#mersenne prime squares

[rbrooksdesign]

"DMT_18," digital + acrylic, April 20, 2024, Reginald Brooks

DMT = Divisor (Factor) Matrix Table

[worlds-smallest-epsilon]

Number ask game! 37, which for some reason is my favorite number. Also, 127 already got asked so I'll go in a similar vein with 2147483647 :))

37 is prime which is fun. It's one more than a square which is neat. Interestingly, $\mathbb{Q}(\sqrt{37})$ has class number 1, meaning its ring of integers has unique factorization. A-

2147483647 is another Mersenne prime, and it's the largest value you can represent with a 32-bit signed integer. That's interesting enough to save it from a D. C-

[afactaday]

#aFactADay2023

#777!!! we made it!! whooo!!! in celebration of seven hundred and seventy-seven facts, here's a fact all about seven! seven is not only a Mersenne Prime (a prime of the form 2^p-1 where p is a prime), but it's the first Double Mersenne Prime (because the exponent p is itself a Mersenne prime formed from 2^2-1). only four double Mersenne primes are known, and the next likely contender is about 10^{10^19}, which is far too large to check. seven is also the first Newman-Shanks-Williams Prime, which has some complicated definition involving square roots. this definition arose from simple group theory and is really intriguing but i'm afraid that can't tell you any more because the only reference i can find (the original paper) seems to have been deleted.

Lucky Numbers, completely unrelated to Euler's Lucky Numbers, are the numbers which remain after a sieving algorithm is applied (find it on the wiki). seven is the second Lucky Prime: a prime lucky number.

seven is also the first Woodall prime, a prime of the form n*2^n-1 (n=2, so it's the second Woodall number). it's also a factorial prime (a prime which is a factorial number plusorminus one), a Safe Prime (a number which is twice plus one a prime; look up Sophie Germain primes), a Happy Prime (a prime which is a happy number, which can be found by an iterative algorithm on its digits), and a Leyland Prime of the Second Kind (a prime of the form x^y-y^x). and just to make this slightly less primey, the regular heptagon is the first regular polygon which can't be constructed with a straight edge and compasses.

link: seven!!! (7-a-side sport innit)

ahh 777, it feels so satisfying

[allofmymaths]

BMO1 2021 Question 6

This is another question from the 2021 BMO1 paper which is pretty neat. 

Start with we know that n = 2^a + 2^b for some distinct integers a and b 

Also, we know that n = 2^p - 1 + 2^q - 1 = 2^p + 2^q - 2 for some p and q such that 2^p - 1 and 2^q - 1 are primes 

First, without loss of generality we can assume that a < b and p < q

For now let’s just focus on the Mersenne Prime, 2^p - 1

Notice that if you can write p = rs for some integers r and s, then without loss of generality if we assume that r > 1 and let x = 2^s then:

If r is even we can write 2^p - 1 = x^r - 1 = (x^(r/2) + 1)(x^(r/2) - 1) 

If r is odd, we can write 2^p - 1 = x^r - 1 = (x - 1)(x^(r-1) + x^(r2) + ... + 1) 

In both cases this violates the primality of 2^p - 1 whenever r < p or s > 1, and therefore we know that we must have r = p and s = 1. This is telling us that p is a prime number. 

Now we may go back to 2^p + 2^q - 2 = n = 2^a + 2^b 

If we consider both of these expression in binary:

- 2^a + 2^b can be written as (10)^a + (10)^b in binary, which tells us that the number has a 1 in the (a+1)th and (b+1)th digits and zeroes everywhere else. In particular, this tells us there are EXACTLY two 1 digits since a =/= b.

-  2^p + 2^q - 2 can be written as (10)^p + (10)^q - 10. When subtracting the 10, starting from the 2nd digit, any zeroes become a one until you reach the first 1. Because you know that p < q this means that you’ll always hit the 1 in the (p+1)th place before the 1 in the (q+1)th place. This is important because this tells us that q = b.

Additionally, since we know that q = b, we know that 2^p - 2 = 2^a and this can only be true if p = 2, a = 1. This is because the left hand side is divisible by 2 but not 4, so a = 1. If a = 1 then p could only be 2. 

Now finally we have: n = 2^q - 2 + 2^2 = 2^q + 2 

= 2^(q-1) + 1 + 2^(q-1) + 1

Notice that q-1 is even since q > 2 is prime and therefore odd. Let q-1 = 2k for some integer k. The we may write

n = [2^(2k) + 2*2^(k) + 1] + [2^(2k) - 2*2^(k) + 1]

-> n = (2^k + 1)^2 + (2^k - 1)^2 

I liked using binary digits to deduce information about a, b, p and q. Ultimately it’s the same as considering various conditions of divisibility, but it was a nice visual aid. 

The last step of adding in the terms 2*2^k - 2*2^k to get two squares was quite fun as well. 

