Selected citations on twin primes. See also http://www.mathsoft.com/asolve/constant/brun/brun.html ============================================================================== From: Mathematical Reviews on the Web _________________________________________________________________ 10,353f 10.0X Clement, P. A. Congruences for sets of primes. Amer. Math. Monthly 56, (1949). 23--25. Using Wilson's theorem, ($(n-1)!+1\equiv 0 (\text{mod}\,n)$ if and only if $n$ is a prime), the author proves that $4[(n-1)!+1]+n\equiv 0 (\text{mod}\,n(n+2))$ if and only if $n$ and $n+2$ are both primes. Similar results are also proved for prime triplets $n$, $n+2$, $n+6$; and prime quadruplets $n$, $n+2$, $n+6$, $n+8$. A method is proposed for obtaining this kind of result for prime sets of "any prescribed type." Reviewed by H. N. Shapiro _________________________________________________________________ 99f:11117 11N05 (11Y35) Richstein, Jörg(D-GSSN-AGI) Unendliche Geschwisterliebe unter Zahlen oder Die Suche nach den Primzahlzwillingen. (German) Überblicke Mathematik 1996/97, 135--143, [Infinite sibling love among numbers or The search for the twin primes] Überbl. Math., Vieweg, Braunschweig, 1997. In this well-written expository paper, the author surveys the history of prime number theory with an emphasis on the twin prime problem; in particular, there is a short biography with a photo of Viggo Brun, a pioneer in the study of this problem. Number-theoretic phenomena related to primes are illustrated by extensive computational data. \{For the entire collection see MR 99d:00011.\} Reviewed by Matti Jutila _________________________________________________________________ 97e:11014 11A41 (11Y11) Nicely, Thomas R. Enumeration to $10\sp {14}$ of the twin primes and Brun's constant. (English. English summary) Virginia J. Sci. 46 (1995), no. 3, 195--204. This is an interesting report on how the author discovered a flaw in the hardware divider of the floating point unit of the early Intel Pentium processors. The background is the enumeration of the number of twin primes up to $10\sp {14}$ and their reciprocal sum. By the pioneering work of V. Brun this reciprocal sum converges but very slowly. One can construct an accelerated sum which converges more rapidly and approximates the original sum if one assumes that the conjectural asymptotics of the number of twin primes holds. The limit of the accelerated sum is also calculated with a precision of $±2.1\times10\sp {-9}$. The method of computation is well explained. Reviewed by Antal Balog _________________________________________________________________ 97e:11004 11-02 (11B13 11P05 11P32) Nathanson, Melvyn B.(1-CUNY7) Additive number theory. (English. English summary) The classical bases. Graduate Texts in Mathematics, 164. Springer-Verlag, New York, 1996. xiv+342 pp. $49.95. ISBN 0-387-94656-X This book, intended for graduate students, is principally concerned with the problems of Waring and Goldbach. The general theory of additive bases receives virtually no discussion. The topics covered are: sums of polygonal numbers; elementary results on Waring's problem, including Hilbert's work; the Hardy-Littlewood method; estimates for primes; Brun's theorem on twin primes; the Selberg sieve; Shnirelman's theorem on the Goldbach problem; Vinogradov's 3 primes theorem; the linear sieve; and Chen's theorem. There is also an appendix on arithmetic functions. Each chapter has notes and exercises, though the latter would have benefitted from some more taxing examples. Occasional results, such as the Bombieri-Vinogradov theorem, are used without proof, but the text is essentially self-contained. It is perhaps surprising to see so much material gathered into little more than 300 pages, but the exposition is efficient, rather than terse. It benefits from the latest ideas across the range of topics, the treatment of the linear sieve, for example, being based on recent unpublished lectures of Iwaniec. The reviewer's main doubt concerns the disparate nature of the topics considered. Not all graduate students are likely to get on top of both the linear sieve and the circle method, for example. However, for those interested in analytic number theory as a whole this book has much to offer. The reviewer found more than one section containing material that was totally new to him, and it is likely that other readers will find it a useful book to read. Reviewed by D. R. Heath-Brown _________________________________________________________________ _________________________________________________________________ Next Review 58 #16472 10-01 Roberts, Joe Elementary number theory---a problem oriented approach. MIT Press, Cambridge, Mass.