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We show that for each positive integer a there exist only finitely many prime numbers p such that a appears an odd number of times in the period of continued fraction of p or 2 p. We also prove that if p is a prime number and D = p or 2 p is such that the length of the period of continued fraction expansion of D is divisible by 4, then 1 appears as a partial quotient in the continued fraction of D. Furthermore, we give an upper bound for the period length of continued fraction expansion of D , where D is a positive non-square, and factorize some family of polynomials with integral coefficients connected with continued fractions of square roots of positive integers. These results answer several questions recently posed by Miska and Ulas [MU]. [ABSTRACT FROM AUTHOR]

In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic p -rational number fields of degree 2 t for any odd prime number p and any integer t ≥ 1. Using the criteria provided by him to check p -rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic p -rational fields have been shown to exist in recent years. In this article, for any integer k ≥ 1 , we build upon the existing work and prove the existence of infinitely many prime numbers p for which the imaginary quadratic fields Q (− (p − 1)) , ... , Q (− (p − k)) and Q (− p (p − 1)) , ... , Q (− p (p − k)) are all p -rational. This can be construed as analogous results in the spirit of Iizuka's conjecture on the divisibility of class numbers of consecutive quadratic fields. We also address a similar question of p -rationality for two consecutive real quadratic fields by proving the existence of infinitely many p -rational fields of the form Q ( p 2 + 1) and Q ( p 2 + 2). The result for imaginary quadratic fields is accomplished by producing infinitely many primes for which the corresponding consecutive discriminants have large square divisors and the same for real quadratic fields is proven using a result of Heath-Brown on the relative density of square-free values of polynomials at prime arguments. [ABSTRACT FROM AUTHOR]

For an odd prime number p , we consider degree p extensions L / K of p -adic fields with normal closure L ˜ such that the Galois group of L ˜ / K is the dihedral group of order 2 p. We shall prove a complete characterization of the freeness of the ring of integers O L over its associated order A L / K in the unique Hopf-Galois structure on L / K , which is analogous to the one already known for cyclic degree p extensions of p -adic fields. We shall derive positive and negative results on criteria for the freeness of O L as A L / K -module. [ABSTRACT FROM AUTHOR]

Let p be an odd prime number and k an imaginary quadratic field in which p splits. In this paper, we consider a weak form of Greenberg's generalized conjecture for p and k , which states that the non-trivial Iwasawa module of the maximal multiple Z p -extension field over k has a non-trivial pseudo-null submodule. We prove this conjecture for p and k under the assumption that the Iwasawa λ -invariants vanish for the Z p -extensions over k in which one of the prime ideals of k lying above p do not ramify and that the characteristic ideal of the Iwasawa module associated to the cyclotomic Z p -extension over k has a square-free generator. [ABSTRACT FROM AUTHOR]

For a number field F and an odd prime number p , let F ˜ be the compositum of all Z p -extensions of F and Λ ˜ the associated Iwasawa algebra. Let G S (F ˜) be the Galois group over F ˜ of the maximal extension which is unramified outside p -adic and infinite places. In this paper we study the Λ ˜ -module X S (− i) (F ˜) : = H 1 (G S (F ˜) , Z p (− i)) and its relationship with X (F ˜ (μ p)) (i − 1) Δ , the Δ : = Gal (F ˜ (μ p) / F ˜) -invariant of the Galois group over F ˜ (μ p) of the maximal abelian unramified pro- p -extension of F ˜ (μ p). More precisely, we show that under a decomposition condition, the pseudo-nullity of the Λ ˜ -module X (F ˜ (μ p)) (i − 1) Δ is implied by the existence of a Z p d -extension L with X S (− i) (L) : = H 1 (G S (L) , Z p (− i)) being without torsion over the Iwasawa algebra associated to L , and which contains a Z p -extension F ∞ satisfying H 2 (G S (F ∞) , Q p / Z p (i)) = 0. As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer i ≡ 1 mod [ F (μ p) : F ]. This existence is fulfilled for (p , i) -regular fields. [ABSTRACT FROM AUTHOR]

We investigate the integer solutions of Diophantine equations related to perfect numbers. These solutions generalize the example, found by Descartes in 1638, of an odd, "spoof" perfect factorization 3 2 ⋅ 7 2 ⋅ 11 2 ⋅ 13 2 ⋅ 22021 1. More recently, Voight found the spoof perfect factorization 3 4 ⋅ 7 2 ⋅ 11 2 ⋅ 19 2 ⋅ (− 127) 1. No other examples appear in the literature. We compute all nontrivial, odd, primitive spoof perfect factorizations with fewer than seven bases—there are twenty-one in total. We show that the structure of odd, spoof perfect factorizations is extremely rich, and there are multiple infinite families of them. This implies that certain approaches to the odd perfect number problem that use only the multiplicative nature of the sum-of-divisors function are unworkable. On the other hand, we prove that there are only finitely many nontrivial, odd, primitive spoof perfect factorizations with a fixed number of bases; this generalizes previous results, which presupposed positivity of the bases. [ABSTRACT FROM AUTHOR]

