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— CH. 1 · INTRODUCTION —

Prime number

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  • The number 5 cannot be broken apart. Write it as a product and the only options are 1 times 5 or 5 times 1, each of which still leans on the 5 itself. The number 4 has no such defense. It folds neatly into 2 times 2, two smaller numbers locked together. This is the dividing line that defines a prime number, a natural number greater than 1 that is not the product of two smaller natural numbers. A number greater than 1 that is not prime is called composite. From this plain rule grows one of the oldest and deepest subjects in mathematics. Why did the ancient Greeks call these numbers protos arithmos, and why did Euclid prove around 300 BC that they never run out? Why does a question as simple as Goldbach's conjecture stay unsolved for centuries, even after being verified for staggering ranges of numbers? And how did a field that mathematicians once prized for having no use at all become the hidden machinery behind public-key cryptography? The answers run from a papyrus dated to around 1550 BC to a Mersenne prime with more than 41 million digits found in October 2024.

  • Every integer larger than 1 can be written as a product of one or more primes, and that is why mathematicians call primes the basic building blocks of the natural numbers. This statement is the fundamental theorem of arithmetic. It says more than that a factorization exists. It says the factorization is unique. Any two prime factorizations of the same number contain the same primes in the same quantities, differing only in the order you write them. A single prime factor can appear more than once, and exponentiation groups those repeated copies together. Many proofs of this uniqueness rest on Euclid's lemma. If a prime divides a product of two integers, then it must divide at least one of those two integers. The reverse holds as well. If a number always divides at least one factor whenever it divides a product, that number must itself be prime. This building-block idea also explains a small set of useful shortcuts about which numbers can be prime. No even number above 2 is prime, since any such number splits off a factor of 2, so every prime past 2 is odd. In the decimal system, every prime larger than 5 ends in 1, 3, 7, or 9, because numbers ending in 0, 2, 4, 6, or 8 are even and numbers ending in 0 or 5 are divisible by 5. The first twenty-five primes, all those below 100, begin 2, 3, 5, 7, 11 and end at 89 and 97.

  • Most early Greeks did not even regard 1 as a number, so they never asked whether it was prime. Scholars including Nicomachus, Iamblichus, Boethius, and Cassiodorus treated primes as a subset of the odd numbers, which left 1 outside the question entirely. Euclid and most other Greek mathematicians, by contrast, did count 1 as prime. Attitudes drifted over centuries. Medieval Islamic mathematicians mostly followed the Greeks in denying that 1 was a number at all. By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and by the 17th century some listed it as the first prime. Christian Goldbach put 1 among the primes in his letters to Leonhard Euler, though Euler did not agree. The habit lingered for a long time. Derrick Norman Lehmer included 1 in his list of primes below ten million, published in 1914, and lists with 1 as a prime were still appearing as late as 1956. The reason 1 was eventually exiled into its own category as a unit is practical. If 1 counted as prime, the fundamental theorem of arithmetic would break, because any number could absorb extra copies of 1 and gain endless factorizations. The sieve of Eratosthenes would collapse too, striking out every multiple of 1 and leaving only the number 1 behind.

  • Euclid's theorem holds that the primes never end, and the first known proof of it is credited to him. The argument is a trap built from any finite list of primes you care to name. Multiply all of them together and add one. By the fundamental theorem of arithmetic, this new number has its own prime factorization. Yet it leaves a remainder of one when divided by any prime on your original list, so none of its prime factors can be among them. Every finite list is therefore incomplete, which means the supply must be infinite. The numbers formed by adding one to the products of the smallest primes carry the name Euclid numbers. The first five of them are prime, but the sixth turns out to be composite. Euclid's was only the first of many proofs. Euler supplied an analytical one, Goldbach built a proof from Fermat numbers, Furstenberg constructed one out of general topology, and Kummer argued by contradiction. The endlessness of the primes raised a harder question once it was settled. If they never stop, how are they spread out, and is there any pattern to where the next one falls?

  • No known simple formula separates the primes from the composites, and there is no non-constant polynomial, even in several variables, that produces only prime values. There are stranger constructions that do encode the primes. One formula built on Wilson's theorem generates the number 2 repeatedly and every other prime exactly once. A system of Diophantine equations in nine variables has a parameter that is prime precisely when the system has a solution in natural numbers, which yields a single formula whose positive values are all prime. Mills' theorem and a theorem of Wright provide real constants that generate primes through the floor function, but these are useless in practice, because you must already know the primes to compute the constants. Where exact formulas fail, statistics succeeds. The prime number theorem, proven at the close of the 19th century, says roughly that a large number's chance of being prime is inversely proportional to its number of digits, that is, to its logarithm. The road to that theorem was long. Legendre and Gauss conjectured the asymptotic count of primes at the start of the 19th century. Pafnuty Chebyshev proved Bertrand's postulate in 1852, guaranteeing a prime in certain intervals. Bernhard Riemann's 1859 paper on the zeta-function sketched the outline that Hadamard and de la Vallee Poussin completed in 1896.

