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Mathematics Illuminated

The Primes Online Textbook

Long studied for their mysteries, many of which remain unsolved, primes were, until relatively recently, solely the concern of mathematicians. With the explosion of digital communications and our dependence on Internet transactions, however, primes now play a pivotal role in other areas, as well.

1. Introduction

“Reason is immortal, all else mortal.”

-Pythagoras

“Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.”

-Leonhard Euler (1707-1783)

In 1974, astronomers sent a message into space from the Arecibo radio telescope in Puerto Rico in an attempt to broadcast the presence of our intelligent life to any other potentially sentient beings that happened to be listening. The message contained information about our planet’s location, the basis of our chemistry, and rudimentary information about our biological form. In sending this message, astronomers had to confront a deep problem: how does one communicate with another intelligent species that one knows nothing about?

ARECIBO MESSAGEItem 1671 / Frank Drake, ARECIBO MESSAGE OF 1974 (1974). Courtesy of Frank Drake.

At the time, astronomers thought that the best way to ensure the comprehensibility of the message was to use a concept fundamental to our own logical understanding of the world, namely, prime numbers. No known natural process creates prime numbers, yet they are at the very center of mathematical thought. The astronomers assumed that any beings intelligent enough to be able to listen to radio broadcasts would know about prime numbers. With this in mind, the message senders encoded a pictogram consisting of 1,679 bits; although the number 1,679 is not itself prime, it is the product of two prime numbers. The general idea was that any alien listener who receives a random number of blips might think to factor that number, just to get a sense of what it is. Choosing a number whose only factors are two primes might prompt the receiver to think two-dimensional,” a rectangular array, in other words, with one factor representing the number of rows and the other representing the number of columns. Using this information to arrange the pixels into a rectangle would reveal the picture. This, of course, assumes that an intelligent alien would recognize prime numbers to be special in some way.

This assumption is actually quite reasonable. Prime numbers, known to humankind for at least 20,000 years, represent a fundamental concept of mathematics that has provided rich grounds for study. Primes have been described as the atoms of arithmetic, the indivisible parts from which all other numbers can be constructed. Seemingly simple, they have provided some of the most challenging problems for those willing to explore their world. Long studied for their mysteries, many of which remain unsolved, primes were, until relatively recently, solely the concern of mathematicians. With the explosion of digital communications and our dependence on Internet transactions, however, primes now play a pivotal role in other areas, as well.

The advent and growth of the Internet and the “information age” have given unprecedented convenience to millions of technology users worldwide. Technological advances and their application in numerous fields have helped to shrink perceived distances and have done much to break down barriers that divide us. Peril often accompanies progress, however, and our modern Internet economy is not immune to risks. For example, when we make a purchase, pay bills, or check our bank balance online, how do we know that someone will not intercept the transmission and steal our personal information? One of the ideas we will see in this unit is how the properties of prime numbers, in combination with modular arithmetic, or “clock math,” are used to help keep our information secure.

In this unit we will see how primes are fundamental to mathematical thought. We will explore the seemingly simple concept that underlies their existence, and we’ll catch a glimpse of the mystery inherent in their distribution on the number line. Finally, we’ll take a brief look at how the modern standard of data security, RSA encryption, uses fundamental properties of primes to create virtually impenetrable codes.

2. Math at the Dawn of Time

TALLY STICKS

  • The earliest evidence of humans using numbers comes in the form of tally sticks.
  • The Ishango bone represents a level of early mathematics more sophisticated than simple counting.

Evidence suggests that the exact beginning of mathematical thought, even the origin of the concept of a “number,” predates the advent of written language. The earliest example of recorded mathematical symbols is a sequence of tally marks on the leg bone of a baboon found in Swaziland, dating to around 35,000 years ago. By contrast, the earliest known language writings date to around 6,500 years ago. It is not known for sure what the tally marks on the bone represent, but it is plausible that they represent a record of an early hunter’s kills. These tally marks may represent numbers in application, but there may also be evidence that early humans were interested in properties of numbers themselves.

ISHANGO BONE

Item 3060 / Oregon Public Broadcasting, created for Mathematics Illuminated, ISHANGO BONE (2008). Courtesy of Oregon Public Broadcasting.

Possible evidence of mathematics more sophisticated than counting comes from another bone dated ten thousand years younger than the Swaziland counting bone. Around 25,000 years ago, by the shores of Lake Edward (which today lies on the border between Uganda and Zaire), the Ishango people lived in a small fishing, hunting, and farming community. This settlement lasted for a few centuries before being buried in a volcanic eruption.

Excavations at this site turned up a bone tool handle with a series of interesting marks. The Ishango bone, as it is now called, has groups of markings, some of which represent primes. Although the exact meaning of these markings is still being debated, the current thought is that they represent some sort of lunar calendar. Regardless of the precise meaning of the markings, the artifact demonstrates that humans were thinking about mathematical concepts, perhaps even the concept of prime numbers, 25,000 years ago, well before the emergence of cities.

CULTURAL MATH

QUIPU USED FOR COUNTING

Item 2251 / Inca, QUIPU. USED FOR COUNTING (fifteen-early sixteenth century). Courtesy of Kathleen Cohen.

  • Mathematics arose independently in different forms across many cultures throughout history.

Jump 10,000 years forward and a bit to the east from the makers of the Ishango bone, and you’re in the emerging Egyptian and Fertile Crescent civilizations. These civilizations had deep understanding of mathematics and used it to achieve unequalled engineering feats.

MATHEMATICS TABLET

Item 2250 / Babylonian, MATHEMATICS TABLET SHOWING CALCULATIONS OF VOLUME (ca. 1635 BCE). Courtesy of Kathleen Cohen.

Babylonian clay tablets show an understanding of Pythagorean triples, centuries before the cult of Pythagoras appeared in Greece. However, these were not the only ancient civilizations to develop, presumably independently, familiarity with numbers and number relationships. Mathematical concepts may have spread naturally throughout Africa, and the Middle East, and Asia, but they also appeared early on in Central and South America.

