The exceptional set for the distribution of primes between consecutive powers

A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of reach at present, even under the assumption of the Riemann Hypothesis. This paper is concerned with the distribution of prime numbers between two consecutive powers of integers, as a natural generalization of the afore-mentioned conjecture.      

s个几乎相等的素数的k次方和(Ⅰ)

假定pθ‖k,当p=2,2|k时，γ=θ 2；其它情况时，γ=θ 1。而R＝П(p-1)|kp^γ。本文在GRH（广义Riemann假设下），证明了当s=2^k 1,1≤k≤11时，任何足够大的整N≡s(modR)都可以表示为s个几乎相等的素数的k次方程。     

Ten consecutive primes in arithmetic progression

In 1967 the first set of 6 consecutive primes in arithmetic progression was found. In 1995 the first set of 7 consecutive primes in arithmetic progression was found. Between November, 1997 and March, 1998, we succeeded in finding sets of 8, 9 and 10 consecutive primes in arithmetic progression. This was made possible because of the increase in computer capability and availability, and the ability to obtain computational help via the Internet. Although it is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression, it is very likely that 10 primes will remain the record for a long time.

     

On exponential sums over primes and application in Waring-Goldbach problem

In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2 log κ/log2, x≥2 and α=a/q λsubject to (a, q) = 1, 1≤a≤q, and λ∈R. Then As an application, we prove that with at most O(N2/8 ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.     

A search for Wieferich and Wilson primes

An odd prime is called a Wieferich prime if

alternatively, a Wilson prime if

To date, the only known Wieferich primes are and , while the only known Wilson primes are , and . We report that there exist no new Wieferich primes , and no new Wilson primes . It is elementary that both defining congruences above hold merely (mod ), and it is sometimes estimated on heuristic grounds that the ``probability" that is Wieferich (independently: that is Wilson) is about . We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and Wilson quotients (mod ).

     

On Diophantine approximation with one prime and three squares of primes

Let λ₁, λ₂, λ₃, λ₄ be non-zero real numbers, not all of the same sign, w real. Suppose that the ratios λ₁/λ₂, λ₁/λ₃ are irrational and algebraic. Then there are in.nitely many solutions in primes p_j, j =1, 2, 3, 4, to the inequality . This improves the earlier result.     

A finiteness result for associated primes of local cohomology modules

We show that the first non-finitely generated local cohomology module of a finitely generated module over a noetherian ring with respect to an ideal has only finitely many associated primes.

     

Diameter of sets and measure of sumsets

For bounded sets A, B of reals we show that wherea=(A),b=(B) andD is the diameter ofB. For large values ofa this yields (A+B)a+D.Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901.     

Large sets in finite fields are sumsets

For a prime p, a subset S of Zp is a sumset if S=A+A for some A⊂Zp. Let f(p) denote the maximum integer so that every subset S⊂Zp of size at least p−f(p) is a sumset. The question of determining or estimating f(p) was raised by Green. He showed that for all sufficiently large p, and proved, with Gowers, that f(p)<cp2/3log1/3p for some absolute constant c. Here we improve these estimates, showing that there are two absolute positive constants c₁,c₂ so that for all sufficiently large p,

     

Diophantine inequalities over Piatetski-Shapiro primes

Let c>1 and : We study the solubility of the Diophantine inequality in Piatetski-Shapiro primes p₁,p₂, .., p_s of the form p_j=[m^{\gamma }] for some m\in \mathbb{N}, and improve the previous results in the cases s = 2, 3, 4.     

Exponential sums over primes in short intervals

In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N = p12 p22 p32 p42 p52, with |pj-(N/5)~(1/2)|≤U = N1/2-1/20 ε, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.     

Properties of the Beurling generalized primes

Text

In this paper, we prove a generalization of Mertens' theorem to Beurling primes, namely that , where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit exists. We also show that this limit coincides with ; for ordinary primes this claim is called Meissel's theorem. Finally, we will discuss a problem posed by Beurling, namely how small |N(x)−[x]| can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c>0 there exists a Q such that and conjecture that this is the best possible bound.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Kw3iNo3fAbk/.     

On various restricted sumsets

For finite subsets A₁,…,A_n of a field, their sumset is given by . In this paper, we study various restricted sumsets of A₁,…,A_n with restrictions of the following forms:

     

Weakly associated primes and primary decomposition of modules over commutative rings

Summary We characterize finitely generated Laskerian modules over a commutative ring in terms of weakly associated prime ideals.     

On the sum of the first <Emphasis Type="Italic">n</Emphasis> primes being a square

Let p _n be the nth prime. In this note, we show that the set of n such that is a square is of asymptotic density zero. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 301–306, July–September, 2007.     

New Fibonacci and Lucas primes

Extending previous searches for prime Fibonacci and Lucas numbers, all probable prime Fibonacci numbers have been determined for and all probable prime Lucas numbers have been determined for . A rigorous proof of primality is given for and for numbers with , , , , , , , , the prime having 3020 digits. Primitive parts and of composite numbers and have also been tested for probable primality. Actual primality has been established for many of them, including 22 with more than 1000 digits. In a Supplement to the paper, factorizations of numbers and are given for as far as they have been completed, adding information to existing factor tables covering .

     

On the structure of the sumsets

Let A be a set of nonnegative integers. For h≥2, denote by hA the set of all the integers representable by a sum of h elements from A. In this paper, we prove that, if k≥3, and A={a₀,a₁,…,a_k−1} is a finite set of integers such that 0=a₀<a₁<?<a_k−1 and (a₁,…,a_k−1)=1, then there exist integers c and d and sets C⊆[0,c−2] and D⊆[0,d−2] such that hA=C∪[c,ha_k−1−d]∪(ha_k−1−D) for all . The result is optimal. This improves Nathanson’s result: h≥max{1,(k−2)(a_k−1−1)a_k−1}.     

On some random thin sets of integers

We show how different random thin sets of integers may have different behaviour. First, using a recent deviation inequality of Boucheron, Lugosi and Massart, we give a simpler proof of one of our results in Some new thin sets of integers in harmonic analysis, Journal d'Analyse Mathématique 86 (2002), 105-138, namely that there exist -Rider sets which are sets of uniform convergence and -sets for all but which are not Rosenthal sets. In a second part, we show, using an older result of Kashin and Tzafriri, that, for , the -Rider sets which we had constructed in that paper are almost surely not of uniform convergence.

     

Primitive densities of certain sets of primes

While it has already been demonstrated that the set of twin primes (primes that differ by 2) is scarce in the $\text{Σ}\text{1}\text{p}$ (all twin primes) converges whereas $\text{Σ}\text{1}\text{p}$ (all primes) diverges, this paper proves in Theorems 1 and 2 the scarcity of twin primes (and, in general, of primes p which differ by any even integer as well as primes p for which yp + z is prime, y positive, z nonzero, (y, z) = 1) in a novel and natural way — by showing that the natural density of such primes compared to the set of all primes is 0, that is, $\text{lim}{}_{\mathrm{n\to \infty }}\text{(}\text{π′(n)}\text{π(n)}\text{) = 0}$, where π′(n) is the number of, say, twin primes between 1 and n for any n, and π(n) is the number of all primes between 1 and n. Theorem 3 then establishes that if a set of primes is scarce in the sense that the sum of the reciprocals of such primes converges, they are also scarce in the natural density sense outlined above.