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Let p be a prime such that p ≡ 3 (mod 4). Then, we show that there is no imaginary cyclic quartic extension K of ℚ whose class number is p. Suppose L / ℚ is a cyclic extension of number fields with an odd degree. Then, we show that 2 does not divide the class number of L if the class group of L is cyclic. We also construct some families of number fields whose class number is not divisible by a fixed prime. [ABSTRACT FROM AUTHOR]

Let Q (n , d) be the number of representations of n by all reduced quadratic forms of discriminant d. We give an asymptotic formula for the number of n ≤ x such that 1 2 w (d) Q (k n + l , d) is odd, where k , l are integers and w (d) = 2 if d < − 4 , w (d) = 4 if d = − 4 and w (d) = 6 if d = − 3. As applications, we improve early results on odd values of the number of four-core partitions, the Rogers–Ramanujan's partitions and five-regular partitions. [ABSTRACT FROM AUTHOR]

In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime p ∑ i , j , k ≥ 1 gcd (i j k , p) = 1 i + j + k = p 1 i j k ≡ − 2 B p − 3 (mod p) , where B n is the nth Bernoulli number. This congruence was generalized by Wang and Cai in 2014, and Cai, Shen and Jia in 2017 by replacing the odd prime p in the summation and modulus with an odd prime power, and a product of two odd prime powers, respectively. In particular, Cai, Shen and Jia proposed a conjectural congruence: for any positive integer n with an odd prime factor p such that p r | | n where r ≥ 1 ∑ i , j , k ≥ 1 gcd (i j k , n) = 1 i + j + k = n 1 i j k ≡ − 2 B p − 3 ⋅ n p ⋅ ∏ prime q | n q ≠ p 1 − 2 q 1 − 1 q 3 (mod p r). In this paper, we establish the following generalization of their conjecture: for any positive integer n with an odd prime factor p such that p r | | n where r ≥ 1 ∑ i , j , k ≥ 1 gcd (i j k , n) = 1 a 1 i + a 2 j + a 3 k = A n 1 i j k ≡ − 2 B p − 3 ⋅ n p ⋅ A g 3 3 1 a 1 2 g 1 2 + 1 a 2 2 g 2 2 + 1 a 3 2 g 3 2 × ∏ prime q | n q ≠ p 1 − 2 q 1 − 1 q 3 (mod p r) , where a 1 , a 2 and a 3 are positive integers coprime to p , and A is a positive common multiple of a 1 , a 2 and a 3 . Also, g 1 = gcd (a 2 , a 3) , g 2 = gcd (a 3 , a 1) , g 3 = gcd (a 1 , a 2) and g = gcd (a 1 , a 2 , a 3). [ABSTRACT FROM AUTHOR]

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 easy to see that t ≥ 3. In this paper, intrigued by De Koninck and Luca's work, we further determine all weakly prime-additive numbers n such that n = 2 2 + p 2 + q 2 , where p , q are distinct odd prime factors of n. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Diophantine equation in the title, where p , q are distinct odd prime numbers and a , b are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of p , q , a and b. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well as the factorization of certain polynomials. [ABSTRACT FROM AUTHOR]

Recently, Hirschhorn and the first author considered the parity of the function a (n) which counts the number of integer partitions of n wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of a (2 m) based solely on properties of m. In this paper, we quickly reprove their result, and then extend it to an explicit characterization of the parity of a (n) for all n ≢ 7 (mod 8). We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case n ≡ 7 (mod 8) , where, interestingly, the behavior of a (n) modulo 2 appears to be entirely different. In particular, we conjecture that, asymptotically, a (8 m + 7) is odd precisely 5 0 % of the time. This conjecture, whose broad generalization to the context of eta-quotients will be the topic of a subsequent paper, remains wide open. [ABSTRACT FROM AUTHOR]

An odd Durfee symbol of n is an array of positive odd integers and a subscript D , a 1 a 2 ⋯ a s b 1 b 2 ⋯ b t D such that 2 D + 1 ≥ a 1 ≥ a 2 ≥ ⋯ ≥ a s > 0 , 2 D + 1 ≥ b 1 ≥ b 2 ≥ ⋯ ≥ b t > 0 , and n = ∑ i = 1 s a i + ∑ j = 1 t b j + 2 D 2 + 2 D + 1. Andrews defined the odd rank of an odd Durfee symbol as (s − t). Let N 0 (a , M ; n) be the number of odd Durfee symbols of n with odd rank congruent to a modulo M. We decompose the generating function of N 0 (a , M ; n) into modular and mock modular parts. Specifically, we derive some special cases of the generating functions of N 0 (a , M ; n) for M ∈ { 3 , 6 , 1 2 }. Some generating functions are related to classical mock theta functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Iwasawa λ -invariant of the cyclotomic ℤ 2 -extension of ℚ (p q) , where p , q are distinct odd prime numbers satisfying certain arithmetic conditions. Moreover, we obtain an upper bound of the 2 -part of the class number of certain quartic number fields by calculating the Sinnott index explicitly. [ABSTRACT FROM AUTHOR]