CALENDAR RELIEF

Item 2253 / Mayan, CALENDAR RELIEF (fifth-ninth century). Courtesy of Kathleen Cohen.

Evidence suggests that the development of mathematics in early cultures was tied to specific purposes. Much of early mathematical thought was focused on representing and understanding the movements of the heavens, as is evident in the Mayan long count calendar and the constellations of the zodiac. Elsewhere, such as in ancient China, mathematics was put to use in bookkeeping and other business activities. Math’s development generally served practical purposes until about 600 BC, when the Greek philosophers began to explore the world of numbers itself.

3. Number for Number's Sake: The Greeks

PLAYING WITH NUMBERS

  • The Greeks studied numbers independent of their application to uses in the real world.

Prime numbers may have been of interest to the people of Ishango, but it was the Greeks who began to ask deep questions about them. The Greeks held high esteem for the pursuit of knowledge and, in particular, mathematical truth. One would not have expected such a high-level fascination from the illiterate and innumerate people who conquered the Aegean peninsula. Their conquests and the knowledge that flowed along their trade routes, however, enabled them to catch up quickly with the rest of the mathematical world.

While the rest of humanity was seemingly occupied with the more practical uses of mathematics, the Greeks were among the first to develop a mathematical world that was not necessarily tied to real-world applications. It was during this time that the concept of axiomatic structure emerged—mathematical proof, in other words. Thales, a mathematician who is said to have astonished his countrymen by correctly predicting a solar eclipse in the year 585 BC, is generally credited as taking the first steps toward focusing on the logical structure and principles behind mathematics. Described as the first philosopher and the first mathematician, he is definitely the first person to whom a specific mathematical “discovery” is ascribed, namely that an angle inscribed in a semicircle is a right angle. As is often the case with the emergence of new ideas, there is some dispute on this, however. Thales is generally given the credit for this discovery, although some claim that he simply re-packaged a previous Babylonian finding.

There is some debate as to whether it was Thales or the Pythagoreans who presided over the shift in mathematics from practical concerns to the development of general principles and ideas. The supposed motto of the Pythagorean group, “All Is Number,” encapsulates their preoccupation with both mathematical and numerological concepts. For example, they ascribed a gender to numbers, odd numbers being male and even numbers being female. Much of the mathematical tradition of ancient Greece, and, thus, of the civilizations that followed, stemmed from the obsessions of the Pythagoreans.

THE PRIMACY OF PROOF

  • Proof has long been one of the central ideas in mathematics.
  • Mathematical theorems, unlike scientific theories, last forever.

Chief among the Pythagorean concerns was the notion of proof. In philosophy it was possible to argue, as the Sophists did, both sides of a scenario and see that neither was a clear winner. Math, however, is different in that “truth” can be proved through a system of assumptions and allowed actions that show that a given statement must follow from initial postulates. In other words, in mathematics at least, there is indisputably a right answer, although it may not always be obvious. This clarity, and the comfort that it often brings, was of central importance to the Pythagoreans, and it represents a distinguishing feature of the field of mathematics.

Theorems proved by, or at least attributed to, early Greeks, such as the Pythagoreans, remain as true today as they were in ancient times. The same cannot be said for any host of other Greek beliefs from non-mathematical disciplines. One of the alluring features of mathematics is that it enables one to say definite things about reality. This aspect was, and continues to be, a major reason why people choose to study mathematics.

FIGURATE NUMBERS

  • Playing with the geometric structure of numbers led to early insights in number theory.

Playing with numbers—the exploration of numbers for their own sake—is perhaps the first step towards mathematical sophistication. The Greeks were fascinated by the different properties that certain numbers exhibited geometrically. If we represent each whole number by a collection of pebbles equal in count to that number, then many interesting relations and properties of numbers can be found by looking at the shapes that one is able to make with different arrangements of the collections.

Square Numbers

Square numbers are whole numbers that, when represented as a collection of pebbles, can form a square array. The first square number is 1, the second is 4, which can be portrayed as a 2 × 2 array, the third square number, 9, forms a 3 × 3 array, etc. The triangular numbers can also be represented by an interesting sequence of dot patterns, which is, in fact, the basis of their classification as “triangular” numbers.

Square Numbers

The squares and triangular numbers, in this context, are examples of “figurate” numbers—numbers that have “shapes.” Figurate numbers hold many interesting properties; for example, consider the 5 × 5 square.

5 x 5 square

Is it evident that the fifth square number, 25, represents the sum of the first five odd numbers?

first five odd numbers

We can decompose the square into nested L-shapes, also called gnomons, as shown above. It is, hopefully, straightforward to state that any square can be broken down in this way. It should also be clear that every gnomon represents an odd number, and that nested gnomons represent consecutive odds. Adding a gnomon to an n × n square increases the dimensions to (n+1) × (n+1), which is still a square. It is reasonable, then, to conclude that the sum of the first n odd numbers is n2.

In the online interactive, you will have the opportunity to play with figurate numbers, such as those we have discussed, and to discover interesting relationships between them. The study of figurate numbers leads quite naturally to primes, which we will investigate further in the next section.

4. Primes

RECTANGLES

  • The rectangle model of multiplication links a number’s geometric structure to its divisors.
  • Primes are numbers that cannot be represented by rectangles with both dimensions whole numbers greater than one.

Exploring a number’s geometric shape leads quite naturally to the notion of prime numbers. Just as before, we can consider a whole number to be a collection of pebbles. We could then address the question of whether it is possible to form a rectangle with the pebbles.

Let’s look at a collection of 12 pebbles:

12 pebbles

We can arrange these 12 pebbles into various rectangular arrays.

rectangular arrays

Note that the dimensions of each rectangle—the height and width—multiply to equal 12. We call each of these numbers representing a possible dimension a “divisor” of 12; so 12 has six divisors: 1, 2, 3, 4, 6, and 12. In general, we say that a is a divisor of N if for some whole number b, a × b = N.

