Prime Numbers

to the implied precision. The estimate was made by computing the reciprocal sum very accurately for twin primes up to 1016 and then extrapolating to the infinite sum using (1.7) to estimate the tail of the sum. (All that is actually proved rigorously about B (by year 2004) is that it is between a number slightly larger than 1.83 and a number slightly smaller than 2.347.) In his earlier (1995) computations concerning the Brun constant, Nicely discovered the now-famous floating-point flaw in the Pentium computer chip, a discovery that cost the Pentium manufacturer Intel millions of dollars. It seems safe to assume that Brun had no idea in 1909 that his remarkable theorem would have such a technological consequence!

1.2.2Prime k-tuples and hypothesis H

The twin prime conjecture is actually a special case of the “prime k-tuples” conjecture, which in turn is a special case of “hypothesis H.” What are these mysterious-sounding conjectures?

The prime k-tuples conjecture begins with the question, what conditions on integers a1, b1, . . . , ak, bk ensure that the k linear expressions a1n + b1, . . . , akn + bk are simultaneously prime for infinitely many positive integers n? One can see embedded in this question the first part of the Dirichlet Theorem 1.1.5, which is the case k = 1. And we can also see embedded the twin prime conjecture, which is the case of two linear expressions n, n + 2.

Let us begin to try to answer the question by giving necessary conditions on the numbers ai, bi. We rule out the cases when some ai = 0, since such a case collapses to a smaller problem. Then, clearly, we must have each ai > 0 and each gcd(ai, bi) = 1. This is not enough, though, as the case n, n + 1 quickly reveals: There are surely not infinitely many integers n for which n and n + 1 are both prime! What is going on here is that the prime 2 destroys the chances for n and n+1, since one of them is always even, and even numbers are not often prime. Generalizing, we see that another necessary condition is that for each prime p there is some value of n such that none of ain + bi is divisible by p. This condition automatically holds for all primes p > k; it follows from the condition that each gcd(ai, bi) = 1. The prime k-tuples conjecture [Dickson 1904] asserts that these conditions are su cient:

Conjecture 1.2.1 (Prime k-tuples conjecture). If a1, b1, . . . , ak, bk are integers with each ai > 0, each gcd(ai, bi) = 1, and for each prime p ≤ k, there is some integer n with no ain + bi divisible by p, then there are infinitely many positive integers n with each ain + bi prime.

Whereas the prime k-tuples conjecture deals with linear polynomials, Schinzel’s hypothesis H [Schinzel and Sierpi´nski 1958] deals with arbitrary irreducible polynomials with integer coe cients. It is a generalization of an older conjecture of Bouniakowski, who dealt with a single irreducible polynomial.

Conjecture 1.2.2 (Hypothesis H). Let f1, . . . , fk be irreducible polynomials with integer coe cients such that the leading coe cient of each fi is positive, and such that for each prime p there is some integer n with none of f1(n), . . . , fk(n) divisible by p. Then there are infinitely many positive integers n such that each fi(n) is prime.

A famous special case of hypothesis H is the single polynomial n2 + 1. As with twin primes, we still do not know whether there are infinitely many primes of the form n2 + 1. In fact, the only special case of hypothesis H that has been proved is Theorem 1.1.5 of Dirichlet.

The Brun method for proving (1.8) can be generalized to get upper bounds of the roughly conjectured order of magnitude for the distribution of the integers n in hypothesis H that make the fi(n) simultaneously prime. See [Halberstam and Richert 1974] for much more on this subject.

For polynomials in two variables we can sometimes say more. For example, Gauss proved that there are infinitely many primes of the form a2 + b2. It was shown only recently in [Friedlander and Iwaniec 1998] that there are infinitely many primes of the form a2 + b4.

1.2.3The Goldbach conjecture

In 1742, C. Goldbach stated, in a letter to Euler, a belief that every integer exceeding 5 is a sum of three primes. (For example, 6 = 2 + 2 + 2 and 21 = 13 + 5 + 3.) Euler responded that this follows from what has become known as the Goldbach conjecture, that every even integer greater than two is a sum of two primes. This problem belongs properly to the field of additive number theory, the study of how integers can be partitioned into various sums. What is maddening about this conjecture, and many “additive” ones like it, is that the empirical evidence and heuristic arguments in favor become overwhelming. In fact, large even integers tend to have a great many representations as a sum of two primes.