[scifigeneration]

Largest known prime number discovered: Why it matters

by Anthony Bonato

In the movie Contact, based on the novel of the same name by Carl Sagan, Dr. Ellie Arroway searches for intelligent extraterrestrial life by scanning the sky with radio telescopes. When Arroway, played by Jodie Foster, recognizes prime numbers in an interplanetary signal, she believes it’s proof that an alien intelligence has sent the human race a message.

A number is considered prime if it is only divisible by one and itself. For example, two, three, five and seven are prime. The number 15, which is three times five, is not prime. It’s no coincidence that Arroway believes the aliens in Contact use prime numbers as a cosmic “hello” — they are building blocks of other numbers. Every number is a product of primes.

In December 2017, the largest known prime number was discovered using a computer search. The prime was discovered by Jonathan Pace, an electrical engineer who currently works at FedEx. Why is this important? Because without prime numbers your banking information, Paypal transactions or Amazon purchases could be compromised.

Large primes, like the one just discovered, play a critical role in cyber-security. Cryptography is the science of encoding and decoding information, and many of its algorithms, such as RSA, rely heavily on prime numbers.

Bitcoin and other crypto-currencies use security that depends on prime numbers. (Shutterstock)

Mersenne primes

While there are infinitely many primes, there is no known formula to generate them all. A race is ongoing to find larger primes using a mixture of math techniques and computation.

One way to get large primes uses a mathematical concept discovered by the 17th-century French monk and scholar, Marin Mersenne.

Marin Mersenne. H Loeffel, Blaise Pascal, Basel: Birkhäuser 1987, CC BY-NC

A Mersenne prime is one of the form 2ⁿ - 1, where n is a positive integer. The first four of these are three, seven, 31 and 127.

Not every number of the form 2ⁿ - 1 is prime, however; for example, 2⁴ - 1 = 15. If 2ⁿ - 1 is prime, then it can be shown that n itself must be prime. But even if n is prime, there is no guarantee the number 2ⁿ - 1 is prime: 2¹¹ - 1 = 2,047, which is not prime becauase it equals 23 times 89.

There are only 50 known Mersenne primes. An unresolved conjecture is that there is an infinite number of them.

The search for new primes

The Great Internet Mersenne Prime Search (or GIMPS) is a collaborative effort of many individuals and teams from around the globe to find new Mersenne primes. George Woltman began GIMPS in 1996, and in 2018 it includes more than 183,000 volunteer users contributing the collective power of over 1.6 million CPUs.

The most recently discovered Mersenne prime is succinctly written as 2⁷⁷²³²⁹¹⁷ - 1; that’s two multiplied by itself 77,232,917 times, minus one. Jonathan Pace’s discovery took six days of computation on a quad-core Intel i5-6600 CPU, and was independently verified by four other groups.

The newly discovered prime has a whopping 23,249,425 digits. To get a sense of how large that is, suppose we filled up a book with digits, each digit counted as a word and each book having 100,000 words. Then the digits of 2⁷⁷²³²⁹¹⁷ - 1 would fill up about 232 books!

How does GIMPS find primes?

GIMPS uses the Lucas-Lehmer test for primes. For this, form a sequence of integers starting with four, and whose terms are the previous term squared and minus two. The test says that the number 2ⁿ - 1 is prime if it divides the (n-2)th term in the sequence.

While the Lucas-Lehmer test looks easy enough to check, the computational bottleneck in applying it comes from squaring numbers. Multiplication of integers is something every school-aged kid can do, but for large numbers, it poses problems, even for computers. One way around this is to use Fast Fourier Transforms (FFT), algorithms that speed up computations.

Anyone can get involved with GIMPS — as long as you have a decent computer with an internet connection. Free software to search for Mersenne primes can be found on the GIMPS website.

While the largest known prime is stunningly massive, there are infinitely many more primes beyond it waiting to be discovered. Like Ellie Arroway did in Contact, we only have to look for them.

Anthony Bonato is Professor of Mathematics at Ryerson University.

This article was originally published on The Conversation.

[quadrantmathematics]

Quadrant Mathematics Book

https://en.wikipedia.org/wiki/Quartic_function

Polynomials can be solved by radicals only up to degree four. It took a very long time to discover the proof for degree four. The fourth is always different. It took until 1824 to discover the proof.

The first four Mersenne primes were known by the ancients. The fourth is different. It took until modern times to discover the fifth. The fourth is always different. The fifth is ultra transcendent.

https://en.wikipedia.org/wiki/Mersenne_prime

3

31

7

127

8191 was discovered in 1461

The FOIL method is used for multiplying binomials in math. The four terms for the FOIL method are

First ("first" terms of each binomial are multiplied together)

Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second)

Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second)

Last ("last" terms of each binomial are multiplied)

https://www.youtube.com/watch?v=S0_qX4VJhMQ

The geometric proof for derivatives in calculus involves four squares in quadrant formation. I learned this proof while watching a teaching company course on Calculus.

https://www.mathsisfun.com/algebra/completing-square.html

The geometric analysis for completing the square involves four squares. Completing the square is essential in algebra. The geometric proof for derivatives in calculus also involves four squares.