Note that any whole number, N, can be represented by a 1 × N rectangle—a single row of pebbles.

1 x N row of pebbles

Some numbers, such as 12, 15, 20, and 100, can be represented by rectangles that are more interesting than a single row of pebbles.

12, 15, 20, 100

Other numbers, however, such as 5, 11, 17, and 101, can be represented only by the single-row type of rectangle.

single-row type of rectangle

In arithmetic, we call a number “prime” if it has precisely two divisors. These primes are the numbers, such as 2, 3, 5, 7, and 11, whose pebble representations can be arranged only into the single-row type of rectangle. Numbers with more than two divisors are called “composite.” Geometrically, these are numbers that can be represented in dot formations by more than one type of rectangle. Note that the number one, which has precisely one divisor, is considered to be neither prime nor composite.

FACTOR TREES

  • Factor trees reveal the prime decompositions of composite numbers.

A number that is prime has exactly two divisors, itself and one. If a number is not prime—composite, in other words—we can “factor” it to find all of its constituent prime “factors.” For example, the number 30 can be written as 6 × 5, which can in turn be written as a product of all-prime factors: 2 × 3 × 5.

2 x 3 x 5

Some numbers have multiple possible factor trees. Let’s consider the number 300, for example:

300

Note, however, that although these factor trees for 300 are different, the set of prime factors generated is the same: 3, 5, 5, 2, and 2.

Any composite number can be decomposed this way into a product of primes. In doing this, we see that primes can indeed be thought of as the “atoms,” or fundamental building blocks, of all numbers. Real atoms are the smallest individual pieces of an element, such as gold, that still retain all the properties of that element. In this analogy, a composite number is like a molecule. Breaking apart a molecule generates a collection of atoms of different elements, each of which cannot be broken down further. We perform an analogous breakdown when we decompose a composite number and express its prime decomposition via a factor tree. It is interesting to note that every composite number breaks down into a unique product of primes. How do we know this?

FUNDAMENTAL THEOREM OF ARITHMETIC

  • The Fundamental Theorem of Arithmetic states that every number has only one prime decomposition.
  • Primes are the “atoms” of arithmetic.

The fact that every composite number has a unique prime decomposition, a concept known as the fundamental theorem of arithmetic, is often taken for granted. In mathematics we should always be careful to question both our own assumptions and the assumptions of those who would tell us something. Such a questioning attitude derives from the previously mentioned importance of proof, pioneered by the Greek philosophers, logicians, and mathematicians. The tools for proving the fundamental theorem of arithmetic were first laid down by the great mathematician Euclid, who lived in Alexandria in the third century BC. These core concepts were not rigorously expressed, however, until Karl Gauss put them on a solid foundation in his Disquisitiones Arithmeticae, first published in 1801. Expressed in modern language, the fundamental theorem of arithmetic states that:

Every natural number greater than 1 can be written as a product of prime numbers in essentially just one way.

This may seem intuitive, even obvious, but it doesn’t have to be the case. We can imagine a number system in which the fundamental theorem of arithmetic does not hold. Take, for example, a set, S, consisting of the numbers {1, 4, 7, 10, 13, 16, 19,…, 3n+1,…}. Each number in this system is one more than a multiple of three. Suppose that these numbers are all we have to work with in performing arithmetic operations in this system. As with the natural numbers, we can have notion of prime and composite in this system—let’s call them “S-prime” and “S-composite” respectively.

s-composite

The number 10 has only two divisors in this system, one and itself. The number 16, on the other hand, has three: 1, 4, and 16. Numbers such as 4, 7, and 10 can be called “S-prime,” because they have exactly two divisors within the number system, S. Numbers such as 28 and 16 can be called “S-composite,” because they have more than two divisors in S. If S obeys the fundamental theorem of arithmetic, then no matter how we draw a factor tree for an S-composite, we should end up with the same set of S-primes. Is this the case? Let’s look at two factor trees of the S-composite number 100:

s-composite

Notice that 100 can be written as a product of S-primes in two different ways, 4 × 25 and 10 × 10. So, this demonstrates that the fundamental theorem of arithmetic does not hold for our number system S. This suggests to us that we cannot take this seemingly obvious property of natural numbers for granted— hence, Gauss’s proof that every natural number greater than one has a unique prime factorization is of great importance in the field of number theory.

Actually, finding a number’s prime decomposition—also known as the prime factorization—is relatively straightforward, provided we are aware that our number is divisible by some factor and we know what that factor is. Suppose, however, that we come across some large number and wish to find its prime factors. How would we do this? How would we know whether it even had prime factors other than itself?

If the situation were different, and we wished to multiply two large numbers, our task would be easy—we have good, efficient algorithms for multiplying numbers of any size. Factoring a large number, however, is exceedingly difficult. Most factoring methods involve some version of the “trial and error” strategy. Even for a relatively small number, such as 527, our only real choice is to try dividing it by potential factors until one of them divides it evenly. In this case, it wouldn’t take us too long to find that 527 = 17 × 31. For a very large number, however, going through all the possibilities could take an intractably long time.

If someone took two large prime numbers and multiplied them together and then asked us to find the prime factorization of that composite number, we would be in trouble. The number we are given has only two prime factors, and if we don’t know either of them, there is no good algorithm for finding them. This “one-way-street” aspect of multiplication and factoring will play a key role in our upcoming discussion of encryption. Before we tackle encryption, however, we should take a closer look at prime numbers, for they themselves hold a great deal of mystery.

5. Questions about Primes

AN INFINITUDE OF PRIMES

  • Euclid proved that, given a finite list of primes, one can always create a new prime not on that list, which implies that there are an infinite number of primes.