Denote the number of Goldbach representations of an even n by

R2(n) = #{(p, q) : n = p + q; p, q P}.

Thinking heuristically as before, one might guess that for even n,

1

since the “probability” that a random number near x is prime is about 1/ ln x. But such a sum can be shown, via the Chebyshev Theorem 1.1.3 (see Exercise 1.40) to be n/ ln2 n. The frustrating aspect is that to settle the Goldbach conjecture, all one needs is that R2(n) be positive for even n > 2. One can tighten the heuristic argument above, along the lines of the argument for (1.7), to suggest that for even integers n,

where C2 is the twin-prime constant of (1.6). The Brun method can be used to establish that R2(n) is big-O of the right side of (1.9) (see [Halberstam and Richert 1974).

Checking (1.9) numerically, we have R2(108) = 582800, while the right side of (1.9) is approximately 518809. One gets better agreement using the asymptotically equivalent expression R2(n) defined as

which at n = 108 evaluates to about 583157.

As with twin primes, [Chen 1966] also established a profound theorem on the Goldbach conjecture: Any su ciently large even number is the sum of a prime and a number that is either a prime or the product of two primes.

It has been known since the late 1930s, see [Ribenboim 1996], that “almost all” even integers have a Goldbach representation p + q, the “almost all” meaning that the set of even natural numbers that cannot be represented as a sum of two primes has asymptotic density 0 (see Section 1.1.4 for the definition of asymptotic density). In fact, it is now known that the number of exceptional even numbers up to x that do not have a Goldbach representation is O x1−c for some c > 0 (see Exercise 1.41).

The Goldbach conjecture has been checked numerically up through 1014 in [Deshouillers et al. 1998], through 4 · 1014 in [Richstein 2001], and through 1017 in [Silva 2005]. And yes, every even number from 4 up through 1017 is indeed a sum of two primes.

As Euler noted, a corollary of the assertion that every even number after 2 is a sum of two primes is the additional assertion that every odd number after 5 is a sum of three primes. This second assertion is known as the “ternary Goldbach conjecture.” In spite of the di culty of such problems of additive number theory, Vinogradov did in 1937 resolve the ternary Goldbach conjecture, in the asymptotic sense that all su ciently large odd integers n admit a representation in three primes: n = p + q + r. It was shown in 1989 by Chen and Y. Wang, see [Ribenboim 1996], that “su ciently large” here can be taken to be n > 1043000. Vinogradov gave the asymptotic representation count of

where Θ is the so-called singular series for the ternary Goldbach problem, namely

It is not hard to see that Θ(n) for odd n is bounded below by a positive constant. This singular series can be given interesting alternative forms (see Exercise 1.68). Vinogradov’s e ort is an example of analytic number theory par excellence (see Section 1.4.4 for a very brief overview of the core ideas).

[Zinoviev 1997] shows that if one assumes the extended Riemann hypothesis (ERH) (Conjecture 1.4.2), then the ternary Goldbach conjecture holds for all odd n > 1020. Further, [Saouter 1998] “bootstrapped” the then current bound of 4 · 1011 for the binary Goldbach problem to show that the ternary Goldbach conjecture holds unconditionally for all odd numbers up to 1020. Thus, with the Zinoviev theorem, the ternary Goldbach problem is completely solved under the assumption of the ERH.

It follows from the Vinogradov theorem that there is a number k such that every integer starting with 2 is a sum of k or fewer primes. This corollary was actually proved earlier by G. Shnirel’man in a completely di erent way. Shnirel’man used the Brun sieve method to show that the set of even numbers representable as a sum of two primes contains a subset with positive asymptotic density (this predated the results that almost all even numbers were so representable), and using just this fact was able to prove there is such a number k. (See Exercise 1.44 for a tour of one proof method.) The least number k0 such that every number starting with 2 is a sum of k0 or fewer primes is now known as the Shnirel’man constant. If Goldbach’s conjecture is true, then k0 = 3. Since we now know that the ternary Goldbach conjecture is true, conditionally on the ERH, it follows that on this condition, k0 ≤ 4. The best unconditional estimate is due to O. Ramar´e who showed that k0 ≤ 7 [Ramar´e 1995]. Ramar´e’s proof used a great deal of computational analytic number theory, some of it joint with R. Rumely.