There are 16 squares in the quadrant model. There are 16 possible Truth Functions of two binaries

Here is an example of a truth table with the four possibilities

The ancient Greeks knew about the first four perfect numbers. The fourth one is different, and it took a lot longer for the ancient Greeks to discover the fourth.

https://en.wikipedia.org/wiki/Perfect_number

for p = 2:   21(22 − 1) = 6

for p = 5:   24(25 − 1) = 496

for p = 3:   22(23 − 1) = 28

for p = 7:   26(27 − 1) = 8128.

Euclid’s Elements was the most popular book behind the Bible. Euclid’s Elements had five postulates. The five postulates are the most famous set of postulates in math history. The fifth postulate was proven to be false. The fifth is always questionable. However, the discovery that the fifth postulate was false lead to the uncovering of different geometries. Ancients saw the fourth postulate as different. Proclus did not think that the fourth postulate should be a postulate. The fourth is always different. Euclid also proposed five axioms. The fifth axiom is false and the fourth is different. Some suggest that Euclid intentionally made the fifth postulate false so that out of its discovery as false, new geometries would be discovered. I learned about Euclid’s postulates on a teaching company course on mathematics, and the professor described how the fifth was incorrect and the fourth was different.

https://blogs.scientificamerican.com/roots-of-unity/whate28099s-the-deal-with-euclide28099s-fourth-postulate/

"To draw a straight line from any point to any point."

"To describe a circle with any centre and distance [radius]."

The third is about circles. A circle has the look of being an object and being whole. The first postulate is about a line. A line is like air, taking up no area. Air is the first square element. The second postulate is a straight line that is extended. The second element is water. Still there is no area being taken up, but it is longer. The first two are a duality. The third, the circle, brings area into the equation. The third square is the most solid. The fourth postulate is on right angles. Right angles are quadrants. The fourth is always transcendent

"To produce [extend] a finite straight line continuously in a straight line."

"That all right angles are equal to one another."

Semiotic squares take the form of a quadrant and are very popular. Semiotic squares are used a lot in academia.

https://en.wikipedia.org/wiki/Semiotic_square

https://www.newyorker.com/magazine/2008/03/03/numbers-guy

Deheane describes how in cultures throughout the world the first three numbers were usually represented similarly, but the fourth differently. For instance Roman numerals has I II and III, but IV is written with a V, different from the first three. There are some cultures that represent the first four numbers similarly. But all cultures represent five differently.

Deheane also points out that when tracking objects, only four can be tracked at one time. With three objects it is difficult, but four very difficult, but five at one time is impossible.

The quadrant model has 16 squares. Also scientists pointed out that a person can only imagine 16 squares at one time.

https://books.google.com/books?id=1p6XWYuwpjUC&pg=PT83&lpg=PT83&dq=Stanislas+Dehaene+one+two+three+four+chinese&source=bl&ots=S6NCR0YXvL&sig=01bEkiLf6Hoj44PrRG4mPsc8C6A&hl=en&sa=X&ved=0ahUKEwjsrfre0N3XAhUY6GMKHeClDXs4ChDoAQgxMAE#v=onepage&q=Stanislas%20Dehaene%20one%20two%20three%20four%20chinese&f=false

https://en.wikipedia.org/wiki/Subitizing

People can make flash judgements on the number of objects from one to three well. At four it becomes difficult, and at five the accuracy makes an exponential decline. The fourth is always different. The fifth is ultra transcendent

https://en.wikipedia.org/wiki/Approximate_number_system

The approximate number system is four numbers greater than four. Numbers one through four are counted through parallel individuation.

There are four fundamental operations in math. The four operations are

https://en.wikipedia.org/wiki/Operation_(mathematics)

+, plus (addition)

÷, obelus (division)

−, minus (subtraction)

×, times (multiplication)

Quartiles are very important in statistics

https://en.wikipedia.org/wiki/Quartile

Quartiles are three points that divide a graph into four equal groups.

Boxplot (with quartiles and an interquartile range) and a probability density function (pdf) of a normal N(0,1σ2) population

Boxplots are also very important in statistics. I learned about boxplots in middle school and in my college statistics course. I also learned about quartiles and interquartile range in middle school and in college. A box plot graphically depicts numerical data through its quartiles.

https://en.wikipedia.org/wiki/Box_plot. The bottom and top of the box are the first and third quartiles. The inside of the box is the second quartile.