Let’s take a look at a partial list of primes, with spaces left to represent the composite numbers that occur between them:

_2 3 _ 5 _ 7 _ _ _ 11 _ 13 _ _ _ 17 _ 19 _ _ _ 23 _ _ _ _ _ 29 _ 31 _ _ _ _ _ 37 _ _ _41 _ 43 _ _ _ 47 _ _ _ _ _ 53…

The primes seem to be haphazardly distributed; that is, there is not a clear pattern to them. It seems also, with the exception of the occasional pair of “twin primes”—primes separated by only one number—that the gaps between adjacent primes generally tend to become larger as the numbers themselves get larger. From this evidence, one could plausibly think that at some point the stream of primes dries up and that there might be no primes above some given number. The question of how many primes there are is an interesting one that arises quite naturally. Is the list finite or infinite?

Euclid thought about this problem and found a way to show that, in fact, there are an infinite number of primes to be found. He did this by demonstrating that from any finite collection of primes it is always possible to find one more. For example, take the finite set {2, 3, 5}. Euclid suggested multiplying all members of the set together and adding one. For our set, this gives us 31 (2 × 3 × 5 + 1 = 31), which cannot be evenly divided by any of the primes on our list, for it always gives a remainder of one. Hence, 31 is a “new” prime.

Remember that in math it’s good to keep a skeptical, questioning attitude concerning pronouncements. Perhaps that example was just an anomaly, and maybe 2, 3, 5, and 31 are the only primes that exist. Applying Euclid’s test strategy to this expanded set, we would multiply them together and get 930, to which we add one to end up with 931. Dividing 931 by 2, 3, 5, or 31 always leaves a remainder of one. Does this mean that 931 is prime? No, not necessarily, but it does mean that none of the numbers 2, 3, 5, and 31 is a factor of 931. It might not be obvious, but 931 is actually the product of 7, 7, and 19. So, even though 931 is not itself prime, it is a composite number whose prime factorization reveals new prime numbers (7 and 19) not in our original list. Again, our list was incomplete.

The point is that Euclid showed that multiplying together a finite list of primes and adding one will always produce a new number which, if not itself prime, factors into new primes not on the list. We can always find a new prime using this method. This implies that any finite list will never include all of the prime numbers; therefore, the list of primes must be infinite!

The method we employed above will always uncover at least one new prime, but it could very well miss some along the way. In our first example above, we missed a few primes in between 5 and 31, namely 11, 13, 17, 23, and 29. This method obviously will not produce an exhaustive list of primes; it serves only to generate a new prime or two, given a list of starting primes.

In the second example above, we found that this method produces a number that may not necessarily be prime. This suggests to us that it would be nice to have an efficient, fail-safe method for determining whether a number of any size is prime or not.

IS IT PRIME?

  • Determining whether or not a number is prime is a non-trivial task.

The most straightforward way to know for sure whether or not a number is prime is to test, systematically, its divisibility by all the numbers less than its square root. For example, if 1001 were composite, then it would have to be the product of two numbers, a and b, neither of which is one. One of these two numbers would have to be smaller than square root 1001—they can’t both be larger—so we need only check for divisors up to 32, which is the square root of the closest square number larger than 1001 (i.e., 1024). Because every number is composed essentially of prime factors, we need only check the prime numbers less than 32; these are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. This is a feasible number of potential divisors to check.

There are a number of familiar divisibility tests that can be used to determine if a number is divisible by most of the single digit numbers. One rule that is widely known concerns divisibility by 3: a number is a multiple of 3 if all of its digits sum to a multiple of 3. For example, the digits of 78 sum to 15, and 78 factors to 3 × 26. What about a number such as 221, which doesn’t fall in line with any of the short-cut divisibility tests? To determine whether or not it is prime, using the brute force method described above, we would have to divide it by all of the primes less than its square root, which is somewhere between 14 and 15. Were we to do this, we would find that our first five tries (checking 2, 3, 5, 7, and 11) would be unsuccessful, all yielding remainders. Only on the sixth try (dividing by 13) would we find that 221 is indeed factorable into 13 × 17.

The brute force method of testing whether or not a number is prime can prove to be quite a headache for large numbers. For example, using a computer that can perform tens of billions of operations per second, the test for a 100-digit number would take about 1040 seconds. This is quite a bit longer than the current estimated age of the universe, about 12 billion years.

There are various tests, other than the brute force method, that can tell us about the primeness of larger numbers. One such test uses Wilson’s theorem, which states that if a number, p, is prime, then (p-1)! + 1 is a multiple of p. For example, we know that the number 7 is prime. Then, according to Wilson’s theorem, (7-1)! + 1 = 6! + 1 = 721 should be a multiple of 7. Indeed it is: 721 = 7 × 103. Unfortunately, as we will see in Combinatorics Counts, computing factorials is practically infeasible for large numbers—e.g., 1000! is a number 2,568 digits long, and 1,000,000! is 5.5 million digits long. Wilson’s theorem is useless for the sizes of numbers about which the question of primeness has not already been settled.

THE STRUCTURE OF THE PRIMES

  • The answer to whether or not there is a pattern behind the primes has eluded mathematicians for millennia.
  • One can find interesting examples of both structure and randomness in the primes.

Because directly testing whether or not an arbitrary large number is prime is either very time-consuming or not very reliable, mathematicians have sought to determine the primeness of a number in a totally different way—that is, by looking for a pattern to the primes. If a pattern can be established, then it should be possible to determine where primes do and do not occur. Describing
exactly where primes should occur on the number line would not only provide an elegant way to know whether or not a number is prime—it would be a beautiful result in itself.

The first person generally credited with attempting to describe a pattern behind the primes was the Greek philosopher and mathematician Eratosthenes. He is credited with creating a method, now called the Sieve of Eratosthenes, for systematically identifying all prime numbers. His method is simple: begin with a finite list of whole numbers and eliminate all the multiples of 2 greater than 2, then all the multiples of 3 greater than 3, then all the multiples of 5 greater than 5, and so on until the only numbers left are the ones that are not multiples of any other numbers: these will be the primes.