Robin's criterion states that the Riemann hypothesis is equivalent to σ (n) < e γ n log log n for all integers n ≥ 5 0 4 1 , where σ (n) is the sum of divisors of n and γ is the Euler–Mascheroni constant. We prove that the Riemann hypothesis is equivalent to the statement that σ (n) < e γ 2 n log log n for all odd numbers n ≥ 3 4 ⋅ 5 3 ⋅ 7 2 ⋅ 1 1 ⋯ 6 7. Lagarias's criterion for the Riemann hypothesis states that the Riemann hypothesis is equivalent to σ (n) < H n + exp H n log H n for all integers n ≥ 1 , where H n is the n th harmonic number. We establish an analogue to Lagarias's criterion for the Riemann hypothesis by creating a new harmonic series H n ′ = 2 H n − H 2 n and demonstrating that the Riemann hypothesis is equivalent to σ (n) ≤ 3 n log n + exp H n ′ log H n ′ for all odd n ≥ 3. We prove stronger analogues to Robin's inequality for odd squarefree numbers. Furthermore, we find a general formula that studies the effect of the prime factorization of n and its behavior in Robin's inequality. [ABSTRACT FROM AUTHOR]

Using convexity properties of reciprocals of zeta functions, especially the reciprocal of the Riemann zeta function, we show that certain weighted Mertens sums are biased in favor of square-free integers with an odd number of prime factors. We study such type of bias for different ranges of the parameters and then consider generalizations to Mertens sums supported on semigroups of integers generated by relatively large subsets of prime numbers. We further obtain a wider range for the parameters both unconditionally and then conditionally on the Riemann Hypothesis. At the same time, we extend to certain semigroups, two classical summation formulas originating from the works of Landau concerning the behavior of derivatives of the reciprocal of the Riemann zeta function at s = 1. [ABSTRACT FROM AUTHOR]

In this paper we prove that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further show that, for n large enough, the two quantities are different and which of the two is bigger depends on the parity of n. This solves a recent conjecture formulated by Bringmann and Mahlburg (2012). [ABSTRACT FROM AUTHOR]

Let s (a , b) denote the classical Dedekind sum and S (a , b) = 1 2 s (a , b). Recently, Du and Zhang proved the following reciprocity formula. If a and b are odd natural numbers, (a , b) = 1 , then S (2 a ∗ , b) + S (2 b ∗ , a) = a 2 + b 2 + 4 2 a b − 3 , where a a ∗ ≡ 1 mod b and b b ∗ ≡ 1 mod a. In this paper, we show that this formula is a special case of a series of similar reciprocity formulas. Whereas Du and Zhang worked with the connection of Dedekind sums and values of L -series, our main tool is the three-term relation for Dedekind sums. [ABSTRACT FROM AUTHOR]

Acquaah and Konyagin showed that if N is an odd perfect number with prime factorization N = p 1 a 1 p 2 a 2 ⋯ p k a k , where p 1 < p 2 ⋯ < p k , then one must have p k < 3 1 / 3 N 1 / 3 . Using methods similar to theirs, we show that p k − 1 < (2 N) 1 / 5 and that p k − 1 p k < 6 1 / 4 N 1 / 2. We also show that if p k and p k − 1 are close to each other, then these bounds can be further strengthened. [ABSTRACT FROM AUTHOR]

Let G be a finite additive abelian group with exponent n and S = g 1 ⋯ g t be a sequence of elements in G. For any element g of G and A ⊆ { 1 , 2 , ... , n − 1 } , let N A , g (S) denote the number of subsequences T = ∏ i ∈ I g i of S such that ∑ i ∈ I a i g i = g , where I ⊆ { 1 , ... , t } and a i ∈ A. In this paper, we prove that N A , 0 (S) ≥ 2 | S | − D A (G) + 1 , when A = { 1 , ... , n − 1 } , where D A (G) is the smallest positive integer l , such that every sequence S over G of length at least l has nonempty subsequence T = ∏ i ∈ I g i such that ∑ i ∈ I a i g i = 0 , I ⊆ { 1 , ... , t } and a i ∈ A. Moreover, we classify the sequences such that N A , 0 (S) = 2 | S | − D A (G) + 1 , where the exponent of G is an odd number. [ABSTRACT FROM AUTHOR]

Let p (n , k) denote the number of partitions of n into parts less than or equal to k. We show several properties of this function modulo 2. First, we prove that for fixed positive integers k and m , p (n , k) is periodic modulo m. Using this, we are able to find lower and upper bounds for the number of odd values of the function for a fixed k. [ABSTRACT FROM AUTHOR]

For an odd prime , we determine a minimal set of topological generators of the pro- Iwahori subgroup of a split reductive group over . In the simple adjoint case and for any sufficiently large regular prime , we also construct Galois extensions of with Galois group between the pro- and the standard Iwahori subgroups of . [ABSTRACT FROM AUTHOR]

