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Let ω ( n ) denote the number of distinct prime factors of n and let τ k ( n ) denote the k-fold divisor function. Adapting a method of Granville and Soundararajan, we evaluate the centralized moments of ω ( n ) , weighted by τ k ( n ) , and deduce a weighted version of the Erdős-Kac Theorem.

We study the normal number of prime factors of a f ( p ) + a g ( p ) with p prime and f , g distinct Hecke eigenforms of weight two. Assuming a quasi-generalized Riemann hypothesis, we show that the normal order is log log p . We also obtain an estimate for the number of primes p for which a f ( p ) + a g ( p ) = 0 .

Let p be a rational prime and q a power of p. Let n be a non-constant monic polynomial in F q [ t ] which has a prime factor of degree prime to q − 1 . In this paper, we define a Drinfeld modular curve Y 1 Δ ( n ) over A [ 1 / n ] and study the structure around cusps of its compactification X 1 Δ ( n ) , in a parallel way to Katz-Mazur's work on classical modular curves. Using them, we also define a Hodge bundle over X 1 Δ ( n ) such that Drinfeld modular forms of level Γ 1 ( n ) , weight k and some type are identified with global sections of its k-th tensor power.

Denote by P + ( n ) the largest prime factor of an integer n. In this paper, we show that the Elliott-Halberstam conjecture for friable integers (or smooth integers) implies three conjectures concerning the largest prime factors of consecutive integers, formulated by Erdős-Turan in the 1930s, by Erdős-Pomerance in 1978, and by Erdős in 1979 respectively. More precisely, assuming the Elliott-Halberstam conjecture for friable integers, we deduce that the three sets E 1 = { n ⩽ x : P + ( n ) ⩽ x s , P + ( n + 1 ) ⩽ x t } , E 2 = { n ⩽ x : P + ( n ) P + ( n + 1 ) x α } , E 3 = { n ⩽ x : P + ( n ) P + ( n + 1 ) } have an asymptotic density ρ ( 1 / s ) ρ ( 1 / t ) , ∫ T α u ( y ) u ( z ) d y d z , 1/2 respectively for s , t ∈ ( 0 , 1 ) , where ρ ( ⋅ ) is the Dickman function, and T α , u ( ⋅ ) are defined in Theorem 2.

Erdős proved that F ( A ) : = ∑ a ∈ A 1 a log ⁡ a converges for any primitive set of integers A and later conjectured this sum is maximized when A is the set of primes. Banks and Martin further conjectured that F ( P 1 ) > ⋯ > F ( P k ) > F ( P k + 1 ) > ⋯ , where P j is the set of integers with j prime factors counting multiplicity, though this was recently disproven by Lichtman. We consider the corresponding problems over the function field F q [ x ] , investigating the sum F ( A ) : = ∑ f ∈ A 1 deg f ⋅ q deg f . We establish a uniform bound for F ( A ) over all primitive sets of polynomials A ⊂ F q [ x ] and conjecture that it is maximized by the set of monic irreducible polynomials. We find that the analogue of the Banks-Martin conjecture is false for q = 2 , 3, and 4, but we find computational evidence that it holds for q > 4 .

In 1976, Mahler posed the question about the existence of a transcendental function f ∈ Z { z } with bounded coefficients and such that f ( Q ‾ ∩ B ( 0 , 1 ) ) ⊆ Q ‾ . In this paper, we prove, in particular, the existence of such a function but with the weaker requirement that the coefficients have only 2 and 3 as prime factors. More generally, we shall prove that any subset of Q ‾ ∩ B ( 0 , 1 ) , which is closed under complex conjugation and which contains the element 0, is the exceptional set of uncountably many transcendental functions in Z { z } with coefficients having only 2 and 3 as prime factors.