Sieve of Eratosthenes

This method, however, is not terribly different or better for larger numbers than the brute force method we discussed earlier. It does, however, provide some possible clues to the structure of the primes available to us if, rather than looking at each number individually, we look at the structure as a whole. For example, can we find a pattern behind the primes by focusing on the spacing between them? This strategy is similar to looking at the negative space around a picture instead of looking directly at the picture itself.

prime deserts

A Cup or two faces?

We can think of the space between primes as “prime deserts,” strings of consecutive numbers, none of which are prime. There is a trick, however, for finding prime deserts of whatever length we can think of. For example, 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040. This number is, of course, not prime—but neither is 5,040 + 2, nor 5,040 + 3, nor 5,040 + 4. … nor 5,040 + 7. In general, for any number N, the string N! + 2, N! + 3,…N! + N identifies a string of N-1 consecutive composite numbers. Setting N = 1,000,000 shows that it is possible to find 999,999 consecutive whole numbers, none of which are prime.

Paradoxically, just as we can find prime deserts of whatever length we choose, we can also find strings of primes—sequences of primes that are evenly spaced on the number line—of whatever length we choose. A good example of strings of primes are the sets called “twin primes.” These are pairs of primes that are spaced two numbers apart, such as 11 and 13, or 17 and 19. The set of 3, 5, and 7 forms another such string; again, each number is equally spaced from the others. In yet another example, the numbers 5, 17, 29, 41, and 53 are all prime and are all spaced evenly at 12 numbers apart. Such strings are called arithmetic progressions. In recently completed research done by Terence Tao at UCLA and Ben Green at the University of Bristol, arithmetic progressions of any finite length were proven to exist. Not only are there prime deserts of arbitrary length, but there are also strings of equally spaced primes of whatever finite length we choose. If there is a pattern to the primes, it would have to account for these paradoxically bizarre features.

A tantalizing clue as to the fundamental structure of the primes was discovered by Marin Mersenne, a French music theoretician and mathematician of the 16th and 17th centuries. Specifically, in his study of number theory he became interested in the powers of 2 and found an odd coincidence.

chart of powers of two

Notice that, in the above chart, one less than a prime power of 2 yields a prime number. This seems to give us a potential way to predict where prime numbers fall, and it also shows a startling relationship. If the zero power is disregarded, these so-called Mersenne primes appear in the 2nd, 3rd, 5th, 7th , etc. positions in the sequence—in other words, in the prime positions. They are prime numbers in prime positions! Alas, this is but a coincidental occurrence in the smaller numbers, as the pattern does not hold for all prime powers of 2. For example 211 – 1 is equal to 2,047, which is the product of 89 and 23.

Mersenne primes, nonetheless, are still very important in the modern study of numbers. In fact, the largest prime number that we know about is a Mersenne prime. As of 2006, this number was 232582657 − 1, a number that is 9,808,358 digits long. It would take an average person more than 100 days just to read it! We should note that in addition to generating non-prime numbers, Mersenne’s method also is not guaranteed to predict the positions of all of the primes.

We have so far looked at a few different approaches to predicting where primes should be found on the number line. From the brute force method of Eratosthenes to the elegant ideas of Tao, Green, and Mersenne, we have seen that the primes do not give up their secrets easily. Perhaps, in asking where exactly each prime falls on the number line, we are asking the wrong question. Gauss, of fundamental theorem of arithmetic fame, took a different approach. Finding a straightforward attack daunting, he examined matters from a different perspective. He examined the density of primes below a certain number.

Graph of Prime Destiny

He found that the density of primes—that is, a measure of the number of primes in relation to all the numbers below any given number—is fairly constant. By looking at tables of the number of primes below a given number, n, Gauss observed that this measure always seems to be approximately the ratio n over log n , which implies that the nth prime is approximately n log n. This finding, while not exactly what we were looking for in terms of predicting where primes appear on the number line, represents a significant result in the search for a “law of the primes.”

RIEMANN’S HYPOTHESIS

  • Riemann, following in the footsteps of Euler, thought that the pattern of the primes was tied to the zeta function, which is a generalization of the harmonic series.

Gauss’s conjecture about the distribution of the primes was an important first step. By looking at the problem from a different perspective, that of prime density, and by examining functions that would give not a prime number directly, but rather the density of primes below a specified number, Gauss opened the exploration of the primes to new disciplines of mathematics. He was not the only one working on this problem, however. The Swiss mathematician Leonhard Euler, working in a similar vein, introduced yet another approach to discovering a pattern behind the primes.

Euler was fascinated by a relative of the harmonic series known as the zeta function. The harmonic series is simply the sum of the reciprocals of all the whole numbers.

harmonic series

What tipped Euler off that the harmonic series’ cousin, the zeta function, might be useful in exploring the structure of the primes was that it contains every natural number.

zeta function

Notice that the zeta function expresses the sum of the reciprocals of the xth power of every number. We can input any value we choose for x and study the behavior of the series. For any value of x less than or equal to one, the series diverges—grows infinitely large. For values of x greater than one, the series converges to a finite value. However, just because we know that the series converges does not mean that we can always find the value upon which it settles.

Euler played with the zeta function and found that he was able to express every element of the infinite sum as a product of the reciprocals of prime numbers. This insight tied the study of prime numbers to the study of infinite series and paved the way for one of the most elusive hypotheses in mathematics.

elusive hypotheses

Euler’s Factorization

Euler's Factorization

Bernhard Riemann, one of the most influential mathematicians of the 19th century, built upon Euler’s discovery, specifically exploring the consequences of using complex, or imaginary, numbers as the inputs to Euler’s version of the zeta function. This effectively constructed a larger landscape of numbers to explore, one that gave a better perspective on how the elusive primes might be distributed along the number line. Riemann proposed that the distribution of the primes was tied to the zeroes of the zeta function when considered over the complex numbers.