Let l be an odd prime number and k = Q (ζ + ζ − 1) where ζ is a primitive l -th root of unity. We provide a sufficient condition for the monogenity of a cyclic extension K s / k of degree l , where s is an integer of k and K s is a field defined by Rikuna's generic cyclic polynomial. As an application, we prove that there exist infinitely many monogenic extensions K s / k for l ≥ 5. [ABSTRACT FROM AUTHOR]

Let p be an odd prime number. Let K be the p -th cyclotomic field. We give general formulae for the root numbers of the Jacobian varieties of the Fermat curves X p + Y p = δ where δ is an integer. As an application of these general formulae, we derive the equidistribution of the root numbers for the families of Jacobian varieties of the Fermat curves. [ABSTRACT FROM AUTHOR]

In this note, we confirm a conjectural formula on the number of representations of the square of an odd prime by a sum of an odd number of squares, i.e., # { (x 1 , ... , x 2 k + 1) ∈ Z 2 k + 1 : x 1 2 + ⋯ + x 2 k + 1 2 = p 2 } , proposed by Cooper. [ABSTRACT FROM AUTHOR]

Recently, Andrews defined a partition function EO (n) which counts the number of partitions of n in which every even part is less than each odd part. He also defined a partition function EO ‾ (n) which counts the number of partitions of n enumerated by EO (n) in which only the largest even part appears an odd number of times. Andrews proposed to undertake a more extensive investigation of the properties of EO ‾ (n). In this article, we prove infinite families of congruences for EO ‾ (n). We next study distribution of EO ‾ (n). We prove that there are infinitely many integers N in every arithmetic progression for which EO ‾ (2 N) is even; and that there are infinitely many integers M in every arithmetic progression for which EO ‾ (2 M) is odd so long as there is at least one. We further prove that EO ‾ (n) is even for almost all n. Very recently, Uncu has treated a different subset of the partitions enumerated by EO (n). We prove that Uncu's partition function is divisible by 2 k for almost all k. We use arithmetic properties of modular forms and Hecke eigenforms to prove our results. [ABSTRACT FROM AUTHOR]

We give an algorithm to enumerate all primitive abundant numbers (PAN) with a fixed Ω, the number of prime factors counted with their multiplicity. We explicitly find all PAN up to Ω = 6 , count all PAN and square-free PAN up to Ω = 7 and count all odd PAN and odd square-free PAN up to Ω = 8. We find primitive weird numbers (PWN) with up to 16 prime factors, the largest of which is a number with 14712 digits. We find hundreds of PWN with exactly one square odd prime factor: as far as we know, only five were known before. We find all PWN with at least one odd prime factor with multiplicity greater than one and Ω = 7 and prove that there are none with Ω < 7. Regarding PWN with a cubic (or higher power) odd prime factor, we prove that there are none with Ω ≤ 7. We find several PWN with 2 square odd prime factors, and one with 3 square odd prime factors. These are the first such examples. We finally observe that these results are in favor of the existence of PWN with arbitrarily many prime factors. For a video summary of this paper, please visit https://youtu.be/kEl20Tyf1zU. [ABSTRACT FROM AUTHOR]

Text This paper focuses on the number of partitions of a positive integer n into k positive summands, where k is an integer between 1 and n . Recently some upper bounds were reported for this number in [Merca14] . Here, it is shown that these bounds are not as tight as an earlier upper bound proved in [Andrews76-1] for k ≤ 0.42 n . A new upper bound for the number of partitions of n into k summands is given, and shown to be tighter than the upper bound in [Merca14] when k is between O ( n ln ⁡ n ) and n − O ( n ln ⁡ n ) . It is further shown that the new upper bound is also tighter than two other upper bounds previously reported in [Andrews76-1] and [Colman82] . A generalization of this upper bound to number of partitions of n into at most k summands is also presented. Video For a video summary of this paper, please visit http://youtu.be/Pb6lKB3MnME . [ABSTRACT FROM AUTHOR]

Suppose that p is an odd prime and α , β are prime to p . We prove that p 2 divides the truncated hypergeometric function F 2 3 [ α β 1 − α − β 1 1 | 1 ] p provided 〈 α 〉 p + 〈 β 〉 p ≤ p , where 〈 α 〉 p denotes the least non-negative residue of α modulo p . [ABSTRACT FROM AUTHOR]

Let ℓ , p be odd primes such that p = 2 n ℓ + 1 . In the paper the sufficient conditions are proved under which prime q does not divide the class number h K of subfields K of Q ( ζ p + ζ p − 1 ) , where [ K : Q ] = ℓ . [ABSTRACT FROM AUTHOR]