Wang and Cai established the following harmonic congruence for odd prime and positive integer : where denote the set of positive integers which are prime to . In this paper, the authors generalize it into super congruence for odd prime and positive integer : where and are positive integers such that [ABSTRACT FROM AUTHOR]

In RSA cryptography numbers of the form , with and two distinct proportional primes play an important role. For a fixed real number we formalize this by saying that an integer is an RSA-integer if and are primes satisfying . Recently Dummit, Granville and Kisilevsky showed that substantially more than a quarter of the odd integers of the form up to , with both prime, satisfy . In this paper, we investigate this phenomenon for RSA-integers. We establish an analogue of a strong form of the prime number theorem with the logarithmic integral replaced by a variant. From this we derive an asymptotic formula for the number of RSA-integers which is much more precise than an earlier one derived by Decker and Moree in 2008. [ABSTRACT FROM AUTHOR]

In this paper, we continue to investigate the properties of those sequences satisfying the condition . As applications we deduce some recurrence relations and congruences for Bernoulli and Euler numbers. [ABSTRACT FROM AUTHOR]

For any non-negative integer , define and by respectively. We mainly prove that for any positive integer and odd prime , which were originally conjectured by Sun. [ABSTRACT FROM AUTHOR]

Let denote a complex variable with . For a positive integer let If we define for each nonnegative integer . In this paper, we determine results of the type [ABSTRACT FROM AUTHOR]

In this paper, we prove that, for any given odd number , the number of integers such that is a square is , where is the Euler totient function. Beyond this, we pose a conjecture and a problem for further research. [ABSTRACT FROM AUTHOR]

In 2014, Wang and Cai established the following harmonic congruence for any odd prime and positive integer , where and denote the set of positive integers which are prime to . In this paper, we obtain an unexpected congruence for distinct odd primes , and positive integers , and the necessary and sufficient condition for Finally, we raise a conjecture that for and odd prime power , , However, we fail to prove it even for with three distinct prime factors. [ABSTRACT FROM AUTHOR]

Let be the number of relatively prime subsets of . In this article, we obtain some local behaviors of and related functions. For instance, we show that is positive for every odd number , and for every . [ABSTRACT FROM AUTHOR]

Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct modulo powers of 3. In this paper, we focus on congruences for (n) modulo 5. For example, for α ≥ 2 and n ≥ 0, we have In addition, applying some equations given by Hirschhorn and Sellers [Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010) 273-284], some new congruences are established. [ABSTRACT FROM AUTHOR]

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2 + dy2 or 4p = x2 + dy2. In this paper we determine x (mod p) for many values of d. For example,... where x is chosen so that x = 1 (mod 3). We also pose some conjectures on supercongruences modulo p concerning binary quadratic forms [ABSTRACT FROM AUTHOR]

An elementary approach for finding non-trivial parametric solutions to homogeneous symmetric Diophantine equations of odd degree in sufficiently many variables is presented. The method is based on studying a model case of quintic symmetric Diophantine equations in six variables. We prove that every symmetric form of odd degree n ≥ 5 in 6 . 2n-5 variables has a rational parametric solution depending on 2n -- 8 parameters. We also present a solution to a problem of Waring type: if F(xi,..., xN) is a symmetric form of odd degree n ≥ 5 in N = 6 . 2n-4 variables, then for any q ∈ Q the equation F(x¡) = q has a rational parametric solution depending on 2n -- 6 parameters. [ABSTRACT FROM AUTHOR]

In this paper, for any fixed odd positive integer a ≥ 1, we prove that all of the solutions of the Diophantine equation (2am - 1)x + (2m)y = (2am + 1)z m,x,y,z ƐN are given by (m,x,y,z) = (2a, 2,2,2), (1,1,1,1), further, (m,x,y,z) = (a½, 2,3,2), when a is square and the additional solution (m,x,y,z) = (1,1,13,2), when a = 45. Using this result, for any fixed odd positive integer b ≥ 5, we prove that the Diophantine equation bx + 2y = (b + 2)z, x,y,zƐN has only the solution (x,y,z) = (1,1,1), if b ≠ 89 and the solutions (x,y,z) = (1,1,1), (1,13, 2), if b = 89. [ABSTRACT FROM AUTHOR]

We prove that a prime factor q of an odd perfect number x satisfies the inequality q < (3x)1/3. [ABSTRACT FROM AUTHOR]

Every Euclidean algorithm is associated with a kind of continued fraction representation of a number. The representation associated with "odd" Euclidean algorithm we will call "odd" continued fraction. We consider the limit distribution function F(x) for sequences of rationals with bounded sum of partial quotients for "odd" continued fractions. In this paper we prove certain properties of the function F(x). Particularly this function is singular and satisfies a number of functional equations. We also show that the value F(x) can be expressed in terms of partial quotients of the "odd" continued fraction representation of a number x. [ABSTRACT FROM AUTHOR]