If you imagine the entire collection of complex numbers to be a landscape with hills and valleys, the zeta function is like a road that traverses it. We can think of the zeroes of the zeta function as being the points at which the elevation of the road is at sea level. Riemann said that these points mark where the primes should occur.

Riemann’s hypothesis was groundbreaking, but it remains a hypothesis, unproven, to this day. Given the history of the problem it purportedly solves, and the centuries of effort that have led to it, the proof of Riemann’s hypothesis will be quite valuable. As of this writing, there is a $1,000,000 reward available to any person who shows definitively whether or not it is true or untrue.

The preceding discussion should have given you a sense that primes are intrinsically fascinating. They are the fundamental basis of arithmetic—yet, they are extremely mysterious. They surprisingly represent both the most basic and the most cutting-edge aspects of mathematics. On a daily basis, however, we are not typically conscious of them. Because of this, it is tempting to believe that playing with primes is the purview of pure mathematics, with little relevance to our daily lives. This couldn’t be further from the truth, especially in our modern age when numbers are used to transmit all types of information, the security of which is of great importance. Primes provide the building blocks of encryption schemes that protect our most sensitive and valuable data.

6. Encryption

HISTORY

• The need to send private messages has ancient roots.
• Most historical encryption schemes used private keys.

When sending sensitive information, such as logging onto a bank account via the World Wide Web, we want the intended receiver to be able to “read” the message, and we also want any unintended recipient (or thief) of the message to be thwarted. To accomplish this, we need a system of encryption, one that will be impervious to even the cleverest of hackers. Online transactions are only the most recent example of situations in which information must be protected while in transit. There have been many different systems of encoding throughout the years, many of them mathematical in nature.

Mathematical encryption relies on concepts such as modular arithmetic and the fundamental properties of prime numbers to make messages incomprehensible to unintended recipients. Obviously, encryption requires operations that are easy to perform one way (i.e., encoding), yet nearly impossible to perform the other way (i.e., decoding) without a key. Prime numbers can help us with this. Recall that multiplying two primes is easy, whereas factoring a product of two unknown primes can be extremely difficult and time-consuming.

The problem of sending a vital message to someone without it being discovered, or deciphered, is an ancient one. This need crops up in a variety of situations, but an all-too-common need for encryption has arisen time after time on the battlefield. When attacking an enemy, it is critical to be able to send coordinating information to your involved units without the enemy discovering your plans. By encrypting messages in some fashion, you hope that, even if a message is intercepted, its meaning will be inaccessible to unintended readers. Encryption schemes have come in many forms throughout the ages. Many early schemes could hardly be called “encryption”—”concealment” would be a more appropriate term. A particularly striking example is the story of Histaiaeus, a 5th-century-BC Greek provincial ruler, or tyrant, who wanted to encourage a fellow tyrant of a neighboring state to revolt against the Persian king, Darius. To convey his instructions securely, Histaiaeus shaved the head of his messenger, wrote the message on his scalp, and then waited for the hair to re-grow. The messenger, apparently carrying nothing of interest, traveled without being harassed. Upon arriving at his destination, the messenger shaved his head and pointed it toward the intended recipient. Although this was perhaps clever for its time, such a system would quickly be compromised as word of its use spread.

CAESAR CIPHERS

• The simplest encryption scheme is to jumble the letters of the alphabet according to some rule.

Clearly, a system of simply concealing one’s message is effective only up to a point—a thorough search is all that is needed to expose the scheme and intercept the message. To ensure message security, one would want an interloper, were he or she to find the communication, to be baffled by what it actually says. A simple scheme, whose first use is attributed to Julius Caesar, is to replace each letter of the alphabet with a different letter according to some rule. For example:

rule of add three

This scheme replaces each letter with the letter that comes three letters later in the sequence of the alphabet. In effect, this encoding scheme just shifts the alphabet three letters to the left, with the substitutions for the last three letters coming from the beginning of the sequence. Our key, let’s call it “k,” is 3. Let’s now shift our thinking from letters to numbers; we can easily assign each letter of the alphabet to a corresponding number as follows:

rule of add three

Applying the same “shift by 3” that was just used in the preceding example to this new system creates the following encryption scheme:

rule of add three

To encrypt a letter, we simply add three to the number with which it was originally paired. For example:

A, originally paired with “1” gets shifted, or encrypted, to “4.”
and
M, originally paired with “13” gets shifted to “16.”

When we get to X, however, we have a problem. Following the current encryption scheme, X, originally associated with “24,” would get shifted to “27.” However, 27 isn’t anywhere in our list of possible numbers. So, we can wrap around and start over at 1 after hitting 26 instead of using more numbers. To express this adjustment mathematically, we could write:

24 + 3 = 1

This strange sort of arithmetic might seem somewhat familiar because it is related to how we apply arithmetic to cyclical time. For example, if it is four o’clock and we add twelve hours, it is again four o’clock, ignoring the A.M./P.M. difference. This “clock math” is known as “modular arithmetic.”

MODULAR ARITHMETIC

  • Modular arithmetic is math that incorporates “wrap-around” effects.

Modular arithmetic is a key to understanding modern forms of encryption, and it also demonstrates interesting properties of prime numbers. It incorporates”wrap around” effects by having some number other than zero play the role of zero in addition. For example, on an analog clock:

Analog Clock

The number 12 behaves like zero, because adding 12 hours to any time (again, ignoring A.M. or P.M. differences) doesn’t change anything. A typical addition problem in this scheme would be:

3 + 12 = 3

To write a number in this system, we have to be conscious of how many multiples of 12 it contains. A number such as 5 has no multiples of 12 in it, so we would simply write 5. The number 17, however, is 5 + 1(12), which is also equal to 5 in this system. Similarly, the number 29 is 5 + 2(12), which is also equal to 5, again because the number twelve is acting like zero in this system.