Generalizing the concept of a perfect number is a Zumkeller or integer perfect number that was introduced by Zumkeller in 2003. The positive integer $n$ is a Zumkeller number if its divisors can be partitioned into two sets with the same sum, which will be $\sigma(n)/2$. Generalizing even further, we call $n$ a $k$-layered number if its divisors can be partitioned into $k$ sets with equal sum. In this paper, we completely characterize Zumkeller numbers with two distinct prime factors and give some bounds for prime factorization in case of Zumkeller numbers with more than two distinct prime factors. We also characterize $k$-layered numbers with two distinct prime factors and even $k$-layered numbers with more than two distinct odd prime factors. Some other results concerning these numbers and their relationship with practical numbers and Harmonic mean numbers are also discussed., Comment: 14 pages, accepted version, to appear in the Journal of Number Theory

Let ω ( n ) denote the number of distinct prime factors of the positive integer n. In 1917, Hardy and Ramanujan showed that for all real numbers x ≥ 2 and all positive integers k, ∑ n ≤ x ω ( n ) = k 1 ≤ C x log ⁡ x ( log ⁡ log ⁡ x + D ) k − 1 ( k − 1 ) ! , where C and D are absolute constants. We derive an analogous result when the summand 1 is replaced by f ( n ) , for many nonnegative multiplicative functions f. Summing on k recovers a frequently-used mean-value theorem of Hall and Tenenbaum. We use the same idea to establish a variant of a theorem of Shirokov, concerning multiplicative functions that are o ( 1 ) on average at the primes.

Denote by P the set of all primes and by P + ( n ) the largest prime factor of integer n ⩾ 1 with the convention P + ( 1 ) = 1 . For each η > 1 , let c = c ( η ) > 1 be some constant depending on η and P a , c , η : = { p ∈ P : p = P + ( q − a ) for some prime q with p η q ⩽ c ( η ) p η } . In this paper, under the Elliott-Halberstam conjecture we prove, for y → ∞ , π a , c , η ( x ) : = | ( 1 , x ] ∩ P a , c , η | ∼ π ( x ) or π a , c , η ( x ) ≫ a , η π ( x ) according to values of η. These are complement for some results of Banks-Shparlinski [1] , of Wu [12] and of Chen-Wu [2] .

In this paper, we will use Maynard's method [5] to prove that for any positive integer m, there exists infinitely many integers n with at most two prime factors that can be written in the form p + 2 l in at least m + 1 different ways.

Bang (1886), Zsigmondy (1892) and Birkhoff and Vandiver (1904) initiated the study of the largest prime divisors of sequences of the form a n − b n , denoted P ( a n − b n ) , by essentially proving that for integers a > b > 0 , P ( a n − b n ) ≥ n + 1 for every n > 2 . Since then, the problem of finding bounds on the largest prime factor of Lehmer sequences, Lucas sequences or special cases thereof has been studied by many, most notably by Schinzel (1962), and Stewart (1975, 2013). In 2002, Murty and Wong proved, conditionally upon the abc conjecture, that P ( a n − b n ) ≫ n 2 − ϵ for any ϵ > 0 . In this article, we improve this result for the specific case where b = 1 . Specifically, we obtain a more precise result, and one that is dependent on a condition we believe to be weaker than the abc conjecture. Our result actually concerns the largest prime factor of the nth cyclotomic polynomial evaluated at a fixed integer a, P ( Φ n ( a ) ) , as we let n grow. We additionally prove some results related to the prime factorization of Φ n ( a ) . We also present a connection to Wieferich primes, as well as show that the finiteness of a particular subset of Wieferich primes is a sufficient condition for the infinitude of non-Wieferich primes. Finally, we use the technique used in the proof of the aforementioned results to show an improvement on average of estimates due to Erdős for certain sums.

We construct several new families of non-congruent numbers with arbitrarily many prime factors congruent to 3 modulo 8. Our results generalize the work of Reinholz, Spearman and Yang [14] . Our methods are based on Monsky's formula on the 2-Selmer rank of the congruent elliptic curve and the recent results by Tian, Yuan and Zhang on the congruent number problem.

We find a complete criterion for a Kummer extension K over the rational function field k = F q ( T ) of degree l to have indivisibility of its divisor class number h K by l, where F q is the finite field of order q and l is a prime divisor of q − 1 . More importantly, when h K is not divisible by l, we have h K ≡ 1 ( mod l ) . In fact, the indivisibility of h K by l depends on the number of finite primes ramified in K / k and whether or not the infinite prime of k is unramified in K. Using this criterion, we explicitly construct an infinite family of the maximal real cyclotomic function fields whose divisor class numbers are divisible by l.