1.2.4The convexity question

One spawning ground for curiosities about the primes is the theoretical issue of their density, either in special regions or under special constraints. Are there regions of integers in which primes are especially dense? Or especially sparse? Amusing dilemmas sometimes surface, such as the following one. There is an old conjecture of Hardy and Littlewood on the “convexity” of the distribution of primes:

Conjecture 1.2.3. If x ≥ y ≥ 2, then π(x + y) ≤ π(x) + π(y).

On the face of it, this conjecture seems reasonable: After all, since the primes tend to thin out, there ought to be fewer primes in the interval [x, x + y] than in [0, y]. But amazingly, Conjecture 1.2.3 is known to be incompatible with the prime k-tuples Conjecture 1.2.1 [Hensley and Richards 1973].

So, which conjecture is true? Maybe neither is, but the current thinking is that the Hardy–Littlewood convexity Conjecture 1.2.3 is false, while the prime k-tuples conjecture is true. It would seem fairly easy to actually prove that the convexity conjecture is false; you just need to come up with numerical values of x and y where π(x+y), π(x), π(y) can be computed and π(x+y) > π(x)+π(y).

1.3Primes of special form

By prime numbers of special form we mean primes p enjoying some interesting, often elegant, algebraic classification. For example, the Mersenne numbers Mq and the Fermat numbers Fn defined by

are sometimes prime. These numbers are interesting for themselves and for their history, and their study has been a great impetus for the development of computational number theory.

1.3.1Mersenne primes

Searching for Mersenne primes can be said to be a centuries-old research problem (or recreation, perhaps). There are various easily stated constraints on exponents q that aid one in searches for Mersenne primes Mq = 2q − 1. An initial result is the following:

Theorem 1.3.1. If Mq = 2q − 1 is prime, then q is prime.

Proof. A number 2c − 1 with c composite has a proper factor 2d − 1, where d is any proper divisor of c.

This means that in the search for Mersenne primes one may restrict oneself to prime exponents q. Note the important fact that the converse of the theorem is false. For example, 211 − 1 is not prime even though 11 is. The practical import of the theorem is that one may rule out a great many exponents, considering only prime exponents during searches for Mersenne primes.

Yet more weeding out of Mersenne candidates can be achieved via the following knowledge concerning possible prime factors of Mq :

Theorem 1.3.2 (Euler). For prime q > 2, any prime factor of Mq = 2q − 1 must be congruent to 1 (mod q) and furthermore must be congruent to ±1 (mod 8).

Proof. Let r be a prime factor of 2q − 1, with q a prime, q > 2. Then 2q ≡ 1 (mod r), and since q is prime, the least positive exponent h with 2h ≡ 1 (mod r) must be q itself. Thus, in the multiplicative group of nonzero residues modulo r (a group of order r − 1), the residue 2 has order q. This immediately implies that r ≡ 1 (mod q), since the order of an element in a group divides the order of the group. Since q is an odd prime, we in fact have q| r−2 1 , so

A typical Mersenne prime search runs, then, as follows. For some set of prime exponents Q, remove candidates q Q by checking whether

2q ≡ 1 (mod r)

for various small primes r ≡ 1 (mod q) and r ≡ ±1 (mod 8). For the survivors, one then invokes the celebrated Lucas–Lehmer test, which is a rigorous primality test (see Section 4.2.1).

As of this writing, the known Mersenne primes are those displayed in Table 1.2.

Table 1.2 Known Mersenne primes (as of Apr 2005), ranging in size from 1 decimal digit to over 7 million decimal digits.

