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For any n\in\mathbb{N}=\{0,1,2,\ldots\}and b,c\in\mathbb{Z} the generalized central trinomial coefficient T_n(b,c)denotes the coefficient of x^nin the expansion of (x^2+bx+c)^n Let pbe an odd prime. In this paper, we determine the summation \sum_{k=0}^{p-1}T_k(b,c)^2/m^kmodulo p^2for integers mwith certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), 1375--1400].

Merging together a result of Nathanson from the early 70s and a recent result of Granville and Walker, we show that for any finite set Aof integers with \min(A)=0and \gcd(A)=1there exist two sets, the "head" and the "tail", such that if m\ge\max(A)-|A|+2 then the mfold sumset mAconsists of the union of these sets and a long block of consecutive integers separating them. We give sharp estimates for the length of the block of integers contained in mA and investigate the corres...

Let \mathbb{F}_qbe a finite field with q=p^nelements. In this paper, we study the number of \mathbb{F}_qrational points on the affine hypersurface \mathcal Xgiven by a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b where b\in\mathbb{F}_q^* A classic well-konwn result from Weil yields a bound for such number of points. This paper presents necessary and sufficient conditions for the maximality and minimality of \mathcal Xwith respect to Weil's bound.

We develop a variety of new techniques to treat Diophantine equations of the shape x^2+D =y^n based upon bounds for linear forms in padic and complex logarithms, the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves, and machinery from Diophantine approximation. We use these to explicitly determine the set of all coprime integers xand y and n \geq 3 with the property that y^n > x^2and x^2-y^nhas no prime divisor exceeding 11

Let S=\left\langle s_1,\ldots,s_n\right\ranglebe a numerical semigroup generated by the relatively prime positive integers s_1,\ldots,s_n Let k\geqslant 2be an integer. In this paper, we consider the following kpower variant of the Frobenius number of Sdefined as ${}^{k\!}g\!\left(S\right):= \text{ the largest } k \text{-power integer not belonging to } S.We investigate the case =2. We give an upper bound for }^{2\!}g\!\left(S_A\right)$ for an infinity family of semigroups...

We study three fundamental questions about 1periods and give complete answers. 1) We give a necessary and sufficient for a period integral to be transcendental. 2) We give a qualitative description of all \overline{\mathbf{Q}}linear relations between 1periods, establishing Kontsevich's period conjecture in this case. 3) For a fixed 1motive, we derive a general formula for the dimension of its space of periods in the spirit of Baker's theorem. The new version fixes a mistake in Append...

For any number field K with non-elementary 3-class group Cl(3,K) = C(3^e) x C(3), e >= 2, the punctured capitulation type kappa(K) of K in its unramified cyclic cubic extensions Li, 1 <= i <= 4, is an orbit under the action of S3 x S3. By means of Artin's reciprocity law, the arithmetical invariant kappa(K) is translated to the punctured transfer kernel type kappa(G2) of the automorphism group G2 = Gal(F(3,2,K)/K) of the second Hilbert 3-class field of K. A classification of finite 3-groups G wi...

We study integral points on the quadratic twists \mathcal{E}_D:y^2=x^3-D^2xof the congruent number curve. We give upper bounds on the number of integral points in each coset of 2\mathcal{E}_D(\mathbb{Q})in \mathcal{E}_D(\mathbb{Q})and show that their total is \ll (3.8)^{\mathrm{rank} \mathcal{E}_D(\mathbb{Q})} We further show that the average number of non-torsion integral points in this family is bounded above by 2 As an application we also deduce from our upper bounds that the s...

A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with both convergent and divergent harmonic series.

We show that for every positive integer k there exist kconsecutive primes having the property that if any digit of any one of the primes, including any of the infinitely many leading zero digits, is changed, then that prime becomes composite.

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