (-1, -2, -3, -4, etc.), and zero. The set of integers is often denoted by the symbol "Z".

Integers are used in many mathematical applications, including arithmetic, algebra, and number theory. In addition to the properties shared by the sets of whole numbers and natural numbers, integers also have additional properties related to their positive and negative values. For example, the sum of two positive integers is always positive, while the sum of a positive and negative integer can be positive, negative, or zero.

The set of integers is well-ordered, meaning that it has a first and a last element, and any non-empty subset of the integers has a smallest element. The set of integers also has the property of closure under addition and subtraction, meaning that the sum or difference of any two integers is always an integer.

Rational Numbers:

Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not equal to zero. A fraction is a rational number if and only if both the numerator and the denominator are integers. The set of rational numbers is often denoted by the symbol "Q".

Examples of rational numbers include 1/2, -3/4, and 5. The number zero is also a rational number, as it can be expressed as 0/1.

Rational numbers play a significant role in many areas of mathematics, including algebra, geometry, and number theory. They are used to represent quantities that can be expressed as a ratio of two integers, such as lengths, distances, and angles.

Rational numbers have several properties that make them unique among other number sets. For example, they are closed under addition, subtraction, multiplication, and division (provided that the denominator is not equal to zero). Additionally, the set of rational numbers is dense, meaning that between any two distinct rational numbers, there exists an infinite number of other rational numbers.

In mathematics, rational numbers are considered as "basic" numbers, as they can be used to construct other number sets, such as the real numbers and the complex numbers. The set of rational numbers is a subset of the set of real numbers, but not all real numbers are rational.

Irrational Numbers:

Irrational numbers are real numbers that cannot be expressed as a ratio of two integers, and therefore cannot be expressed as a simple fraction. The decimal representation of an irrational number is non-repeating and non-terminating. Some famous examples of irrational numbers include the `\sqrt2`, `\sqrt3`, and π (pi).

Irrational numbers play an important role in many areas of mathematics, including geometry, calculus, and number theory. They are used to represent quantities that cannot be expressed as a simple fraction, such as the length of a diagonal line segment in a square, the circumference of a circle, and the value of certain trigonometric functions.

The set of irrational numbers is a subset of the set of real numbers and has several important properties. For example, irrational numbers are dense in the set of real numbers, meaning that between any two distinct real numbers, there exists an infinite number of irrational numbers. Additionally, the set of irrational numbers is uncountable, meaning that there is no one-to-one correspondence between the irrational numbers and the positive integers.

Real Numbers:

Real numbers are numbers that represent quantities on the number line, including both rational and irrational numbers. The set of real numbers is often denoted by the symbol "R".

Rational numbers are real numbers that can be expressed as a ratio of two integers, such as 1/2, -3/4, and 5. Irrational numbers are real numbers that cannot be expressed as a ratio of two integers, such as the `\sqrt2`, `\sqrt3`, and π (pi).

Real numbers have several important properties that make them useful in many areas of mathematics, including geometry, calculus, and analysis. For example, the set of real numbers is complete, meaning that any non-empty set of real numbers that has an upper bound also has a least upper bound. Additionally, the set of real numbers is ordered, meaning that for any two real numbers a and b, either a < b, a = b, or a > b.

The set of real numbers is also uncountably infinite, meaning that there is no one-to-one correspondence between the real numbers and the positive integers. This property makes the set of real numbers much larger than the set of rational numbers, and allows for the representation of quantities that cannot be expressed as a simple fraction.

Complex Numbers:

Complex numbers are numbers that consist of a real part and an imaginary part, and are represented in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. Complex numbers are used to represent quantities in the complex plane, which is a two-dimensional graph with the real part represented on the x-axis and the imaginary part represented on the y-axis.

Complex numbers have several important properties and applications in many areas of mathematics, including algebra, calculus, and engineering. For example, they can be used to solve equations that have no real solutions, and they play a significant role in the study of alternating current (AC) electrical circuits.

The set of complex numbers is closed under addition, subtraction, multiplication, and division, meaning that the sum, difference, product, and quotient of two complex numbers is also a complex number. Additionally, complex numbers have a notion of magnitude and phase, which can be represented as a vector in the complex plane.

History of Numbers

 Why we need numbers ?

Numbers are essential to our daily lives and play a crucial role in many aspects of human existence. Some of the reasons why we need numbers include:

• Measurement: Numbers are used to measure quantities, such as length, weight, volume, and time. This allows us to quantify and compare objects, events, and phenomena in a consistent and meaningful way.

• Calculation: Numbers are used to perform mathematical operations, such as addition, subtraction, multiplication, and division. This allows us to make quantitative predictions, solve problems, and make decisions.

• Communication: Numbers are used to communicate information, such as telephone numbers, addresses, and dates. They are also used to represent data and statistics, which help us to make informed decisions and understand the world around us.

• Science and Technology: Numbers play a crucial role in many fields of science, including physics, chemistry, and engineering. They are used to describe natural phenomena and to model complex systems, such as the weather, the economy, and the behavior of living organisms.

• Commerce and Finance: Numbers are used to represent monetary values, to keep track of financial transactions, and to make investment decisions. They are also used to calculate taxes, loans, and other financial obligations.

In short, numbers provide us with a way to quantify, describe, and understand the world, making them an essential tool in many aspects of human life.