-London, 1977. vii+647 pp. (not consecutively paged). $12.50. This text is an introduction to elementary number theory---problem oriented rather than theory oriented. The first $40%$ of the text consists of numbered paragraphs containing one or more statements (for the reader to prove) culminating in a theorem, and the remaining $60%$ is a section with proofs or hints to proofs of the statements in the first section. To make sure the reader does not go to the last section too frequently, sections and paragraphs are not clearly indicated on the top of the page. The text is meant for self-study and something like $75%$ of it can be read without knowledge of college mathematics (of course to solve the problems you had better be ingenious). The remaining $25%$ requires the knowledge of calculus and some advanced calculus, but no algebra or geometry and no complex variables. The text covers most topics to be found in every elementary text on number theory since Gauss's Disquisitiones arithmeticae (although a casual scanner may have trouble finding a particular topic) excluding Diophantine equations (descent methods of Fermat) and quadratic forms and quadratic fields. On the other hand, quaternions, transcendence, uniform distribution, Brun's theorem on twin primes and Dirichlet's theorem on primes in arithmetic progression do appear. The author has searched the literature well to find interesting and unusual problems which fit with his view of number theory. Many have delightful and unexpected proofs and some unconventional proofs are given of old warhorse problems. As a collection of problems, this text is quite unique; a large percentage will be a challenge to all but the best professionals and even they occasionally might be stumped for a while. A large bibliography is given and frequent referral is made to it. However, no attempt is made to let the reader in on the fact that number theory is more than further problems along the same line. The reader is left in the dark regarding both the variety and type of problem being studied, and the depth and sophistication of the tools being used in modern number theory. The author feels that "style and notation are integral to mathematical meaning" and "that the format of a mathematics book is of great importance". For that reason he chose to have the text produced wholly by calligraphy. In the reviewer's opinion, the advantages of calligraphy were not utilized, nothing is highlighted, there is a greyness and evenness page after page. Some results are more significant than others (even if easier to prove), and deserve to stand out, even to appear in boldface or to be begun with an embellished colophon as in old handwritten texts, or, at the minimum, begun with the word "Theorem" as is done conventionally. The seductive ease with which one can make conjectures in elementary number theory has lured many to fritter away their lives on trivia---a disease no worse than addiction to crossword puzzles and no more rewarding. A serious author, especially one concerned with style and format, should assume some responsibility to develop the reader's taste and sense of significance. In the reviewer's opinion, this has not been done here. This text will produce its share of trivia addicts, but, to be fair, it must also be said that hidden within it is the potential to challenge the innovative and imaginative reader to do battle with the unsolved problems of number theory. Reviewed by D. J. Lewis _________________________________________________________________ 51 #5522 10H15 (10-04) Brent, Richard P. Irregularities in the distribution of primes and twin primes. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 43--56. Let $\pi(x)$ denote the number of primes $p\leq x$, $L(x)=\intslash\sb 0\sp x(\log t)\sp {-1}\,dt$, $R(x)=\sum\sb {k=1}\sp \infty k\sp {-1}µ(k)L(x\sp {1/k})$, and let $\langle y\rangle$ denote the integer closest to the real number $y$. The author tabulates the maximum and minimum values of $r\sb 1(x)=\langle L(x)\rangle-\pi(x)$, $r\sb 2(x)=\langle R(x)\rangle-\pi(x)$ in various intervals up to $x=8\times 10\sp {10}$, and discusses certain conjectures about possible asymptotic mean-values of $s\sb i(n)=r\sb i(n)n\sp {-1/2}\log n (i=1,2)$, which are suggested by the numerical results. In a similar way, he then examines the difference $\langle L\sb 2(n)\rangle-\pi\sb 2(n)$, where $L\sb 2(x)=2c\sb 2\int\sb 2\sp x(\log t)\sp {-2}\,dt$, $c\sb 2=0.66016\cdots$ is the "twin-prime" constant of Hardy-Littlewood, and $\pi\sb 2(x)$ is the number of twin primes not exceeding $x$. In particular, on the basis of the latter computations, it is estimated that the value of Brun's twin-prime constant $\sum\{q\sp {-1}+(q+2)\sp {-1}\}$ is likely to lie in the range $1.9021604±5\times 10\sp {-7}$. \{For the entire collection, see MR 50 #1791.\} Reviewed by J. Knopfmacher _________________________________________________________________ _________________________________________________________________ 30 #1079 10.01 (10.05) Rademacher, Hans Lectures on elementary number theory. A Blaisdell Book in the Pure and Applied Sciences Blaisdell Publishing Co. Ginn and Co.