W. Colquitt and L. Welsh, Jr., in 1988. The record for consecutive Mersenne

S. Kurowski, et al. (further verified by D. Slowinski as prime in a separate machine/program run). Then in 1999 the prime 26972593 − 1 was found by N. Hajratwala, Woltman, and Kurowski, then verified by E. Mayer and D. Willmore. The case 213466917 − 1 was discovered in November 2001 by M. Cameron, Woltman, and Kurowski, then verified by Mayer, P. Novarese, and G. Valor. In November 2003, M. Shafer, Woltman, and Kurowski found 220996011 − 1. The Mersenne prime 224036583 − 1 was found in May 2004 by J. Findley, Woltman, and Kurowski. Then in Feb 2005, M. Nowak, Woltman and Kurowski found 225964951 − 1. Each of these last two Mersenne primes has more than 7 million decimal digits.

The eight largest known Mersenne primes were found using a fast multiplication method—the IBDWT—discussed in Chapter 8.8 (Theorem

9.5.18 and Algorithm 9.5.19). This method has at least doubled the search e ciency over previous methods.

It should be mentioned that modern Mersenne searching is sometimes of the “hit or miss” variety; that is, random prime exponents q are used to check accordingly random candidates 2q − 1. (In fact, some Mersenne primes were indeed found out of order, as indicated above). But much systematic testing has also occurred. As of this writing, exponents q have been checked for all q ≤ 12830000. Many of these exponents are recognized as giving composite Mersennes because a prime factor is detected. For example, if q is a prime that is 3 (mod 4), and p = 2q + 1 is prime, then p|Mq . (See also Exercises 1.47, 1.81.) For the remaining values of q, the Lucas–Lehmer test (see Section 4.2.1) was used. In fact, for all q ≤ 9040000 for which a factor of Mq was not found, the Lucas–Lehmer test was carried out twice (see [Woltman 2000], which website is frequently updated).

As mentioned in Section 1.1.2, the prime M25964951 is the current record holder as not only the largest known Mersenne prime, but also the largest explicit number that has ever been proved prime. With few exceptions, the record for largest proved prime in the modern era has always been a Mersenne prime. One of the exceptions occurred in 1989, when the “Amdahl Six” found the prime [Caldwell 1999]

391581 · 2216193 − 1,

which is larger than 2216091 − 1, the record Mersenne prime of that time. However, this is not the largest known explicit non-Mersenne prime, for Young found, in 1997, the prime 5· 2240937 +1, and in 2001, Cosgrave found the prime

3 · 2916773 + 1.

Actually, the 5th largest known explicit prime is the non-Mersenne

5359 · 25054502 + 1,

found by R. Sundquist in 2003.

Mersenne primes figure uniquely in the ancient subject of perfect numbers. A perfect number is a positive integer equal to the sum of its divisors other than itself. For example, 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14 are perfect numbers. An equivalent way to define “perfection” is to denote by σ(n) the sum of the positive divisors of n, whence n is perfect if and only if σ(n) = 2n. The following famous theorem completely characterizes the even perfect numbers.

Theorem 1.3.3 (Euclid–Euler). An even number n is perfect if and only if it is of the form

n = 2q−1Mq ,

where Mq = 2q − 1 is prime.

Proof. Suppose n = 2am is an even number, where m is the largest odd

n is M (D + m). If M is prime and M = m, then D = 1, and the sum of all the divisors of n is M (1 + m) = 2n, so that n is perfect. This proves the first half of the assertion. For the second, assume that n = 2am is perfect. Then M (D + m) = 2n = 2a+1m = (M + 1)m. Subtracting M m from this equation, we see that

m = M D.

If D > 1, then D and 1 are distinct divisors of m less than m, contradicting the definition of D. So D = 1, m is therefore prime, and m = M = 2a+1 − 1.

The first half of this theorem was proved by Euclid, while the second half was proved some two millennia later by Euler. It is evident that every newly discovered Mersenne prime immediately generates a new (even) perfect number. On the other hand, it is still not known whether there are any odd perfect numbers, the conventional belief being that none exist. Much of the research in this area is manifestly computational: It is known that if an odd perfect number n exists, then n > 10300, a result in [Brent et al. 1993], and that n has at least eight distinct prime factors, an independent result of E. Chein and P. Hagis; see [Ribenboim 1996]. For more on perfect numbers, see Exercise 1.30.

