Prime Numbers

≤

which leads immediately to the fact that for all x ≥ 8,

ln x π(x) > 2 ln ln x .

Though this bound is relatively poor, it does prove the infinitude of primes directly from the fundamental theorem of arithmetic.

The idea of Euclid in the proof of Theorem 1.1.2 is to generate new primes from old primes. Can we generate all of the primes this way? Here are a few possible interpretations of the question:

(1)Inductively define a sequence of primes q1, q2, . . ., where q1 = 2, and qk+1 is the least prime factor of q1 · · · qk + 1. Does the sequence (qi) contain every prime?

(2)Inductively define a sequence of primes r1, r2, . . ., where r1 = 2, and rk+1 is the least prime not already chosen that divides some d +1, where d runs over the divisors of the product r1 · · · rk. Does the sequence (ri) contain every prime?

(3)Inductively define a sequence of primes s1, s2, . . ., where s1 = 2, s2 = 3, and sk+1 is the least prime not already chosen that divides some sisj + 1, where 1 ≤ i < j ≤ k. Does the sequence (si) contain every prime? Is the sequence (si) infinite?

The sequence (qi) of problem (1) was considered by Guy and Nowakowski and later by Shanks. In [Wagsta 1993] the sequence was computed through the 43rd term. The computational problem inherent in continuing the sequence further is the enormous size of the numbers that must be factored. Already, the number q1 · · · q43 + 1 has 180 digits.

The sequence (ri) of problem (2) was recently shown in unpublished work of Pomerance to contain every prime. In fact, for i ≥ 5, ri is the i-th prime. The proof involved a direct computer search over the first (approximately) one million terms, followed by some explicit estimates from analytic number theory, about more of which theory we shall hear later in this chapter. This proof is just one of many examples that manifest the utility of the computational perspective.

The sequence (si) of problem (3) is not even known to be infinite, though it almost surely is, and almost surely contains every prime. We do not know whether anyone has attacked the problem computationally; perhaps you, the reader, would like to give it a try. The problem is due to M. Newman at the Australian National University.

Thus, even starting with the most fundamental and ancient ideas concerning prime numbers, one can quickly reach the fringe of modern research.

1.1.4Asymptotic relations and order nomenclature

At this juncture, in anticipation of many more asymptotic density results and computational complexity estimates, we establish asymptotic relation nomenclature for the rest of the book. When we intend

f (N ) g(N )

to be read “f is asymptotic to g as N goes to infinity,” we mean that a certain limit exists and has value unity:

lim f (N )/g(N ) = 1.

N →∞

When we say

f (N ) = O(g(N )),

to be read “f is big-O of g,” we mean that f is bounded in this sense: There exists a positive number C such that for all N , or for all N in a specified set,

|f (N )| ≤ C|g(N )|.

The “little-o” notation can be used when one function seriously dominates another; i.e., we say

f (N ) = o(g(N ))

to mean that

lim f (N )/g(N ) = 0.

N →∞

Some examples of the notation are in order. Since π(x), the number of primes not exceeding x, is clearly less than x for any positive x, we can say

π(x) = O(x).

On the other hand, it is not so clear, and in fact takes some work to prove (see Exercises 1.11 and 1.13 for two approaches), that

Equation (1.1) can be interpreted as the assertion that at very high levels the primes are sparsely distributed, and get more sparsely distributed the higher one goes. If A is a subset of the natural numbers and A(x) denotes the number of members of A that do not exceed x, then if limx→∞ A(x)/x = d, we call d the asymptotic density of the set A. Thus equation (1.1) asserts that the set of primes has asymptotic density 0. Note that not all subsets of the natural numbers possess an asymptotic density; that is, the limit in the definition may not exist. As just one example, take the set of numbers with an even number of decimal digits.