In this paper, we will evaluate the values of the Dedekind zeta functions at s= - 1 of the simplest quartic fields. We first introduce Siegel's formula for the values of the Dedekind zeta function of a totally real number field at negative odd integers, and will apply Siegel's formula to the simplest quartic fields. In the second, we will develop the basic arithmetic properties of the simplest quartic fields which will be necessary in our computation. We will compute the discriminant, ring of integers, and different of the simplest quartic fields. In the third, we will give a full description for a Siegel lattice of the simplest quartic fields, and develop a method of computing sum of divisors function for an ideal. Finally, by combining these results, we compute the values of the Dedekind zeta functions at s = - 1 of the simplest quartic fields. [ABSTRACT FROM AUTHOR]

Recently, mixed polynomial-exponential equations similar to the one in the title have been considered by many authors. In these results a certain non-coprimality condition plays an important role. In this paper we completely solve the title equation for odd positive integers x with x < 50. Since we avoid the mentioned non-coprimality condition, this can be considered as a partial completion of the above mentioned results. It seems that the deep effective tools (such as Baker's method) alone are not capable to handle the problem. We combine local arguments and Baker's method to prove our results. [ABSTRACT FROM AUTHOR]

Let b2a(n) denote the number of 2a-regular partitions of n. Suppose j is a positive integer and i is odd. If a ∈ {1,2}, we show that#{0 ≤ n ≤ X:b2a(n) ≡ i(mod2j)} ≫ jX. This improves a result of Ono and Penniston [6]. For a ≥ 3 odd, we show that#{0 ≤ n ≤ X:b2a(n) ≡ i(mod2j)} ≫ a,jXlogX. Finally for a ≥ 4 even, we prove that#{0 ≤ n ≤ X:b2a(n) ≡ i(mod2j)} ≫ a,jXlogX(loglogX) &#948 (a), where &#948 (a)=2a2 - 1 - 2. [ABSTRACT FROM AUTHOR]

Abstract: We study integers satisfying the relation , where and are the sum of divisors and the product of distinct primes dividing n, respectively. If the prime dividing a solution n is congruent to 3 modulo 8 then it must be greater than 41, and every solution is divisible by at least the fourth power of an odd prime. Moreover at least 2/5 of the exponents a of the primes dividing any solution have the property that is a prime power. Lastly we prove that the number of solutions up to is at most , for any and all . [Copyright &y& Elsevier]

Abstract: Let be a square-free polynomial of degree ⩾3 and be an odd positive integer. Based on our earlier investigations we prove that there exists a function such that the Jacobians of the curves have all positive ranks over . Similarly, we prove that there exists a function such that the Jacobians of the curves have all positive ranks over . Moreover, if for some , we prove the existence of a function such that the Jacobians of the curves have all positive ranks over . We present also some applications of these results. Finally, we present some results concerning the torsion parts of the Jacobians of the superelliptic curves and for a prime p and and and apply our result in order to prove the existence of a function such that the Jacobians of the curves have both positive rank over . [Copyright &y& Elsevier]

Abstract: Let be a real quadratic field with odd class number and its fundamental unit satisfies . We give some congruence relations about explicitly. [Copyright &y& Elsevier]

Abstract: In this paper we computed all the 379 118 solutions of the equation in distinct odd natural numbers for . We provide a lower bound for the number of solutions of . Also, we answer similar questions about the equation . Finally, we prove some results concerning optimization of and . [Copyright &y& Elsevier]

Abstract: The Apéry polynomials are given by (Those are Apéry numbers.) Let p be an odd prime. We show that and that for any p-adic integer . This enables us to determine explicitly , and in the case . Another consequence states that We also prove that for any prime we have where are Bernoulli numbers. [Copyright &y& Elsevier]

Abstract: Let n, k be positive integers. In this paper, we prove that if k is an odd prime with , then the product is not a powerful number. [Copyright &y& Elsevier]

Abstract: In this paper we consider the integers of the forms and , which are ever focused by F. Cohen, P. Erdős, J.L. Selfridge, W. Sierpiński, etc. We establish a general theorem. As corollaries, we prove that (i) there exists an infinite arithmetic progression of positive odd numbers for each term k of which and any nonnegative integer n, each of four integers , , and has at least two distinct odd prime factors; (ii) there exists an infinite arithmetic progression of positive odd numbers for each term k of which and any nonnegative integer n, each of ten integers , , , , , , , , and has at least two distinct odd prime factors; (iii) there exists an infinite arithmetic progression of positive odd numbers for each term k of which and any nonnegative integer n, each of ten integers , , , , , , , , and has at least two distinct odd prime factors. Furthermore, we pose several related open problems in the introduction and three conjectures in the last section. [Copyright &y& Elsevier]

Let -D<-4 denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of Q(√ of -D) exists. Let d be a fundamental discriminant prime to D. Let 2k-1 be an odd natural number prime to the class number of Q(√ of -D). Let χ be the twist of the (2k-1)th power of a canonical Hecke character of Q(√ of -D) by the Kronecker's symbol n↠(d/n). It is proved that the vanishing order of the Hecke L-function L(s,χ) at its central point s=k is determined by its root number when |d|, where the constant implied in the symbol < depends only on k and ε, and is effective for L-functions with root number -1. [ABSTRACT FROM AUTHOR]