For positive integers k and n with k ⩽ n − 1 , let P n , k ( x ) denote the polynomial ∑ j = 0 k ( n j ) x j , where ( n j ) = n ! j ! ( n − j ) ! . In 2011, Khanduja, Khassa and Laishram proved the irreducibility of P n , k ( x ) over the field Q of rational numbers for those n , k for which 2 ≤ 2 k ≤ n ( k + 1 ) 3 . In this paper, we extend the above result and prove that if 2 ≤ 2 k ≤ n ( k + 1 ) e + 1 for some positive integer e and the smallest prime factor of k is greater than e, then there exists an explicitly constructible constant C e depending only on e such that the polynomial P n , k ( x ) is irreducible over Q for k ≥ C e .

Given $k, \ell \in {\bf N}^+$, let $x_{i,j}$ be, for $1 \le i \le k$ and $0 \le j \le \ell$, some fixed integers, and define, for every $n \in {\bf N}^+$, $s_n := \sum_{i=1}^k \prod_{j=0}^\ell x_{i,j}^{n^j}$. We prove that the following are equivalent: (a) There are a real $\theta > 1$ and infinitely many indices $n$ for which the number of distinct prime factors of $s_n$ is greater than the super-logarithm of $n$ to base $\theta$. (b) There do not exist non-zero integers $a_0,b_0,\ldots,a_\ell,b_\ell $ such that $s_{2n}=\prod_{i=0}^\ell a_i^{(2n)^i}$ and $s_{2n-1}=\prod_{i=0}^\ell b_i^{(2n-1)^i}$ for all $n$. We will give two different proofs of this result, one based on a theorem of Evertse (yielding, for a fixed finite set of primes $\mathcal S$, an effective bound on the number of non-degenerate solutions of an $\mathcal S$-unit equation in $k$ variables over the rationals) and the other using only elementary methods. As a corollary, we find that, for fixed $c_1, x_1, \ldots,c_k, x_k \in \mathbf N^+$, the number of distinct prime factors of $c_1 x_1^n+\cdots+c_k x_k^n$ is bounded, as $n$ ranges over $\mathbf N^+$, if and only if $x_1=\cdots=x_k$., Comment: 10 pp., no figures. Fixed various mistakes around Lemma 2. To appear in Journal of Number Theory

We improve Irving's method of the double-sieve [7] by using the DHR sieve. By extending the upper and lower bound sieve functions into their respective non-elementary ranges, we are able to make improvements on the previous records on the number of prime factors of irreducible polynomials at prime arguments. In particular, we prove that irreducible quadratics over Z satisfying necessary local conditions are P 4 infinitely often.

There is a large literature on the asymptotic distribution of numbers free of large prime factors, so-called $\textit{smooth}$ or $\textit{friable}$ numbers. But there is very little known about this distribution that is numerically explicit. In this paper we follow the general plan for the saddle point argument of Hildebrand and Tenenbaum, giving explicit and fairly tight intervals in which the true count lies. We give two numerical examples of our method, and with the larger one, our interval is so tight we can exclude the famous Dickman-de Bruijn asymptotic estimate as too small and the Hildebrand-Tenenbaum main term as too large., Comment: 19 pages

In this work, we give upper bounds for n on the title equation. Our results depend on assertions describing the precise exponents of 2 and 3 appearing in the prime factorization of T k ( x ) = ( x + 1 ) k + ( x + 2 ) k + . . . + ( 2 x ) k . Further, on combining Baker's method with the explicit solution of polynomial exponential congruences (see e.g. [6] ), we show that for 2 ≤ x ≤ 13 , k ≥ 1 , y ≥ 2 and n ≥ 3 the title equation has no solutions.

In this paper, we consider the representation of a large positive integer N ≡ 4 ( mod 24 ) in the form p 2 + x 1 2 + x 2 2 + x 3 2 where p is a prime number and x 1 , x 2 , x 3 are almost-primes. A positive integer is called a P r -number if its number of prime factors counted according to multiplicity is at most r. We establish the above representation in the following two different forms. (i) x 1 is a P 2 -number and x 2 x 3 is a P 182 -number; (ii) x 1 x 2 x 3 is a P 12 -number.

Text A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors p 1 , … , p t of n and positive integers α 1 , … , α t such that n = p 1 α 1 + ⋯ + p t α t . It is clear that t ≥ 3 . In 1992, Erdős and Hegyvari proved that, for any prime p, there exist infinitely many weakly prime-additive numbers with t = 3 which are divisible by p. In this paper, we prove that, for any positive integer m, there exist infinitely many weakly prime-additive numbers with t = 3 which are divisible by m if and only if 8 ∤ m . We also present some related results and pose several problems for further research. Video For a video summary of this paper, please visit https://youtu.be/WC_VRFtY07c .