Figure 1. Box plot of data from the Michelson–Morley experiment

Boxplot (with an interquartile range) and a probability density function (pdf) of a Normal N(0,σ2) Population

https://en.bywiki.com/wiki/Lagrange%27s_four-square_theorem#Historical_development

There is a four square theorem in mathematics

https://en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity

There is a sixteen square identity in mathematics. There are sixteen squares in the quadrant model

Jade Mirror of the Four Unknowns solved equations for up to four unknowns. The book also had equations of three unknowns. In the book we thus see the three, four dynamic.

https://en.wikipedia.org/wiki/Yuan_dynasty

https://en.wikipedia.org/wiki/Jade_Mirror_of_the_Four_Unknowns

The first four problems illustrate the method of the four unknowns. The four unknowns are

Heaven

Man

Earth

Matter

Jade mirror of the four unknowns consists of four books.

The introduction shows The Square of the Sum of the Four Quantities of a Right Angle Triangle

The illustration is a sixteen square quadrant model. The first illustration of the book is a sixteen square quadrant model

The book ends with the equation of the four unknowns

The famous Rhind Mathematical papyrus has four sections

https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus

https://en.wikipedia.org/wiki/Lo_Shu_Square

The Lho Shu Magic Square contains a cross of odd numbers. The magic square is used to structure Jain Temples. It is seen as very sacred

There is a cross/quadrant pattern within the Sator Square. A cross/quadrant is depicted with the word Tenet. The sator square is a “four time palindrome”. People have claimed that it has magical properties.

https://en.wikipedia.org/wiki/Sator_Square

By repositioning the letters around the central letter Ν (en), a Greek cross can be made that reads Pater Noster (Latin for "Our Father", the first two words of the "Lord's Prayer") both vertically and horizontally. The remaining letters – two each of A and O – can be taken to represent the concept of Alpha and Omega, a reference in Christianity to the omnipresence of God. Thus the square might have been used as a covert symbol for early Christians to express their presence to each other

In Plato’s Meno, Plato tries to prove that knowledge is innate and comes from the World of the Forms. Socrates claims that people come from the World of the Forms before they are born, and that whenever somebody learns something, he is merely recalling what he already knows from the World of the Forms. Socrates proves that knowledge is innate by showing that a slave boy knows geometry. According to Socrates, the slave boy has received no education, since he is a slave. But because the slave boy understands the geometry, Socrates claims, the slave boy must have already known it from the World of the Forms

Socrates draws diagrams in the sand. The diagrams that Socrates draws to prove the world of the Forms are what I call the Form of the Good, the Form of Existence. Socrates draws the quadrant model 16 squares. I watched a teaching company course where the Professor drew the diagrams that Socrates drew in the sand.

https://tothereal.wordpress.com/2013/06/16/are-we-slaves-to-socrates/

Here are the diagrams that Socrates draws

In Roman numerals X is the number 10. Philosophers connect the number 10 and the tetractys. I have seen Jungian philosophers propose that because the x is fourfold, and the tetractys is related to four, the reason why the Romans used X as 10 is because of the tetractys.

https://en.wikipedia.org/wiki/The_Garden_of_Cyrus

The Garden of Cyrus is a book by Sir Thomas Browne. Browne sees the quincunx in so many places in reality that he says that it is evidence of intelligent design. Thomas Browne relates the quincunx to the Garden of Cyrus. The Garden of Cyrus took a quincunx pattern.

Frontispiece to 'The Garden of Cyrus' (1658)

In high school trigonometry my class learned the unit circle. The unit circle is a quadrant

http://locusacademy.org/unit-circles-and-trigonometric-equations/

An acronym that my class used to remember the four quadrants of the unit circle was All Stations to Central

http://locusacademy.org/unit-circles-and-trigonometric-equations/

In the upper left quadrant, only the sin values are positive, while the others are negative

The upper right quadrant's values of sin, cos and tan are all positive.

In the lower left quadrant, the tan values are positive, while the others are negative

The fourth quadrant is where the cos values are positive while the others are negative.

There is a famous puzzle called the Nine Dots Puzzle. My Grandpa showed me the puzzle. Also in my psychology class at UCSD the Professor showed us the puzzle. You are told to draw four lines that connect nine dots. The nine dots are in three columns. The way to solve the solution is to go outside of the limits of the nine dots. If you do so, you go into the region of a 16 square quadrant model. Moveroever, at the center of the solution is an X. An X is a quadrant. The solution employs what psychologists call “lateral thinking”. In order to solve the problem you must think outside of the box.

https://en.wikipedia.org/wiki/Thinking_outside_the_box

The Wason Selection task is also known as the “four card problem”. Developed by Peter Cathcart Wason, the four card problem is one of the most famous tasks in the study of deductive reasoning. I remember learning about the four card problem in a psychology class I took at UCSD.

https://en.wikipedia.org/wiki/Wason_selection_task

The Cartesian coordinate system is used extensively in mathematics. The Cartesian coordinate system is made up of four quadrants

https://en.wikipedia.org/wiki/Cartesian_coordinate_system

The complex number plane is similar to the Cartesian coordinate system, but it uses the imaginary number i on the y axis. The complex number plane is called an argand diagram, and it has four quadrants. Pauli saw the use of imaginary numbers and complex number planes in quantum mechanics as further evidence that physics was based around the quatenary.

