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On the incompatibility of two conjectures concerning prime numbers
 Proc. Symp. Pure Math. (Analytic Number Theory
, 1972
"... Introduction. This talk is about the interplay between computers and theoretical research, as experienced by someone who is not a computer expert. The story involves, among other things, a measure of good luck. Several instances of this will emerge in due course, but one example now may give the ide ..."
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Introduction. This talk is about the interplay between computers and theoretical research, as experienced by someone who is not a computer expert. The story involves, among other things, a measure of good luck. Several instances of this will emerge in due course, but one example now may give the idea: The speaker and his coworker, Douglas Hensley,
COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ
"... ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each ..."
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ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c> 0. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of φ at p, and on a result of HeathBrown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.
Large families of pseudorandom sequences of k symbols and their complexity
 Part I, General Theory of Information Transfer and Combinatorics, eds. R. Ahlswede et al., LNCS 4123
, 2006
"... In earlier papers we introduced the measures of pseudorandomness of finite binary sequences [13], introduced the notion of f–complexity of families of binary sequences, constructed large families of binary sequences with strong PR (= pseudorandom) properties [6], [12], and we showed that one of the ..."
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In earlier papers we introduced the measures of pseudorandomness of finite binary sequences [13], introduced the notion of f–complexity of families of binary sequences, constructed large families of binary sequences with strong PR (= pseudorandom) properties [6], [12], and we showed that one of the earlier constructions
A Generalization Of A Conjecture Of Hardy And Littlewood To Algebraic Number Fields
 Rocky Mountain J. Math
, 1998
"... We generalize conjectures of Hardy and Littlewood concerning the density of twin primes and ktuples of primes to arbitrary algebraic number fields. In one of their great Partitio Numerorum papers [7], Hardy and Littlewood advance a number of conjectures involving the density of pairs and ktuples ..."
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We generalize conjectures of Hardy and Littlewood concerning the density of twin primes and ktuples of primes to arbitrary algebraic number fields. In one of their great Partitio Numerorum papers [7], Hardy and Littlewood advance a number of conjectures involving the density of pairs and ktuples of primes separated by fixed gaps. For example, if d is even, we define P d (x) = {0 < n < x : n, n + d are both prime}. They conjecture both that lim x## P d (x) P 2 (x) = # odd pd p  1 p  2 and that P 2 (x) is asymptotic to 2 # p>2 # 1  1 (p  1) 2 # # x 2 dy (log y) 2 . We will refer to the first equation as the "relative conjecture" and the second as the "absolute conjecture." There has been much numerical verification of these conjectures, and many attempts at proofs. Balog [1] proves a result that implies that the conjectures are true "on average," where the average is taken over the possible shapes of the ktuples. Golubev [6] compares these conjectures with provable analogous limit results for patterns of numbers prime to n. Turan [18] relates such theorems to zeroes of the #function, using the large sieve rather than Hardy and Littlewood's circle method. There are also many generalizations to specific fields. Most of those generalizations use "Conjecture H" of Sierpinski and Schinzel [14,15]. For example, Sierpinski [17] shows that Conjecture H implies the existence of infinitely many prime Gaussian integers di#ering by 2. Bateman and Horn [2,3] quote a quantitative form of Conjecture H which allows them to estimate the density of rational twin primes. Shanks [16] numerically verifies that the density of prime pairs of the form a + i, a + 2 + i in the Gaussian integers matches that of the quantitative form of Conjecture H. Rieger ...
Dense Admissible Sets
"... . Call a set of integers fb1 ; b2 ; : : : ; bk g admissible if for any prime p, at least one congruence class modulo p does not contain any of the b i . Let ae (x) be the size of the largest admissible set in [1; x]. The Prime ktuples Conjecture states that any for any admissible set, there are inf ..."
