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.
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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.
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