In such a system, 12 is called the “modulus.” To describe the number 29, we would say that it is “congruent to 5 modulo 12”, or 29 ≡ 5 mod 12.

Note that:
5 ≡ 5 mod 12
17 ≡ 5 mod 12
29 ≡ 5 mod 12
etc.

Any number of the form 5 + n(12) will be congruent to 5 in this system.

With this new foundation in modular arithmetic, let’s return to the Caesar cipher from before. Previously, the integrity of the code was dependent on the security of our key (i.e., only we and our intended correspondent can know it). The integrity of our code is vulnerable, however, to anyone who suspects that we are simply shifting letters. If our message is sufficiently long, one could rather easily figure out our key by examining letter frequencies. That is, it is highly likely that the letter that appears most frequently in our code represents the letter “E,” which is the most commonly used letter in English. After the code is “cracked” in this way, the whole scheme crumbles because every letter is shifted by the same amount!

Let’s shift our thinking back to a number code again. To increase security and make our encryption “uncrackable” by a simple shift, we could multiply numbers instead. In this case, the code key would be the number by which we choose to multiply the original numbers. Multiplying in modular arithmetic is just like multiplying in normal arithmetic, except that whenever we encounter a product that is greater than some multiple of our modulus, we use the wraparound effect. For example, using our clock from before (mod 12), 4 × 2 ≡ 8 mod 12 (no surprise), but 4 × 3 ≡ 0 mod 12, and 4 × 4 ≡ 4 mod 12. A multiplication table for mod 12 looks like this:

Mod 12 Multiplication Chart

Notice first that we are using zero in place of the number twelve. Also, notice that some of the columns do not contain all of the numbers in our system, namely all of the numbers between 0 and 11 inclusive. As you can see, this is true for the rows, as well. If we were to choose these numbers as our key for encrypting a message, we would be setting our recipient up for confusion. For example, let’s say that our encryption scheme involves multiplying the numbers that make up our message by 4 in this mod 12 system. For simplicity’s sake, let’s also say we are using a 12-letter alphabet.

Corresponding 12 Letter Alphabet

This would serve as the top row of our mod 12 table. Because we are using k = 4 as our key, we are going to look to the 4th row of the table.

Table Mod 12 Row 4 Key

The letter D, which is 4, would be encoded as:
4(letter) × 4(key) ≡ 4 (encrypted letter) mod 12

G would be:
7(letter) × 4(key) ≡ 4 (encrypted letter) mod 12

J would be:
10(letter) × 4(key) ≡ 4 (encrypted letter) mod 12

The same goes for the letter A.

So, suppose that the message we want to encrypt is “GD JA.” Following this encryption scheme, the message would be encoded as “4444.” How would the recipient know which 4’s represent D’s and which are supposed to be J’s, G’s, or A’s? It would be impossible for our recipient to decode our message. For this reason, choosing 4, or any other key for which the row does not contain every number is a bad idea!

Note, however, that 7 contains all the numbers in its corresponding row and column. This means that using 7 as a key would give each letter a unique number in our encryption scheme.

Seven is not the only number we could choose, of course. A quick check of the table shows that both 5 and 11 would work, as well. Notice also that 5, 7, and 11, the three numbers that have all the potential code numbers present in their rows and columns, share no common factors with the modulus, 12. This is because they are prime but are not factors of 12. The numbers 2 and 3 are also prime, but they are factors of 12, so their rows and columns include repetitions. This suggests that prime numbers have some interesting properties in modular arithmetic, especially as it relates to making sure that every letter has a unique encryption. The power of prime numbers in codes, however, does not stop there.

PRIME MODULI

  • Using a prime modulus ensures that all reciprocals exist.
  • Arithmetic with prime moduli forms the theoretical underpinnings of RSA encryption.

To maximize the potential number of encoding keys, we would ideally like to have a table that includes every possible value in every column and every row. With such a table, any key we choose will accurately transmit our message— that is, our recipient will be able to decode the message with no ambiguity. We noticed that keys that share factors with the modulus, or “size of our clock,” are no good. So, what if we choose a modulus that is unlikely to share factors with other numbers? A prime number has no factors other than 1 and itself, so it seems that any prime would be a good choice of modulus in this situation.

For example, the mod 7 multiplication table looks like this:

Mod 7 Multiplication Table

This modulus would correspond to a seven-letter alphabet:

Mod 7 Alphabet

Would a system such as this be harder to crack? Looking at our mod 7 table, we can see that for a key of 5, which again means that we look at row 5, the letter D gets mapped, or encoded, to 6. If someone were to figure out, perhaps by analyzing letter frequencies, that D is represented by 6, and they knew that our modulus, the size of our alphabet, was 7, then they could set up the following congruence: 4 × k ≡ 6 mod 7. With a little more thought, they could figure out that k must be represented by 5. So, it seems that this encryption system is somewhat more difficult to crack than the additive Caesar cipher, but it is by no means impenetrable!

An advantage of having a prime modulus is that every “fraction,” mod p, will exist. Think for a minute back to mod 12.

Mod 12 Multiplication

Can we find a number, that, when multiplied by 5, for example, yields 1? In other words, does 5 have a reciprocal? This number would have to be our equivalent of “one fifth.” Looking at our mod 12 table, we can see that this number is 5. So, 5 plays the role of “one fifth” because 5 × 5 ≡ 1 mod 12. Does every number in mod 12 have such a reciprocal fraction? Consider the number 2, and notice that nowhere in the rows or columns marked by 2 does the number 1 appear. This means that in the mod 12 system, there is no number that you can multiply 2 by and get an answer of 1—the 2 has no reciprocal. In other words, in the mod 12 system, not all fractions exist!

Now let’s look at our mod 7 table again.