Throughout the book, when we speak of computational complexity of algorithms we shall stay almost exclusively with “O” notation, even though

some authors denote bit and operation complexity by such as Ob, Oop respectively. So when an algorithm’s complexity is cast in “O” form, we shall endeavor to specify in every case whether we mean bit or operation complexity. One should take care that these are not necessarily proportional, for it matters whether the “operations” are in a field, are adds or multiplies, or are comparisons (as occur within “if” statements). For example, we shall see in Chapter 8.8 that whereas a basic FFT multiplication method requires O(D ln D) floating-point operations when the operands possess D digits each (in some appropriate base), there exist methods having bit complexity O(n ln n ln ln n), where now n is the total number of operand bits. So in such a case there is no clear proportionality at work, the relationships between digit size, base, and bit size n are nontrivial (especially when floating-point errors figure into the computation), and so on. Another kind of nontrivial comparison might involve the Riemann zeta function, which for certain arguments can be evaluated to D good digits in O(D) operations, but we mean full-precision, i.e., D-digit operations. In contrast, the bit complexity to obtain D good digits (or a proportional number of bits) grows faster than this. And of course, we have a trivial comparison of the two complexities: The product of two large integers takes one (high-precision) operation, while a flurry of bit manipulations are generally required to e ect this multiply! On the face of it, we are saying that there is no obvious relation between these two complexity bounds. One might ask,“if these two types of bounds (bitand operationbased bounds) are so di erent, isn’t one superior, maybe more profound than the other?” The answer is that one is not necessarily better than the other. It might happen that the available machinery—hardware and software—is best suited for all operations to be full-precision; that is, every add and multiply is of the D-digit variety, in which case you are interested in the operationcomplexity bound. If, on the other hand, you want to start from scratch and create special, optimal bit-complexity operations whose precision varies dynamically during the whole project, then you would be more interested in the bit-complexity bound. In general, the safe assumption to remember is that bitversus operation-complexity comparisons can often be of the “apples and oranges” variety.

Because the phrase “running time” has achieved a certain vogue, we shall sometimes use this term as interchangeable with “bit complexity.” This equivalence depends, of course, on the notion that the real, physical time a machine requires is proportional to the total number of relevant bit operations. Though this equivalence may well decay in the future—what with quantum computing, massive parallelism, advances in word-oriented arithmetic architecture, and so on—we shall throughout this book just assume that running time and bit complexity are the same. Along the same lines, by “polynomial-time” complexity we mean that bit operations are bounded above by a fixed power of the number of bits in the input operands. So, for example, none of the dominant factoring algorithms of today (ECM, QS, NFS) is polynomial-time, but simple addition, multiplication, powering, and so on are polynomial-time. For example, powering, that is computing xy mod z, using

10 Chapter 1 PRIMES!

naive subroutines, has bit complexity O(ln3 z) for positive integer operands x, y, z of comparable size, and so is polynomial-time. Similarly, taking a greatest common divisor (gcd) is polynomial-time, and so on.

1.1.5How primes are distributed

In 1737, L. Euler achieved a new proof that there are infinitely many primes: He showed that the sum of the reciprocals of the primes is a divergent sum, and so must contain infinitely many terms (see Exercise 1.20).

In the mid-19th century, P. Chebyshev proved the following theorem, thus establishing the true order of magnitude for the prime-counting function.

For example, Theorem 1.1.3 is true with A = 1/2 and B = 2. This was a spectacular result, because Gauss had conjectured in 1791 (at the age of fourteen!) the asymptotic behavior of π(x), about which conjecture little had been done for half a century prior to Chebyshev. This conjecture of Gauss is now known as the celebrated “prime number theorem” (PNT):

Theorem 1.1.4 (Hadamard and de la Vall´ee Poussin). As x → ∞,

π(x) lnxx .

It would thus appear that Chebyshev was close to a resolution of the PNT. In fact, it was even known to Chebyshev that if π(x) were asymptotic to some Cx/ ln x, then C would of necessity be 1. But the real di culty in the PNT is showing that limx→∞ π(x)/(x/ ln x) exists at all; this final step was achieved a half-century later, by J. Hadamard and C. de la Vall´ee Poussin, independently, in 1896. What was actually established was that for some positive number C,

where li (x), the logarithmic-integral function, is defined as follows (for a variant of this integral definition see Exercise 1.36):

Since li (x) x/ ln x, as can easily be shown via integration by parts (or even more easily by L’Hˆopital’s rule), this stronger form of the PNT implies the form in Theorem 1.1.4. The size of the “error” π(x) − li (x) has been a subject of intense study—and refined only a little—in the century following the proof of the PNT. In Section 1.4 we return to the subject of the PNT. But for the moment, we note that one useful, albeit heuristic, interpretation of the PNT

is that for random large integers x the “probability” that x is prime is about 1/ ln x.