In this paper, we show that for any given integer k ≥ 2 , there are infinitely many cube-free integers n having exactly k prime divisors such that n is a sum of two rational cubes. This is a cubic analogue of the work of Tian [Ti] , which proves that there are infinitely many congruent numbers having exactly k prime divisors for any given integer k ≥ 1 .

B. Sury proved the following Menon-type identity, ∑ a ∈ U ( Z n ) , b 1 , ⋯ , b r ∈ Z n gcd ⁡ ( a − 1 , b 1 , ⋯ , b r , n ) = φ ( n ) σ r ( n ) , where U ( Z n ) is the group of units of the ring for residual classes modulo n, φ is the Euler's totient function and σ r ( n ) is the sum of r-th powers of positive divisors of n with r being a non-negative integer. Recently, C. Miguel extended this identity from Z to any residually finite Dedekind domain. In this note, we will give a further extension of Miguel's result to the case with many tuples of group of units. For the case of Z , our result reads as follows ∑ a 1 , ⋯ , a s ∈ U ( Z n ) , b 1 , ⋯ , b r ∈ Z n gcd ⁡ ( a 1 − 1 , ⋯ , a s − 1 , b 1 , ⋯ , b r , n ) = φ ( n ) ∏ i = 1 m ( φ ( p i k i ) s − 1 p i k i r − p i k i ( s + r − 1 ) + σ s + r − 1 ( p i k i ) ) , where n = p 1 k 1 ⋯ p m k m is the prime factorization of n.

We show that there are infinitely many primes $p$ such that not only does $p + 2$ have at most two prime factors, but $p + 6$ also has a bounded number of prime divisors. This refines the well known result of Chen., 22 pages; errors with numerical calculations corrected after referee report; main result somewhat improved

Text Define χ ( n ) recursively by χ ( 1 ) = 1 and χ ( n ) = ϕ ( n ) + χ ( n / q ) for all integers n > 1 , where q is the least prime factor of n, and where ϕ is the Euler totient function. We show that χ ( n ) = ϕ ( d ) ( χ ( l ) − 1 ) + χ ( d ) , where n = d l and the prime factors of d are greater than the prime factors of l. We also show χ ( n m ) ≤ χ ( n ) χ ( m ) when n and m are coprime numbers. As an application, we show that for all primes p ≥ 11 , χ ( p 2 − p ) > χ ( p 2 − 1 ) . We discuss the interpretation of χ as the clique number of the power graph of a finite cyclic group and the significance of the inequality in this context. Video For a video summary of this paper, please visit https://youtu.be/p8finzAEJps .

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that for every large odd integer N , and 5 ≤ a ≤ 8 , a ≤ b , 13 60 1 a + 1 b ≤ 1 3 , the equation N = x 2 + p 2 + p 1 3 + p 2 4 + p 3 5 + p 4 a + p 5 b is solvable with x being an almost-prime P r ( a , b ) and the other variables primes, where r ( a , b ) is defined in the Theorem, in particular, r ( 6 , 7 ) = 5 . This result constitutes an refinement upon that of J. Brűdern.

It was proved that, for k = 332 , every pair of large positive odd integers satisfying some necessary conditions can be represented in the form of a pair of one prime, two prime squares and k powers of 2. In this paper, we sharpen the value of k to 128.

Contrary to the case of the irreducible polynomials of degreem over a finite field, say Fq, up to present there are no known (to the knowledge of the authors) explicit formulas for the number of primitive polynomials of degree m over Fq with a prescribed coefficient. For instance, whereas the number of irreducible polynomials with a prescribed trace coefficient (the coefficient of x m 1 ) can be described by a simple beautiful formula due to Carlitz, no analogue is known in the case of primitives, even in any specific non-trivial case. In this paper we derive an explicit formula for the number of primitive elements, in quartic extensions of Mersenne prime fields, having absolute trace zero. We also give a simple formula in the case when Q = (q m − 1)/(q − 1) is prime. More generally, in the former case, for a positive integer N whose prime factors divide Q and satisfy the so called semi-primitive condition, we give an explicit formula for the number of N-free elements with trace zero. In addition we show that if all the prime factors of q − 1 divide m, then the number of primitive elements in Fqm, with prescribed non-zero trace, is uniformly distributed. Finally we explore the related number, Pq,qm(N,c), of elements in Fqm with multiplicative order N and having trace c ∈ Fq. By showing a connection between N-free and order in the special case when LQ | N, where LQ is the largest divisor of q m − 1 with the same radical as that of Q, we are able to derive the number of elements in Fp4, p being a Mersenne prime, with absolute trace zero and having the corresponding large order LQ.