Mod 7 Multiplication

We figured out that because 7 is prime, every number appears in every row of the table. Therefore, the number 1 appears in every row, which implies that every number has an associated reciprocal. In addition to a maximal choice of keys, the existence of every reciprocal is necessary in order to create an encryption scheme that is much harder to break than anything we have seen so far. Reciprocals are critical because instead of adding or multiplying in a given modulus, we will need to use exponents to create a strong encryption scheme. Specifically, we will be raising powers to powers, such as (23)4, which is equivalent to 23 × 4, or 212.

Suppose that our message is some number, M. In this strong encryption scheme, we could raise M to a power, e, to encrypt it. If another number, d, is the reciprocal of e, then it can be used to decrypt Me as follows:

(Me)d = Me × d = M1 = M

In order for this strategy to work in the modulus that we are using, every number, e, must have a reciprocal, d. This will be true only if our modulus is prime. We’ve now caught a glimpse of the unique properties of primes in modular arithmetic. This understanding is an important base from which we can launch our examination of the modern standard for data encryption, RSA , in the final section.

7. RSA Encryption

PUBLIC KEY ENCRYPTION

• RSA encryption is the modern standard for sending secure information.
• RSA is based on the modular arithmetic of primes.

In the mid 1970s, three MIT researchers, Ron Rivest, Adi Shamir, and Leonard Adleman, discovered a new method of encryption that relies on the properties of primes and modular arithmetic. This system, RSA (initials of the discoverers), has remained secure for over 20 years, although countless people have attempted to breach it. In 1982, Rivest, Shamir, and Adleman founded RSA Security, a company that would go on to provide the standard in data encryption used worldwide on the Internet.

RSA encryption relies not on one key, as in our previous Caesar cipher examples, but on two. One of these keys is made public, and the other is kept private. If you wish someone to send you an encrypted message, you simply tell them your public key, and they can then encrypt their message so that only you can read it. They do not need to know your private key for this system to work. Here is a brief, and somewhat simplified, description of the process.

As we have seen throughout this unit, multiplying two primes together is easy to do, but factoring a large number into primes is a nightmarish trial-and-error scenario if the number has only two large primes as factors. This property of primes provides the basis of the RSA encryption scheme.

In general, to encrypt a message using the RSA method, we choose two large primes, p and q, and multiply them to produce a number N. The number N will be the modulus of our system and will be made public. This is fine; because p and q are both large and prime, their product, N, will be practically impossible to factor and can be confidently made public knowledge. We will now use p and q to make our public and private keys.

To make our keys, we first subtract 1 from both p and q and then multiply the results: (p-1)(q-1) = T. At this point, we will choose our public key, which can be any number less than T that shares no factors with it. This public key is what we referred to in the previous section as e. To construct our private key, we need to identify a number, d, such that when it is multiplied by e, the public key, it is congruent to 1 mod T. In other words, the following congruence must hold:

d × e ≡ 1 mod T

Now, if we want someone to send us a message that only we can read, we tell them the modulus, N, and the encryption key, e. They then convert their message from letters to a series of number strings, or “words,” just as we have been doing in previous sections of this text. We must be careful that none of the words are larger than the modulus, which would result in some words being indecipherable. Let’s say this word is the number M. The encrypter raises each word to the “eth” power, mod N. This new word, let’s call it C, is now encrypted and can be sent to us. In mathematical terms, we have:

C ≡ Me mod N

We receive the coded number C. To decrypt it, all we have to do is raise it to the “dth” power, mod N. This works because Cd ≡ (Me)d mod N, and Rivest, Shamir, and Adleman were able to show that Med ≡ M mod N. Recall that we chose e and d such that, d times e triple equal 1 mod T (d × e = 1 mod T), so there is some mathematics hidden in this statement! Thus, our private key, known
only to us, undoes the encryption of the public key. Anyone who knows e and N can send us a message that only we, because we know d, can decrypt. To get a better idea, let’s look at an example.

A WORKED EXAMPLE OF RSA ENCRYPTION

Real RSA encryption uses very large numbers, which would be very difficult— not to mention less than illuminating—to use here. In this example, we’ll use smaller numbers that are not at all realistic but that illustrate how the method works.

First, let’s choose the primes that form the foundation of our scheme, p and q:
p = 17 and q = 19

By multiplying p and q, we get our modulus, N:
N = 17 × 19 = 323

Now, we find T by subtracting 1 from both p and q and multiplying:

T = (p-1) × (q-1) = (16) × (18) = 288
Next, we can select the encryption key, the public key, e, so that it has no common factors with 288.
Let’s let e = 11.

To find the decryption key, d, we need a number that, multiplied by e, gives a product that is congruent to 1 mod 288. Expressed mathematically, we need to find a number d such that:
11 × d = 1 + n(288)

Using trial and error, we find that 131 works because 11 × 131 = 1 + 5(288).

So, our public key is N = 323 and e = 11.

The private key is d = 131.

If someone wants to send us the message “ABC” securely, using this scheme, we tell them N and e. They first convert “ABC” to “123” and then do the following arithmetic:

123e = (123)11 mod 323 = 81

The sender could then send the message, “81,” over a public line of communication with confidence.

To decrypt the code, “81,” we can use N and d as follows:

81d = (81)131 mod 323 = 123

The message “81” becomes “123” after decryption, which we can then easily convert to “ABC,” which was the intended message.

Notice that in order to break this scheme, a hacker would have to find the two numbers that when multiplied together yield 323. The square root of 323 is a little less than 18, so they would have to try a maximum of 7 divisors before they would be guaranteed to break the modulus into the original primes that were used to find the public and private keys. Using a computer, this would not be difficult, so real RSA encryption uses numbers that are sufficiently large so that even the fastest computers would take longer than a human lifespan to factor them. It is upon this foundation, the difficulty of factoring products of two large prime numbers, that modern data encryption rests.

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