http://www.peterstone.name/Maplepgs/complex.html

The Missing Squares Puzzle is a famous puzzle in mathematics

https://en.wikipedia.org/wiki/Pythagorean_hammers

Pythagoras used four hammers to discover the foundations of musical tuning. The four hammers are known as the Pythagorean hammers. He came across five men hammering with hammers, and four of them were harmonious, but the fifth was not. Again, the fourth is always transcendent, and the fifth is always questionable.

https://www.amazon.com/Fifth-Hammer-Pythagoras-Disharmony-World/dp/193540816X

https://mitpress.mit.edu/books/fifth-hammer

https://en.wikipedia.org/wiki/Anscombe%27s_quartet

Ascombe’s quartet is four datasets that have very similar descriptive statistics, but seem very different when graphed.

https://en.wikipedia.org/wiki/P_versus_NP_problem

NP Hard is different from the other three, P, NP, and NP complete. P v NP is a major unsolved problem in computer science

P

NP-Complete

NP

NP hard

In a math class that I sat in on at UCSD, the Professor drew the bijection, injection, surjection quadrant on the board. The fourth quadrant is different from the other three. The fourth quadrant does not even have a name

I sat in on a math class as well at UCSD where the four types of Fourier Transforms were brought up. The four types fit a quadrant pattern. Fourier transforms were talked about a lot in different classes that I studied. The four types fit a quadrant pattern.

https://dsp.stackexchange.com/questions/28020/formulas-of-the-fourier-transform-family

https://en.wikipedia.org/wiki/Quartic_function

A quartic function is the highest degree such that every polynomial can be solved by radicals. Quartic means four

Graph of a polynomial of degree 4, with 3 critical points and four real roots(crossings of the x axis) (and thus no complex roots).

Ferrari discovered the quartic solution in 1540. The proof that the quartic is the highest degree polynomial that could be solved by radicals was given by the Abel- Ruffini theorem in 1824.

In order to complete the proof on the quadrature of the parabola, Archimedes must prove that

His proof involves a diagram that resembles a quadrant model

Archimedes' proof that 1/4 + 1/16 + 1/64 + ... = 1/3

The Missing Square Puzzle is a famous puzzle in mathematics. It invovles four colors

https://en.wikipedia.org/wiki/Missing_square_puzzle

Kryptos is a sculpture by the American artist Jim Sanborn at the Central Intelligence Agency (CIA) in Langley, Virginia.

It has four encrypted messages. Three of the four messages have been encoded. The fourth is different and transcendent.

https://en.wikipedia.org/wiki/Kryptos

Pyramids in Egypt are quadrants, and the pyramids of Giza reflect the pythagorean theorem.

http://africancreationenergy.blogspot.com/2014/12/the-ptah-horus-pythagoras-theorem.html

http://africancreationenergy.blogspot.com/2014/12/the-ptah-horus-pythagoras-theorem.html

Each side of the Pythagorean theorem represented for the Egyptians a different God. The four by four square is the quadrant model square. The four by four 16 square of the simplest pythagorean triple, the three four five triangle, is represented by Isis

http://africancreationenergy.blogspot.com/search/label/African%20Mathematics

The proof of the Pythagorean theorem attributed to pythagoras involved four squares. Look at the diagram to the right above, and notice the four segments.

http://theopenscroll.blogspot.com/2013/06/part-3-signs-of-horus-worship-345-on-map.html

64 is four quadrant models. 16 is the quadrant model. The eye of Osiris has the ratios of the quadrant model

The right side of the eye =  1⁄2

The pupil =  1⁄4

The eyebrow =  1⁄8

The left side of the eye =  1⁄16

The curved tail =  1⁄32

The teardrop =  1⁄64

The Rhind Mathematical Papyrus contains tables of "Horus Eye Fractions".[16]

https://en.wikipedia.org/wiki/Eye_of_Horus

https://en.wikipedia.org/wiki/Missing_square_puzzle

The missing squares puzzle is a famous mathematical optical illusion. The demonstration involves four figures of four colors.

Sam Lloyd’s paradoxical dissection also has four shapes and colors.

Sam Loyd's paradoxical dissection

https://en.wikipedia.org/wiki/Missing_square_puzzle

Mitsunobu Matsuyama's "Paradox" uses four congruent quadrilaterals and a small square, which form a larger square. The diagram makes a quadrant.

https://en.wikipedia.org/wiki/Missing_square_puzzle

The 15 puzzle involves 16 squares. The 15 puzzle is a four by four quadrant model

https://en.wikipedia.org/wiki/15_puzzle

https://en.wikipedia.org/wiki/Mathematical_puzzle

Four fours is another mathematical puzzle

https://en.wikipedia.org/wiki/Four_fours

https://en.wikipedia.org/wiki/Tower_of_Hanoi

The three peg tower of Hanoi puzzle has a simple recursive solution. The optimal solution for a four peg tower of hanoi puzzle was not discovered until 2014. For the case of more than four pegs the problem is an open problem. The fourth is always transcendent and the fifth is questionable.