Let m be a fifth power free integer greater than one. Let K be an algebraic number field generated by a fifth root of m over the rational number field. If m has a prime factor p congruent to −1 modulo five, the class number of K is a multiple of five.

Let q ≥ 2 be an integer and S q ( n ) denote the sum of the digits in base q of the positive integer n . It is proved that for every real number α and β with α β , lim x ⟶ + ∞ ⁡ 1 x ♯ { n ≤ x : α ≤ v ( φ ( n ) ) − 1 2 b ( log ⁡ log ⁡ n ) 2 1 3 b ( log ⁡ log ⁡ n ) 3 2 ≤ β } = 1 2 π ∫ α β e − t 2 2 d t , where v ( n ) is either ω ˜ ( n ) or Ω ˜ ( n ) , the number of distinct prime factors and the total number of prime factors p of a positive integer n such that S q ( p ) ≡ a mod b ( a , b ∈ Z , b ≥ 2 ). This extends the results known through the work of P. Erdős and C. Pomerance, M.R. Murty and V.K. Murty to primes under digital constraint.

Under Baker's explicit abc conjecture, we completely solve a conjecture of Hickerson when a product of two or more factorials is equal to n! for a given positive integer n. We also prove it unconditionally when n

We investigate the pair correlation of the sequence of fractional parts of α x n , n ∈ N , where x n is rational valued and α is a real number. As examples, we offer two classes of sequences x n whose pair correlation behaves as that of random sequences for almost all real numbers α. First, sequences of the form x n = a n / b n where a n is lacunary and b n satisfies a certain growth condition. Second, sequences of the form x n = g n / 2 ω ( n ) for a positive integer g which is not a power of 2 and where ω ( n ) denotes the number of distinct prime factors of n.

For a positive integer n, let σ(n) and γ(n) denote the sum of divisors and the product of distinct prime divisors of n, respectively. It is known that, if σ(n)=γ(n)2, then at most two exponents of odd primes are equal to 1 in the prime factorization of n. In this paper, we prove that, if σ(n)=γ(n)2 and only one exponent is equal to 1 in the prime factorization of n, then (1) n is divisible by 3; (2) n is divisible by the fourth powers of at least two odd primes; (3) at least two exponents of odd primes are equal to 2. We also prove that, if σ(n)=γ(n)2, then at least half of the exponents α of the primes have the property that the numbers α+1 must be either primes or prime squares.

A generalization of the well-known Fibonacci sequence is the k-generalized Fibonacci sequence ( F n ( k ) ) n ≥ 2 − k for some integer k ≥ 2 , whose first k terms are 0 , … , 0 , 1 and each term afterwards is the sum of the preceding k terms. In this paper, we look at the prime factors of the reduced rational number F n ( k ) / F m ( l ) as max ⁡ { m , n , k , l } tends to infinity.

Hugh Williams posed an interesting problem of whether there exists a Carmichael number N with p+1|N+1 for all primes p|N. Othman Echi calls such numbers Williams numbers (more precisely, 1-Williams numbers). Carl Pomerance gave a heuristic argument that there are infinitely many counterexamples to the Baillie–PSW probable prime test. Based on some reasonable assumptions there exist infinitely many Williams numbers. There are no examples less than 264≈2×1019. Williams proved that any such numbers must have more than three prime factors. In this paper we prove that there are only finitely many Williams numbers N=∏i=1dpi with a given set of d−3 prime factors p1,…,pd−3. Several methods for the organization of a search for Williams numbers are given. We report that if there are any Williams numbers with exactly four prime factors, then the smallest prime factor is greater than 2×104.