It is interesting to ponder how Gauss arrived at his remarkable conjecture. The story goes that he came across the conjecture numerically, by studying a table of primes. Though it is clearly evident from tables that the primes thin out as one gets to larger numbers, locally the distribution appears to be quite erratic. So what Gauss did was to count the number of primes in blocks of length 1000. This smoothes out enough of the irregularities (at low levels) for a “law” to appear, and the law is that near x, the “probability” of a random integer being prime is about 1/ ln x. This then suggested to Gauss that a reasonable estimate for π(x) might be the logarithmic-integral function.

Though Gauss’s thoughts on π(x) date from the late 1700s, he did not publish them until decades later. Meanwhile, Legendre had independently conjectured the PNT, but in the form

with B = 1.08366. No matter what choice is made for the number B, we have x/ ln x x/(ln x − B), so the only way it makes sense to include a number B in the result, or to use Gauss’s approximation li (x), is to consider which option gives a better estimation. In fact, the Gauss estimate is by far the better one. Equation (1.2) implies that |π(x) − li (x)| = O(x/ lnk x) for every k > 0 (where the big-O constant depends on the choice of k). Since

P.Zimmermann) that

π1021 = 21127269486018731928,

while on the other hand

li 1021 ≈ 21127269486616126181.3

and

1021

ln(1021) − 1

≈ 21117412262909985552.2 .

Table 1.1 Values of the prime-counting function π(x). In recent times, distributed computation on networks has been brought to bear on the π(x) counting problem.

It is astounding how good the li (x) approximation really is!

We will revisit this issue of the accuracy of the li approximation later in the present chapter, in connection with the Riemann hypothesis (RH) (see Conjecture 1.4.1 and the remarks thereafter).

The most recent values in Table 1.1, namely π(1022), π(4 · 1022), are due to X. Gourdon and P. Sebah [Gourdon and Sebah 2004]. These researchers, while attempting to establish the value of π(1023), recently discovered an inconsistency in their program, a numerical discrepancy in regard to local sieving. Until this problem has been rectified or there has been a confirming independent calculation, their values for π(1022) and π(4·1022) should perhaps be considered tentative.

Another question of historical import is this: What residue classes a mod d contain primes, and for those that do, how dense are the occurrences of primes in such a residue class? If a and d have a common prime factor, then such a prime divides every term of the residue class, and so the residue class cannot contain more than this one prime. The central classical result is that this is essentially the only obstruction for the residue class to contain infinitely many

{a, a + d, a + 2d, . . .} contains infinitely many primes. In fact, the sum of

Erd˝os–Tur´an conjecture: A subset S of the natural numbers that contains a positive proportion of the natural numbers (that is, the limsup of the proportion of S ∩ [1, x] in {1, 2, . . . , x } is positive) must contain arbitrarily long arithmetic progressions. This result appears not to apply, since the primes do not form a positive proportion of the natural numbers. However, Green and Tao actually prove a version of Szemer´edi’s theorem where the universe of natural numbers is allowed to be somewhat generalized. They then proceed to give an appropriate superset of the primes for which the Szemer´edi analogue is valid and for which the primes form a positive proportion. Altogether, the Green–Tao development is quite amazing.

1.2Celebrated conjectures and curiosities

We have indicated that the definition of the primes is so very simple, yet questions concerning primes can be so very hard. In this section we exhibit various celebrated problems of history. The more one studies these questions, the more one appreciates the profundity of the games that primes play.

1.2.1Twin primes

Consider the case of twin primes, meaning two primes that di er by 2. It is easy to find such pairs, take 11, 13 or 197, 199, for example. It is not so easy, but still possible, to find relatively large pairs, modern largest findings being

the pair

835335 · 239014 ± 1,

found in 1998 by R. Ballinger and Y. Gallot, the pair

361700055 · 239020 ± 1,

found in 1999 by H. Lifchitz, and (see [Caldwell 1999]) the twin-prime pairs

discovered in 2000:

2409110779845 · 260000 ± 1,

by H. Wassing, A. J´arai, and K.-H. Indlekofer, and

665551035 · 280025 ± 1,

by P. Carmody. The current record is the pair

154798125 · 2169690 ± 1,

reported in 2004 by D. Papp.