The famous T puzzle consists of four polygonal shapes. The puzzle seems easy but it is not “because of the irregular piece”. The fourth piece is different than the previous three. Again we see the three plus one pattern. Few people are able to solve the T puzzle in under five minutes.

https://en.wikipedia.org/wiki/T_puzzle

https://en.wikipedia.org/wiki/T_puzzle

The latin cross puzzle uses a cross. A cross is a quadrant.

https://en.wikipedia.org/wiki/T_puzzle

http://www.cropcircleconnector.com/2017/cleyhill/comments.html

https://en.wikipedia.org/wiki/Cantor_set

Cantor Dust is made of quadrants

http://mathworld.wolfram.com/CantorDust.html

https://en.wikipedia.org/wiki/Menger_sponge

A Jerusalem cube is a fractal object described by Eric Baird in 2011. It is created by recursively drilling Greek cross-shaped holes into a cube.[8][9]The name comes from a face of the cube resembling a Jerusalem cross pattern

http://www.critcrim.org/redfeather/chaos/029management.html

Chaos research, as mentioned, tracks the transformations of dynamical systems from one behavioral regime (attractor state) to another. In such transformations, management science has much to learn and much to ponder. As key parameters of systems reach each one of four feigenbaum numbers (F1-F4 discussed below), the system displays an orderly procession from one dynamical state to another. The procession ceases to be orderly and becomes very chaotic at F4. As a system becomes more chaotic, i.e., it transforms from a simple outcome basin to a much more complex causal field.

Sierpinski triangle in logic: The first 16 conjunctions of lexicographicallyordered arguments. The columns interpreted as binary numbers give 1, 3, 5, 15, 17, 51... (sequence A001317 in the OEIS)

https://en.wikipedia.org/wiki/Sierpinski_triangle#Analogues_in_higher_dimensions

The Sierpinski triangle is created by dividing an equilateral triangle into four equal triangles, and continually dividing triangles into four equal triangles.

There is also the Sierpinski tetrahedron. Tetra is four

https://en.wikipedia.org/wiki/Sierpinski_triangle

https://en.wikipedia.org/wiki/Kakuro

Kakuro is a logic puzzle that involves quadrants

Sujiko is a logic based problem that involves quadrants

https://en.wikipedia.org/wiki/Sujiko

https://en.wikipedia.org/wiki/Mathematical_puzzle

Ken ken is a logic puzzle that involves quadrants

https://en.wikipedia.org/wiki/KenKen

https://www.target.com/p/otrio-board-game/-/A-52338707?sid=1307S&ref=tgt_adv_XS000000&AFID=google_pla_df&CPNG=PLA_Toys+Shopping_Local&adgroup=SC_Toys&LID=700000001170770pgs&network=g&device=c&location=9031022&gclid=Cj0KCQiA38jRBRCQARIsACEqIese4XJFLR0_GIN7ppWFN4YZ4q2QxHsQD6F4ry5zAbr1NqyoAG95cg4aAruJEALw_wcB&gclsrc=aw.ds

Wang tiles are quadrants

https://en.wikipedia.org/wiki/Wang_tile

https://en.wikipedia.org/wiki/Level_of_measurement

I learned the four levels of measurement at UCSD. The first thing that was taught in my statistics class was the four levels of measurement.

Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables.[1] The best known classification of levels of measurement was developed by psychologist Stanley Smith Stevens: nominal, ordinal, interval, and ratio.

The four levels are

nominal

interval

ordinal

ratio

https://ncatlab.org/nlab/show/normed+division+algebra

The figure above shows the fibonacci spiral

Approximate and true golden spirals: the green spiral is made from quarter-circles tangent to the interior of each square. The length of the side of a larger square to the next smaller square is in the golden ratio. Quarters are one fourths.

A Fibonacci spiral approximates the golden spiral using quarter-circle arcs inscribed in squares of integer Fibonacci-number side, shown for square sizes 1, 1, 2, 3, 5, 8, 13 and 21.

https://en.wikipedia.org/wiki/Golden_spiral

https://ncatlab.org/nlab/show/normed+division+algebra

Over the real numbers there are only four normed division algebras up to isomorphism: the algebras of

Real numbers

Quaternions

Complex numbers

Octonions

http://mathworld.wolfram.com/TetramagicSquare.html

https://en.wikipedia.org/wiki/Multimagic_square

Tetra is four. A tetramagic square is a magic square such that the first, second, third, and fourth powers of the elements all yield magic squares. The tetramagic square is transcendent. The first tetramagic square was discovered in 1983.

https://en.wikipedia.org/wiki/Euler%27s_four-square_identity

In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.

https://en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity

Pfister’s sixteen square identity shows that, in general, the product of two sums of sixteen squares is the sum of sixteen rational squares. This is a popular theorem.