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, it is proved that for each integer k with 4 ≤ k ≤ 5 , and for every sufficiently large even integer N satisfying the congruence condition N ≢ 2 ( mod 3 ) for k = 4 , the equation N = x 2 + p 1 2 + p 2 3 + p 3 4 + p 4 4 + p 5 k is solvable with x being an almost-prime P r and the other variables primes, where r = 6 for k = 4 , and r = 9 for k = 5 . This result constitutes an improvement upon that of R.C. Vaughan.

Text In this paper, the following results are proved: (i) For any odd integer l with at most two distinct prime factors and any positive integer n , the product (1l+1)(2l+1)⋯(nl+1)(1l+1)(2l+1)⋯(nl+1) is not a powerful number; (ii) For any integer r≥1r≥1, there exists a positive integer TrTr such that, if l is a positive odd integer with at most r distinct prime factors and n is an integer with n≥Trn≥Tr, then (1l+1)(2l+1)⋯(nl+1)(1l+1)(2l+1)⋯(nl+1) is not a powerful number. Video For a video summary of this paper, please visit http://youtu.be/nU-nkxNX1BA.

Zhao established a curious congruence, i.e., for any prime p ≥ 3 , ∑ i + j + k = p i , j , k > 0 1 i j k ≡ − 2 B p − 3 ( mod p ) . In this note we prove that for any prime p ≥ 3 and positive integer r, ∑ i + j + k = p r i , j , k ∈ P p 1 i j k ≡ − 2 p r − 1 B p − 3 ( mod p r ) , where P n denotes the set of positive integers which are prime to n.

Text For any positive integer n, let n = q 1 ⋯ q s be the prime factorization of n with q 1 ≥ ⋯ ≥ q s > 1 . A positive integer n is said to be ordinary if the smallest positive integer with exactly n divisors is p 1 q 1 − 1 ⋯ p s q s − 1 , where p k denotes the kth prime. Let ⌊ x ⌋ be the largest integer not exceeding x. In 2006, Brown proved that all square-free integers are ordinary and the set of all ordinary integers has asymptotic density one. In this paper, we prove that, if q ⌊ s ⌋ ≥ 9 ( log s ) 2 , then n is ordinary. Furthermore, the set of such integers n has asymptotic density one. We also determine all ordinary integers which are not divisible by any fifth power of a prime. Video For a video summary of this paper, please visit http://youtu.be/UeIMWjRFUnA .

For a fixed integer g ≥ 2 , we consider the distribution of powers g n in the residue ring modulo a large power of a fixed prime. We also give an application of this result to the exponential pseudorandom number generator modulo such prime powers.

Let $M(x)$ denote the median largest prime factor of the integers in the interval $[1,x]$. We prove that $$M(x)=x^{\frac{1}{\sqrt{e}}\exp(-\text{li}_{f}(x)/x)}+O_{\epsilon}(x^{\frac{1}{\sqrt{e}}}e^{-c(\log x)^{3/5-\epsilon}})$$ where $\text{li}_{f}(x)=\int_{2}^{x}\frac{\{x/t\}}{\log t}dt$. From this, we obtain the asymptotic $$M(x)=e^{\frac{\gamma-1}{\sqrt{e}}}x^{\frac{1}{\sqrt{e}}}(1+O(\frac{1}{\log x})),$$ where $\gamma$ is the Euler Mascheroni constant. This answers a question posed by Martin, and improves a result of Selfridge and Wunderlich., Comment: 7 pages

We first prove that if $a$ has a prime factor not dividing $b$ then there are infinitely many positive integers $n$ such that $\binom {an+bn} {an}$ is not divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover, we provide some new divisibility properties of binomial coefficients: for example, we prove that $\binom {12n} {3n}$ and $\binom {12n} {4n}$ are divisible by $6n-1$, and that $\binom {330n} {88n}$ is divisible by $66n-1$, for all positive integers $n$. As we show, the latter results are in fact consequences of divisibility and positivity results for quotients of $q$-binomial coefficients by $q$-integers, generalizing the positivity of $q$-Catalan numbers. We also put forward several related conjectures., 16 pages, add a note that Conjectures 7.2 and 7.3 are proved, to appear in J. Number Theory

Let G be a finite cyclic group. Every sequence S over G can be written in the form S = ( n 1 g ) ⋅ ⋯ ⋅ ( n k g ) where g ∈ G and n 1 , … , n k ∈ [ 1 , ord ( g ) ] , and the index ind S of S is defined to be the minimum of ( n 1 + ⋯ + n k ) / ord ( g ) over all possible g ∈ G such that 〈 g 〉 = G . A conjecture says that if G is finite such that gcd ( | G | , 6 ) = 1 , then ind ( S ) = 1 for every minimal zero-sum sequence S. In this paper, we prove that the conjecture holds if | G | has two prime factors.