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. Call a set of integers fb1 ; b2 ; : : : ; bk g admissible if for any prime p, at least one congruence class modulo p does not contain any of the b i . Let ae (x) be the size of the largest admissible set in [1; x]. The Prime ktuples Conjecture states that any for any admissible set, there are infinitely many n such that n+b1 ; n+b2 ; : : : n+bk are simultaneously prime. In 1974, Hensley and Richards [3] showed that ae (x) ? ß(x) for x sufficiently large, which shows that the Prime ktuples Conjecture is inconsistent with a conjecture of Hardy and Littlewood that for all integers x; y 2, ß(x + y) ß(x) + ß(y): In this paper we examine the behavior of ae (x), in particular, the point at which ae (x) first exceeds ß(x), and its asymptotic growth.
AN AMAZING PRIME HEURISTIC
"... The record for the largest known twin prime is constantly changing. For example, in October of 2000, David Underbakke found the record primes: ..."
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The record for the largest known twin prime is constantly changing. For example, in October of 2000, David Underbakke found the record primes:
HYPOTHESIS H AND AN IMPOSSIBILITY
"... Abstract. Dirichlet’s 1837 theorem that every coprime arithmetic progression a mod m contains infinitely many primes is often alluded to in elementary number theory courses but usually proved only in special cases (e.g., when m=3 or m=4), where the proofs parallel Euclid’s argument for the existence ..."
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Abstract. Dirichlet’s 1837 theorem that every coprime arithmetic progression a mod m contains infinitely many primes is often alluded to in elementary number theory courses but usually proved only in special cases (e.g., when m=3 or m=4), where the proofs parallel Euclid’s argument for the existence of infinitely many primes. It is natural to wonder whether Dirichlet’s theorem in its entirety can be proved by such “Euclidean ” arguments. In 1912, Schur showed that one can construct an argument of this type for every progression a mod m satisfying a 2 ≡ 1 (mod m), and in 1988 Murty showed that these are the only progressions for which such an argument can be given. Murty’s proof uses some deep results from algebraic number theory (in particular the Chebotarev density theorem). Here we give a heuristic explanation for this result by showing how it follows from Bunyakovsky’s conjecture on prime values of polynomials. We also propose a widening of Murty’s definition of a Euclidean proof. With this definition, it appears difficult to classify the progressions for which such a proof exists. However, assuming Schinzel’s Hypothesis H, we show that again such a proof exists only when a 2 ≡ 1 (mod m).
A RATIO ASSOCIATED WITH 0(x) = n
, 1984
"... Let $(x) be Euler's totient function. The literature on solving the equation cj)0) = n (see [1, pp. 221223], [25], [6, pp. 5055, problems B36B42], [711], [12, pp. 228256], and the references therein) can be viewed as a collection of open problems. For n = 2 a, we essentially have the problem ..."
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Let $(x) be Euler's totient function. The literature on solving the equation cj)0) = n (see [1, pp. 221223], [25], [6, pp. 5055, problems B36B42], [711], [12, pp. 228256], and the references therein) can be viewed as a collection of open problems. For n = 2 a, we essentially have the problem of factoring the
BOUNDED GAPS BETWEEN PRIMES
"... Abstract. Recently, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs pn, pn 1 of consecutive primes for which pn 1 pn ¤ B. This can be seen as a massive breakthrough on the subject of twin primes and other delicate questions about prime numbers that had ..."
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Abstract. Recently, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs pn, pn 1 of consecutive primes for which pn 1 pn ¤ B. This can be seen as a massive breakthrough on the subject of twin primes and other delicate questions about prime numbers that had previously seemed intractable. In this article we will discuss Zhang’s extraordinary work, putting it in its context in analytic number theory, and sketch a proof of his theorem. Zhang even proved the result with B 70 000 000. A cooperative team, polymath8, collaborating only online, has been able to lower the value of B to 4680, and it seems plausible that these techniques can be pushed somewhat further, though the limit of these methods seem, for now, to be B 12. Contents