[destinimott]

Number 8 is the symbol of harmony and balance. Number 8 symbolizes the ability to make decisions. Number 8 symbolizes abundance and power. The Pythagoreans called the number eight “Ogdoad” and considered it the “little holy number”.

The number 8 in the Bible represents a new beginning, meaning a new order or creation, and man's true 'born again' event when he is resurrected from the dead into eternal life.

the number eight there are 88 keys on the piano and the chess board is 8 times 8 squares, 

Sixty-four is the square of 8, the cube of 4, and the sixth power of 2. It is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime. 64 is the sum of Euler's totient function for the first fourteen integers.

Being the cube of 4, the number 64 represents the physical world of the four elements to the third power, in the plenitude of its expansion

Tessering, a significant part of the fiction of A Wrinkle in Time, is a mode of travel in the fifth dimension that utilizes the shortest route. It can be used in space and in time.

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You may also enjoy taking part in our school’s code breaking website. There are 8 levels of coding difficulty – with each code giving you a password to access the next clue.  There are Maths Murder Mysteries, Spy games and more.  Solve all the clues in a level to make it onto the leaderboard.  The 2 hardest levels – Level 6 and Level 7 are particularly tough – are you good enough to crack them?

Algebra and number

1)

Modular arithmetic

– This technique is used throughout Number Theory. For example, Mod 3 means the remainder when dividing by 3.

2)

Goldbach’s conjecture:

“Every even number greater than 2 can be expressed as the sum of two primes.” One of the great unsolved problems in mathematics.

3) Probabilistic number theory

4) Applications of

complex numbers

: The stunning graphics of Mandelbrot and Julia Sets are generated by complex numbers.

5)

Diophantine equations

: These are polynomials which have integer solutions.

Fermat’s Last Theorem

is one of the most famous such equations.

6)

Continued fractions

: These are fractions which continue to infinity. The great Indian mathematician

Ramanujan

discovered some amazing examples of these.

7)

Patterns in Pascal’s triangle

: There are a large number of patterns to discover – including the Fibonacci sequence.

8)

Finding prime numbers

: The search for prime numbers and the twin prime conjecture are some of the most important problems in mathematics. There is a $1 million prize for solving the

Riemann Hypothesis

and $250,000 available for anyone who discovers a new, really big prime number.

9) Random numbers

10)

Pythagorean triples

: A great introduction into number theory – investigating the solutions of Pythagoras’ Theorem which are integers (eg. 3,4,5 triangle).

11)

Mersenne primes

: These are primes that can be written as 2^n -1.

12)

Magic squares and cubes

: Investigate magic tricks that use mathematics. Why do magic squares work?

13) Loci and complex numbers

14)

Egyptian fractions

: Egyptian fractions can only have a numerator of 1 – which leads to some interesting patterns. 2/3 could be written as 1/6 + 1/2. Can all fractions with a numerator of 2 be written as 2 Egyptian fractions?

15) Complex numbers and transformations

16)

Euler’s identity:

An equation that has been voted the most beautiful equation of all time, Euler’s identity links together 5 of the most important numbers in mathematics.

17)

Chinese remainder theorem

. This is a puzzle that was posed over 1500 years ago by a Chinese mathematician. It involves understanding the modulo operation.

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"DMT_9" digital + acrylic, April 11, 2024, Reginald Brooks

DMT = Divisor (Factor) Matrix Table

[ramialkarmi]

Google hid another nerdy math joke in its latest earnings report (GOOG, GOOGL)

Google announced a very specific share buyback in its earnings report.

The number given is a "perfect number."

Google has a history of making mathematical references.

In its Q4 earnings report on Thursday, Google's parent company Alphabet announced plans to buy back some of its stock — $8,589,869,056 of stock to be exact.

There's a likely reason Google chose that specific value for their planned buyback: It's what mathematicians call a "perfect number."

Perfect numbers are whole numbers with a fun property — they are the sum of all their proper divisors, or the whole numbers that evenly divide into the original number, except itself. The smallest perfect number is 6: 6 is divisible by 1, 2, 3, and itself. But then take those first three divisors and add them up, and you get 1 + 2 + 3 = 6.

8,589,869,056 is such a number. It has a few too many divisors to write out here — 33 proper divisors in total — but the sum of its proper divisors does in fact come back up to itself.

In addition to their defining curious property, perfect numbers are relatively rare. They are closely connected to a certain class of prime numbers called Mersenne primes, or prime numbers that are one less than 2 raised to some prime number. Each even perfect number has a corresponding Mersenne prime.