\,New York-Toronto-London 1964 ix+146 pp. This is an introduction to the theory of numbers, written in the charming and inimitable style of its well-known author. The style is original. Note the Theorem (p. 1, Introduction): "Outside any given finite set of prime numbers there is another one". On p. 3 we learn that "Gauss conjectured $\pi(x)\sim x/\log\sb ex$ at the age of 15". On p. 4 "A. Selberg has shown, amongst the numbers having at most three prime divisors exist infinitely many pairs of difference 2". Chapter 1, Theorem 1: For two consecutive fractions $h/k,l/m$ of the Farey sequence of order $N$ we have $hm=kl-1$, or, in another formulation, $\left\vert \matrix h & l \\ k & m \endmatrix \right\vert =-1$. Chapter 2: Euclid's Lemma, Uniqueness of Prime Factorization. Theorem 7: If $p$ is a prime then $\surd p$ is irrational. Chapter 3: Congruences (Euler's Function, Chinese Remainder Theorem). Chapter 4: Decimal Fractions (e.g., Theorem 15: The residue classes $\text{mod}\,41$ containing 1, 10, 18, 16 and 37 form a multiplicative group). Chapter 5 contains, amongst other topics, a proof that a prime of the form $4k+1$ is a sum of two squares. Chapter 6: Better Rational Approximation of Irrational Numbers; Ford Circles and Hurwitz's Theorem (begins with "goodness of approximation"). Chapter 7: Primitive Congruence Roots; the Regular Heptadecagon. Chapter 8: Solution of Cyclotomic Equations. Chapter 9: Gaussian Sums as Special Lagrange Resolvents. Chapter 10: The Law of Quadratic Reciprocity. In Chapter 11, "The Product Formula for the Gaussian Sums", the author solves "the problem of the sign of a Gaussian sum". In Chapter 12, "Lattice Points", he gives the results of Gauss and Dirichlet on the number of lattice points in the regions $x\sp 2+y\sp 2\leq N$ and $1\leq xy\leq N$, respectively. In Chapter 13 he treats sums involving prime numbers, e.g., $$ \sum\sb {p\leq x}\frac 1{p}=\log\text{}\log x+C+O\left(\frac 1{\log x}\right). $$ In Chapter 14 he proves Dirichlet's famous theorem on primes in an arithmetic progression. In the final Chapter 15 he discusses the Sieve of Eratosthenes, and proves, in easy steps, the difficult theorem of V. Brun that the sum of the reciprocals of twin primes is convergent. Reviewed by S. Chowla _________________________________________________________________ 9,136b 10.0X Ren\cprime i, A. A. On the representation of an even number as the sum of a single prime and a single almost-prime number. (Russian) Doklady Akad. Nauk SSSR (N.S.) 56, (1947). 455--458. A set of integers $S$, with the property that there exists an absolute constant $K$ such that each $x\in S$ has at most $K$ distinct prime factors, is called an almost-prime set. Each $x\in S$ is called an almost-prime number. The author indicates the proof, to be given in detail elsewhere, that each even integer is the sum of an almost-prime number (taken from a fixed set $S$) and a prime number. He also states that he can prove that there exist infinitely many primes $p$ such that $p+2$ is almost-prime (being in a fixed set $S\sp *$). The first result, regarding the representation of an even number, is an approximation to the unproved Goldbach conjecture and supersedes an earlier proof of the same proposition by Estermann [J. Reine Angew. Math. 168, 106--116 (1932)] which made use of an unproved generalized Riemann hypothesis for all Dirichlet $L$-series. The second result is an approximation to the conjecture of the existence of infinitely many twin primes and is apparently a new result. In order to formulate the basic result enabling the author to dispense with the Riemann hypothesis, recall that if $(p,q)=1$ and $D=pq$ then any character $\chi\sb D(n) \text{modulo}\,D$ can be uniquely decomposed into the product $\chi\sb p(n)\chi\sb q(n)$, where the new characters are to the moduli $p$ and $q$, respectively. The author calls $\chi\sb D(n)$ primitive relative to $p$ if $\chi\sb p(n)$ is not the principal character $p$. The author's result, for which no proof is indicated, is the following. Let $q$ be a square-free integer and $c\sb 1>0$ an absolute constant. Then there exists a constant $\delta>0$ such that, if $A\geq c\sb 1$, $k=(\log q)/(\log A)+1\leq\log\sp 3A$, $p$ is any prime such that $(p,q)=1$ and $A\leq p\leq 2A$ (there being, asymptotically for large $A$, $A/\{\varphi(q)\log A\}$ such $p$) with the possible exception of $A\sp {3/4}$ values, and if $\chi(n)$ is any character $\text{mod}\,pq$ which is primitive relative to $p$, then $L(\sigma+it,\chi)=\sum\sb {n=1}\sp \infty\chi(n)n\sp {-\sigma-it}$ has no zeros in the rectangle $1-\delta/(k+1)\leq\sigma\leq 1$, $0\leq\vert t\vert \leq\log\sp 3pq$. To prove his result on the Goldbach conjecture, the author considers $H(2N)=\sum\log p·\exp\,\{-p(\log 2N)/(2N)\}$, extended over those primes $p<2N$ such that $(2N-p,B)=1$; here $B=\prod p\sp *$, extended over those primes $p\sp *$ such that $c\sb 2\leq p\sp *\leq(2N)\sp {1/R}$, where $R$ is a suitably chosen integer. It is clear that $P=2N-p$ is almost-prime and hence, for the weak Goldbach theorem, it is sufficient to prove that there exists a $c\sb 7>0$ such that if $N\geq c\sb 7$ then $H(2N)>0$; for then we have that there exists an almost-prime $P$ and a prime $p$ such that $2N=P+p$. The author states that an application of Brun's sieve method [Skrifter Videnskapsselskapets i Kristiania. I. Mat.-Nat. Kl. 1920, no. 3] gives $$ H(2N)>c\sb 3N/\log\sp 2N-\sum\sb {Q\in E}\vert R\sb Q(2N)\vert , $$ where $E$ is a certain set and $$ R\sb Q(x)=\sum\log p·\exp\,(-p\log x/x)-x/\{\varphi(Q)\log x\}, $$ where the sum extends over all the primes $p0$ for $N\geq c\sb 7$. Reviewed by L. Schoenfeld © Copyright American Mathematical Society 1999