  • All four of Landau's problems from 1912 remain unsolved, a reminder that simple-sounding questions about primes can resist proof for over a century. Goldbach's conjecture is among them. It claims that every even integer greater than 2 is the sum of two primes, and it has been verified for enormous ranges of numbers without ever failing or being proven. Weaker results have fallen. Vinogradov's theorem shows that every sufficiently large odd integer is a sum of three primes, and Chen's theorem shows every sufficiently large even number is a prime plus a semiprime. The twin prime conjecture asks whether infinitely many pairs of primes differ by 2, and Polignac's conjecture extends the idea to every fixed even gap. Prime gaps, the distances between consecutive primes, hold their own surprises. Arbitrarily large gaps must exist, yet they arrive far sooner than the simplest argument predicts. The first prime gap of length 8 sits between 89 and 97. Progress on these questions has been real. The Green-Tao theorem of 2004 showed there are arbitrarily long arithmetic progressions made entirely of primes. Yitang Zhang proved in 2013 that infinitely many prime gaps are bounded in size.

  • British mathematician G. H. Hardy prided himself on doing work with absolutely no military significance, and for a long time the study of primes was the textbook case of pure mathematics. The only application anyone could point to was using a prime number of gear teeth to spread wear evenly. That self-image broke apart in the 1970s, when it became public that primes could anchor public-key cryptography and the RSA cryptosystem. RSA leans on a stark imbalance. Multiplying two large primes is easy, but recovering those primes from the product alone is hard, and that asymmetry guards the system. The Diffie-Hellman key exchange relies on a similar gap, between fast modular exponentiation and the difficult reverse problem of the discrete logarithm. Today 2048-bit primes are common in such systems. The work of generating large random primes drove the study of primality testing. Trial division divides a candidate by each integer up to its square root, declaring it composite at the first clean division and prime otherwise. To test 37, you divide only by the primes 2, 3, and 5, each leaving a remainder, so 37 is prime. That method is too slow for large numbers, where the number of tests grows exponentially with the digit count, so it now serves mainly to strip out small factors before heavier methods take over.

  • Olivier Messiaen used the primes 41, 43, 47, and 53 in the third etude of his Quatre etudes de rythme, titled Neumes rythmiques, layering motifs of those lengths to build rhythms that never quite repeat. The French composer called this approach inspired by the movements of nature, movements of free and unequal durations, and he reached for it again in works such as La Nativite du Seigneur from 1935. Primes have drawn writers as much as composers. In his novel Contact, Carl Sagan proposed prime factorization as a way to build image planes in messages to aliens, an idea he had first sketched informally with American astronomer Frank Drake in 1975. Mark Haddon ordered the chapters of The Curious Incident of the Dog in the Night-Time by consecutive primes to mirror the mind of his narrator. Paolo Giordano cast them as outsiders in The Solitude of Prime Numbers, a metaphor for loneliness. Nature uses them too. Cicadas of the genus Magicicada emerge from underground only after 7, 13, or 17 years, prime-numbered cycles that biologists believe evolved to keep predators from synchronizing with them. The same logic of indecomposability reappears in knot theory, where any knot breaks uniquely into prime knots, just as the energy levels of quantum systems have been speculatively linked to the zeros of the Riemann zeta function ever since the 1970s work of Hugh Montgomery and Freeman Dyson.

Common questions

What is a prime number?

A prime number is a natural number greater than 1 that is not the product of two smaller natural numbers. For example, 5 is prime because the only ways to write it as a product, 1 times 5 or 5 times 1, involve 5 itself, while 4 is composite because it equals 2 times 2.

Why is the number 1 not considered a prime number?

The number 1 is excluded from the primes and placed in its own category as a unit because counting it as prime would break key results. The fundamental theorem of arithmetic would fail, since every number could gain endless factorizations using copies of 1, and the sieve of Eratosthenes would eliminate all multiples of 1 and output only 1.

How do you prove there are infinitely many prime numbers?

Euclid proved around 300 BC that the primes are infinite by showing every finite list of them is incomplete. Multiplying the listed primes together and adding one gives a number that leaves a remainder of one when divided by any prime on the list, so its prime factors must lie outside the list.

What is Goldbach's conjecture about prime numbers?

Goldbach's conjecture asserts that every even integer greater than 2 can be written as the sum of two primes. Christian Goldbach formulated it in a 1742 letter to Leonhard Euler, and it has been verified for large ranges of numbers but remains unproven.

How are prime numbers used in cryptography?

Prime numbers are the basis of public-key cryptography, including the RSA cryptosystem and the Diffie-Hellman key exchange, with 2048-bit primes common. RSA relies on the difficulty of factoring large numbers into their prime factors, since multiplying two large primes is easy but recovering them from the product is hard.

What is the largest known prime number?

The largest known prime is the Mersenne prime 2 to the power 136,279,841 minus 1, which has 41,024,320 decimal digits. It was found on the 12th of October 2024 by Luke Durant and the Great Internet Mersenne Prime Search, and since 1992 the largest known prime has always been a Mersenne prime.

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