Mersenne primes are interesting because it's relatively computationally easy to check whether a candidate Mersenne prime is in fact prime. Because of that, most of the largest prime numbers yet discovered are Mersenne primes, including the largest currently known prime number, 277,232,917 - 1.

Only 50 Mersenne primes have been found so far, which means because of their correspondence, only 50 perfect numbers are known so far. That makes Google's buyback choice quite a special number.

Google has a history of making mathematical references. In 2011, the company bid a multiple of pi in an auction for Nortel Networks' tech patents. Shortly after renaming itself Alphabet in 2015, the company announced a buyback in the amount of $5,099,019,513.59, which happens to be the square root of 26 x 1018 — 26, of course, being the number of letters in the alphabet. Google's name itself comes from the massively large number called a "googol" — 10^100, or a 1 followed by 100 zeros.

SEE ALSO: If you can solve one of these 6 major math problems, you'll win a $1 million prize

Join the conversation about this story »

NOW WATCH: Microsoft President Brad Smith says the US shouldn't get 'too isolationist'

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What is Primecoin Coin?

introduction

Primecoin (symbol: Ψ; ticker: XPM) is an electronic payment system with the same peer-to-peer cryptocurrency . The proof-of-work system is the main difference of this cryptocurrency , and its usefulness lies in the proof of work calculations, which look for chains of prime numbers. The Primecoin source code is copyrighted by a person or group called “Primecoin Developers” and is distributed with the limited MIT / X11 software license. Primecoin was described as the main cause of dedicated server bottlenecks, as it was only possible to mine the crypto currencies with CPUs at that time .

Primcecoin – The Features

Comparing Primecoin with Bitcoin , the most widely used cryptocurrency today , there are some notable differences:

The scarcity is determined by the type of prime distribution

Almost all existing cryptocurrencies define their scarcity properties only through a set of predefined values ​​in the source code, while the scarcity of coins is purely based on a relationship of a simple function and a mathematical property of the natural occurrence of primer chains in the set of integers.

 Read Also  : What is Crystal Clear Coin?

 Read Also  :  What is Condensate Coin?

 Read Also  :  What is CrowdCoin?

No predefined final number of coins

Instead of setting the final number of coins in your code like many other alternative cryptocurrencies, the number of primecoins released per block is always equal to 999 divided by the square of the difficulty. There have been some attempts to approximate this number. The number of primecoins being gemined is determined by the progress of their adaptation and by the mining community, further by the improvements made to the mining algorithms and finally by the Moore Law.

Difficulty adjustment is more common

The Primecoin protocol slightly adjusts its difficulty after each block. The difficulty change that occurs in each block aims to reach the goal of a new block being created once a minute. By way of comparison, the Bitcoin protocol adjusts its difficulty in each 2016 block or about every two weeks.

 Read Also  : What is Unitus Coin?

 Read Also  : What is Xios Coin?

 Read Also  : What is 1337 Coin?

Faster transaction confirmations

Since Primecoin blocks are generated 8 to 10 times faster than Bitcoin blocks on average, Primecoin transactions are confirmed about 8 to 10 times faster.

Primecoin – the proof-of-work system

Primecoin, used in its PoW labor records for finding primary chains, the composition of Cunningham chains and bi-twin chains, the result can lead to useful byproducts. The Primecoin proof-of-work system has the following features:

Primecoin’s work takes the form of prime chains

Finding the prime number chains becomes exponentially harder as the chain length is increased

The verification of the prime-number chains of an appropriate size can be efficiently implemented by all network nodes

Mersenne primes are excluded due to their extreme size

Three types of prime chains are accepted as proof of the work: Cunningham chain of the first kind. Cunningham second-grade necklace, bi-twin necklace. Read Also  : What is SmileyCoin?

 Read Also  : What is UltraCoin?

 Read Also  :  What is Unify Coin?

Other cryptocurrencies, including Bitcoin, typically use a Hashcash PoW work test based on SHA-256 hash calculations that have no value outside their own economy. The list of the largest known Cunningham chains of given length contains several results generated by prime miners.

For More  Digital Coins Information  here

The post What is Primecoin Coin? appeared first on Technology Geeks | Gadgets | IoT | News.

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"Janometry: 6--Butterfly Fractal Resonance," digital + acrylic, 2024, Reginald Brooks

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"Janometry: 12--Butterfly Fractal Resonance," digital + acrylic, 2024, Reginald Brooks

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"DMT_17,"digital + acrylic, April 19, 2024, Reginald Brooks

DMT = Divisor (Factor) Matrix Table

[rbrooksdesign]

"DMT_13," digital + acrylic, April 15, 2024, Reginald Brooks

DMT = Divisor (Factor) Matrix Table

[rbrooksdesign]

"DMT_19," digital + acrylic, April 21, 2024, Reginald Brooks

DMT = Divisor (Factor) Matrix Table