We study the uniform upper bound for the least prime that is a primitive root. Let g ⁎ ( q ) be the least prime primitive root (mod q) where q is a prime power or twice a prime power of a prime p. The upper bound for g ⁎ ( q ) is studied by many authors who succeeded in establishing various conditional upper bounds. However, no uniform bounds were known other than Linnikʼs bound on the least prime in an arithmetic progression. In this paper, we prove that g ⁎ ( q ) ≪ p 3.1 . The exponent 3.1 is improved from the known exponent 4.5 from Linnikʼs bound for the prime modulus.

Text Let the prime factorization of n be n = q 1 q 2 ⋯ q a with q 1 ⩾ q 2 ⩾ ⋯ ⩾ q a ⩾ 2 . A positive integer n is said to be ordinary if the smallest positive integer with exactly n divisors is p 1 q 1 − 1 p 2 q 2 − 1 ⋯ p a q a − 1 , where p k denotes the kth prime. In this paper I prove that all integers of the form ql are ordinary, where l is a square-free positive integer and q is a prime. This confirms a conjecture of Yong-Gao Chen. Video For a video summary of this paper, please click here or visit http://youtu.be/WTY4wr8L_U0 .

A primitive prime divisor of an element an of a sequence (am)m⩾0 is a prime p that divides an, but does not divide am for all m

Text In this paper, the following results are proved: (i) For any odd integer l with at most two distinct prime factors and any positive integer n, the product ( 1 l + 1 ) ( 2 l + 1 ) ⋯ ( n l + 1 ) is not a powerful number; (ii) For any integer r ≥ 1 , there exists a positive integer T r such that, if l is a positive odd integer with at most r distinct prime factors and n is an integer with n ≥ T r , then ( 1 l + 1 ) ( 2 l + 1 ) ⋯ ( n l + 1 ) is not a powerful number. Video For a video summary of this paper, please visit http://youtu.be/nU-nkxNX1BA .

A method is given for generating families of non-congruent numbers with arbitrarily many prime factors. We then use this method to construct an infinite set of new families of these numbers with prime factors of the form 8 k + 3 .

We consider a variant of a question of N. Koblitz. For an elliptic curve $E/\Q$ which is not $\Q$-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes $p$ such that $N_p(E)=#E(\F_p)=p+1-a_p(E)$ is also a prime. We consider a variant of this question. For a newform $f$, without CM, of weight $k\geq 4$, on $\Gamma_0(M)$ with trivial Nebentypus $\chi_0$ and with integer Fourier coefficients, let $N_p(f)=\chi_0(p)p^{k-1}+1-a_p(f)$ (here $a_p(f)$ is the $p^{th}$-Fourier coefficient of $f$). We show under GRH and Artin's Holomorphy Conjecture that there are infinitely many $p$ such that $N_p(f)$ has at most $[5k+1+\sqrt{\log(k)}]$ distinct prime factors. We give examples of about hundred forms to which our theorem applies., Comment: 22 pages

Text Let A 3 be the set of all positive integers pqr, where p, q, r are primes such that at least two of them are not equal. Denote by P ( n ) the largest prime factor of n. For n = p q r ∈ A 3 , define w ( n ) : = P ( p + q ) P ( p + r ) P ( q + r ) . In 2006, Wushi Goldring proved that for any n ∈ A 3 , there exists an i such that w i ( n ) ∈ { 20 , 98 , 63 , 75 } , where w 0 ( n ) = n and w i ( n ) = w ( w i − 1 ( n ) ) ( i ⩾ 1 ). If w ( m ) = n , then m is called a parent of n. Let B 3 be the set of all positive integers p q 2 of A 3 . In this paper, we study the function w extensively. For example, one of our results is that there exist infinitely many n ∈ B 3 which have at least n 1.1886 parents in B 3 . Several open problems are posed. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=8FWvR8_KoHA .