History of Numbers

The history of numbers, or numerals, dates back to the earliest civilizations. People have been counting and recording numbers for thousands of years in various forms.

Tally Marking

Bone artifacts bearing incised markings seem to indicate that the people of the Old Stone Age had devised a system of tallying by groups as early as 30,000 B.C. The most impressive example is a shinbone from a young wolf, found in Czechoslovakia in 1937; about 7 inches long, the bone is engraved with 55 deeply cut notches, more or less equal in length, arranged in groups of ve. (Similar recording notations are still used, with the strokes bundled. Voting results in small towns are still counted in the manner devised by our remote ancestors.) For many years such notched bones were interpreted as hunting tallies and the incisions were thought to represent kills. The markings on bones discovered in French cave sites in the late 1880s are grouped in sequences of recurring numbers that agree with the numbers of days included in successive phases of the moon. One might argue that these incised bones represent lunar calendars.

Ancient Egyptians Number System (Hieroglyphic)

The ancient Egyptians used a hieroglyphic numeral system to represent numbers. This system was based on the use of symbols to represent quantities, similar to the way we use numerals today. In the Egyptian numeral system, there were symbols for the numbers 1, 10, 100, and 1,000. To write larger numbers, these symbols were combined in a similar way to the way we use place value in our modern decimal numeral system. For example, the symbol for 100 was used to represent 100, while the symbol for 10 was used to represent 10, 20, 30, etc.

For example:- the number 142,136 = 1*100,000 + 4*10,000 + 2*1,000 + 1*100 + 3*10 + 6*1

is represented in hieroglyphic as

The Egyptian numeral system was a decimal system, meaning that it was based on the number 10. This made it possible for the ancient Egyptians to perform mathematical operations, such as addition, subtraction, multiplication, and division, using the symbols.

Lets see the example of addition

It is worth noting that while the hieroglyphic numeral system was used for mathematical purposes, it was also used to represent quantities in other contexts, such as measuring quantities of grain, recording the number of people in a population, or counting the number of objects in a collection.

Overall, the hieroglyphic numeral system was an important tool for the ancient Egyptians, allowing them to perform calculations, record information, and communicate quantitative information in a consistent and meaningful way.

Greek Alphabetic Numeral System

Around the fifth century B.C., the Greeks of Ionia also developed a ciphered numeral system, but with a more extensive set of symbols to be memorized. They ciphered their numbers by means of the 24 letters of the ordinary Greek alphabet, augmented by three obsolete Phoenician letters. The resulting 27 letters were used as follows. The initial nine letters were associated with the numbers from 1 to 9; the next nine letters represented the next nine integral multiples of 10; the next nine letters were used for the next nine integral multiples of 100. The following table shows how the letters of the alphabet (including the special forms) were arranged for use as numerals.

Because the Ionic system was still a system of additive type, all numbers between 1 and 999 could be represented by at most three symbols. The principle is shown by ψπδ = 700 + 80 + 4 =784

Multiplication example

Hindu-Arabic Numeral

The Hindu-Arabic numeral system is a decimal numeral system that is used widely in the world today. It is also known as the base-10 numeral system and the decimal system. The Hindu-Arabic numeral system uses ten symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, to represent numbers. The symbols are combined to represent larger numbers and to perform mathematical operations. The use of zero is indeed one of the key features of the Hindu-Arabic numeral system that sets it apart from other numeral systems. The concept of zero as a number in its own right was first introduced by Indian mathematicians, and it was incorporated into the Hindu-Arabic numeral system as a symbol that represented the absence of value.

The Hindu-Arabic numeral system originated in India and was developed from the earlier Indian numeral system. It was later transmitted to the Islamic world and eventually to Europe, where it was adopted and became widely used. The Hindu-Arabic numeral system has several advantages over other numeral systems, such as the ability to perform arithmetic operations quickly and easily, the flexibility to represent very large and very small numbers, and the ease of use and compactness of the symbols.

The Hindu-Arabic numeral system has been adopted as the standard numeral system in most countries and has become an essential tool in commerce, science, and technology. Its widespread use has also helped to promote communication and collaboration between people and cultures, and has made it possible to perform complex calculations and to analyze data in a way that was not possible with earlier numeral systems.

Above we show some of the civilization and their numeral system but in the human history we see many civilization that use their own numeral system like

• One of the earliest known numeral systems was developed by the Sumerians in Mesopotamia around 4000 BCE. They used a system of cuneiform symbols to represent numbers.

• The Babylonians, who lived in Mesopotamia around 2000 BCE, used a sexagesimal (base-60) numeral system, which was later adopted by the ancient Mayans in Central America.

• The ancient Romans used a system of letters and symbols, known as Roman numerals, which is still used today for certain purposes, such as in the numbering of chapters in books and in the numbering of movie sequels.

• The Chinese numeral system is a traditional numeral system used in China and other East Asian countries, such as Japan and Korea. It is a non-positional numeral system, which means that the value of a symbol depends on its position relative to other symbols. One unique aspect of the Chinese numeral system is that it has two sets of symbols: one for counting objects, and one for representing numbers in financial transactions. This is because the characters for the numbers 1 to 9 have different forms in the two sets of symbols, with the financial forms being more complex and ornate.

Throughout history, people have continued to develop new methods of representing numbers and calculating with them. The invention of computers and the development of digital technology has allowed us to perform complex mathematical operations on a massive scale, revolutionizing the way we use and understand numbers.