There are many interesting open problems concerning Mersenne primes. We do not know whether there are infinitely many such primes. We do not even know whether infinitely many Mersenne numbers Mq with q prime are composite. However, the latter assertion follows from the prime k-tuples Conjecture 1.2.1. Indeed, it is easy to see that if q ≡ 3 (mod 4) is prime and 2q + 1 is also prime, then 2q + 1 divides Mq . For example, 23 divides M11. Conjecture 1.2.1 implies that there are infinitely many such primes q.

Various interesting conjectures have been made in regard to Mersenne numbers, for example the “new Mersenne conjecture” of P. Bateman, J. Selfridge, and S. Wagsta , Jr. This stems from Mersenne’s original assertion in 1644 that the exponents q for which 2q −1 is prime and 29 ≤ q ≤ 257 are 31, 67, 127, and 257. (The smaller exponents were known at that time, and it was also known that 237 −1 is composite.) Considering that the numerical evidence below 29 was that every prime except 11 and 23 works, it is rather amazing that Mersenne would assert such a sparse sequence for the exponents. He was right on the sparsity, and on the exponents 31 and 127, but he missed 61, 89, and 107. With just five mistakes, no one really knows how Mersenne e ected such a claim. However, it was noticed that the odd Mersenne exponents below 29 are all either 1 away from a power of 2, or 3 away from a power of 4 (while the two missing primes, 11 and 23, do not have this property), and Mersenne’s list just continues this pattern (perhaps with 61 being an “experimental error,” since Mersenne left it out). In [Bateman et al. 1989] the authors suggest a new

Mersenne conjecture, that any two of the following implies the third: (a) the prime q is either 1 away from a power of 2, or 3 away from a power of 4, (b) 2q − 1 is prime, (c) (2q + 1)/3 is prime. Once one gets beyond small numbers, it is very di cult to find any primes q that satisfy two of the statements, and probably there are none beyond 127. That is, probably the conjecture is true, but so far it is only an assertion based on a very small set of primes.

It has also been conjectured that every Mersenne number Mq , with q prime, is squarefree (which means not divisible by a square greater than 1), but we cannot even show that this holds infinitely often. Let M denote the set of primes that divide some Mq with q prime. We know that the number of members of M up to x is o(π(x)), and it is known on the assumption of the generalized Riemann hypothesis that the sum of the reciprocals of the members of M converges [Pomerance 1986].

It is possible to give a heuristic argument that supports the assertion that there are c ln x primes q ≤ x with Mq prime, where c = eγ / ln 2 and γ is Euler’s constant. For example, this formula suggests that there should be, on average, about 23.7 values of q in an interval [x, 10000x]. Assuming that the machine checks of the Mersenne exponents up to 12000000 are exhaustive, the actual number of values of q with Mq prime in [x, 10000x] is 23, 24, or 25 for x = 100, 200, . . . , 1200, with the count usually being 24. Despite the good agreement with practice, some still think that the “correct” value of c is 2/ ln 2 or something else. Until a theorem is actually proved, we shall not know for sure.

We begin the heuristic with the fact that the probability that a random number near Mq = 2q − 1 is prime is about 1/ ln Mq , as seen by the prime number Theorem 1.1.4. However, we should also compare the chance of Mq being prime with a random number of the same size. It is likely not the same, as Theorem 1.3.2 already indicates. Let us ignore for a moment the intricacies of this theorem and use only that Mq has no prime factors in the interval [1, q]. Here q is about lg Mq (here and throughout the book, lg means log2). What is the chance that a random number near x whose least prime factor exceeds lg x is prime? We know how to answer this question rigorously. First consider the chance that a random number near x has its least prime factor exceeding lg x. Intuitively, this probability should be

≤

since each prime p has probability 1/p of dividing a random number, and these should be at least roughly independent events. They cannot be totally independent, for example, no number in [1, x] is divisible by two primes in the interval (x1/2, x], yet a purely probabilistic argument suggests that a positive proportion of the numbers in [1, x] actually have this property! However, when dealing with very small primes, and in this case only those up to lg x, the heuristic guess is provable. Now, each prime near x survives this sieve; that is, it is not divisible by any prime p ≤ lg x. So, if a number n near x has already