Are there infinitely many pairs of twin primes? Can we predict, asymptotically, how many such pairs there are up to a given bound? Let us try to think heuristically, like the young Gauss might have. He had guessed that the probability that a random number near x is prime is about 1/ ln x, and thus came up with the conjecture that π(x) ≈ 2x dt/ ln t (see Section 1.1.5). What if we choose two numbers near x. If they are “independent prime

events,” then the probability they are both prime should be about 1/ ln2 x. Thus, if we denote the twin-prime-pair counting function by

π2(x) = #{p ≤ x : p, p + 2 P},

where P is the set of all primes, then we might guess that

x 1

π2(x) 2 ln2 t dt.

However, it is somewhat dishonest to consider p and p + 2 as independent prime events. In fact, the chance of one being prime influences the chance that the other is prime. For example, since all primes p > 2 are odd, the number p + 2 is also odd, and so has a “leg up” on being prime. Random odd numbers have twice the chance of being prime as a random number not stipulated beforehand as odd. But being odd is only the first in a series of “tests” a purported prime must pass. For a fixed prime q, a large prime must pass the “q-test” meaning “not divisible by q.” If p is a random prime and q > 2, then the probability that p+2 passes the q-test is (q−2)/(q−1). Indeed, from (1.5), there are ϕ(q) = q − 1 equally likely residue classes modulo q for p to fall in, and for exactly q−2 of these residue classes we have p+2 not divisible by q. But the probability that a completely random number passes the q-test is (q − 1)/q. So, let us revise the above heuristic with the “fudge factor” 2C2, where C2 = 0.6601618158 . . . is the so-called “twin-prime constant”:

or perhaps, more simply, that

x π2(x) 2C2 ln2 x .

The two asymptotic relations are equivalent, which can be seen by integrating by parts. But the reason we have written the more ungainly expression in (1.7) is that, like the estimate π(x) ≈ li (x), it may be an extremely good approximation.

Let us try out the approximation (1.7) at x = 5.4 · 1015. It is reported, see [Nicely 2004], that

Let’s hear it for heuristic reasoning! Very recently P. Sebah found

π2 1016 = 10304195697298,

as enunciated in [Gourdon and Sebah 2004].

As strong as the numerical evidence may be, we still do not even know whether there are infinitely many pairs of twin primes; that is, whether π2(x) is unbounded. This remains one of the great unsolved problems in mathematics. The closest we have come to proving this is the theorem of Chen Jing-run in 1966, see [Halberstam and Richert 1974], that there are infinitely many primes p such that either p + 2 is prime or the product of two primes.

A striking upper bound result on twin primes was achieved in 1915 by V. Brun, who proved that

(see [Halberstam and Richert 1974]). Thus, in some sense, the twin prime conjecture (1.7) is partially established. From (1.8) one can deduce (see Exercise 1.50) the following:

Theorem 1.2.1 (Brun). The sum of the reciprocals of all primes belonging to some pair of twin primes is finite, that is, if P2 denotes the set of all primes p such that p + 2 is also prime, then

1 + 1 < ∞. p p + 2

p P2

(Note that the prime 5 is unique in that it appears in two pairs of twins, and in its honor, it gets counted twice in the displayed sum; of course, this has nothing whatsoever to do with convergence or divergence.) The Brun theorem is remarkable, since we know that the sum of the reciprocals of all primes diverges, albeit slowly (see Section 1.1.5). The sum in the theorem, namely

B = (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + · · · ,

is known as the Brun constant. Thus, though the set of twin primes may well be infinite, we do know that they must be significantly less dense than the primes themselves.

An interesting sidelight on the issue of twin primes is the numerical calculation of the Brun constant B . There is a long history on the subject, with the current computational champion being Nicely. According to the paper [Nicely 2004], the Brun constant is likely to be about

B ≈ 1.902160583,