Mixed sums of squares and triangular numbers (III)

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if Tm=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p²=x²+8(y²+z²) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2Tm(m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4.     

On Delannoy numbers and Schröder numbers

The nth Delannoy number and the nth Schröder number given by

     

Sums and products with smooth numbers

We estimate the sizes of the sumset A+A and the productset A⋅A in the special case that A=S(x,y), the set of positive integers n?x free of prime factors exceeding y.     

A remark on sociable numbers of odd order

Write s(n) for the sum of the proper divisors of the natural number n. We call n sociable if the sequence n, s(n), s(s(n)), … is purely periodic; the period is then called the order of sociability of n. The ancients initiated the study of order 1 sociables (perfect numbers) and order 2 sociables (amicable numbers), and investigations into higher-order sociable numbers began at the end of the 19th century. We show that if k is odd and fixed, then the number of sociable n?x of order k is bounded by as x→∞. This improves on the previously best-known bound of , due to Kobayashi, Pollack, and Pomerance.     

On congruences related to central binomial coefficients

It is known that

     

Multiperfect numbers on lines of the Pascal triangle

A number n is said to be multiperfect (or multiply perfect) if n divides its sum of divisors σ(n). In this paper, we study the multiperfect numbers on straight lines through the Pascal triangle. Except for the lines parallel to the edges, we show that all other lines through the Pascal triangle contain at most finitely many multiperfect numbers. We also study the distribution of the numbers σ(n)/n whenever the positive integer n ranges through the binomial coefficients on a fixed line through the Pascal triangle.     

q-Analogs of some congruences involving Catalan numbers

We present some variations on the Greene–Krammer?s identity which involve q-Catalan numbers. Our method reveals an intriguing analogy between these new identities and some congruences modulo a prime.     

On the quadratic character of quadratic units

Let be a prime. Let a,b∈Z with p?a(a²+b²). In the paper we mainly determine by assuming p=c²+d² or p=Ax²+2Bxy+Cy² with AC−B²=a²+b². As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1).     

Unique expansions of real numbers

It was discovered some years ago that there exist non-integer real numbers q>1 for which only one sequence (ci) of integers ci∈[0,q) satisfies the equality . The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation.In this paper we consider for each fixed q>1 the set Uq of real numbers x having a unique representation of the form with integers ci belonging to [0,q). We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases q for which Uq is closed or even a Cantor set. We also study the set consisting of all sequences (ci) of integers ci∈[0,q) such that . We determine the numbers r>1 for which the map (defined on (1,∞)) is constant in a neighborhood of r and the numbers q>1 for which is a subshift or a subshift of finite type.     

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have

     

On consecutive happy numbers

Let e?1 and b?2 be integers. For a positive integer with 0?aj<b, define

     

On congruences of Euler numbers modulo powers of two

In this paper, we establish some identities involving the Euler numbers, the Euler numbers of order 2 and the central factorial numbers, and give a new proof of a classical result due to M.A. Stern.

Video abstract

For a video summary of this paper, please visit http://www.youtube.com/watch?v=kdNsdTDA-FE.     

Algebraic identities for the Nijenhuis tensors

The general algebraic identities are discovered for the Nijenhuis and Haantjes tensors on an arbitrary manifold Mn. For n=3, the special algebraic identities involving the symmetric bilinear form H(u,v) are derived.     

Algebraic identity for the Schouten tensor and bi-Hamiltonian systems

A (0,3)-tensor Tijk is introduced in an invariant form. Algebraic identities are derived that connect the Schouten (2,1)-tensor and tensor Tijk with the Nijenhuis tensor . Applications to the bi-Hamiltonian dynamical systems are presented.     

Sums of squares on real algebraic surfaces

Consider real polynomials g₁, . . . , g_r in n variables, and assume that the subset K = {g₁≥0, . . . , g_r≥0} of ℝⁿ is compact. We show that a polynomial f has a representation in which the s_e are sums of squares, if and only if the same is true in every localization of the polynomial ring by a maximal ideal. We apply this result to provide large and concrete families of cases in which dim (K) = 2 and every polynomial f with f|_K≥0 has a representation (*). Before, it was not known whether a single such example exists. Further geometric and arithmetic applications are given. Support by DFG travel grant KON 1823/2002 and by the European RAAG network HPRN-CT-2001-00271 is gratefully acknowledged. Part of this work was done while the author enjoyed a stay at MSRI Berkeley. He would like to thank the institute for the invitation and the very pleasant working conditions.     

On Euler numbers modulo powers of two

To determine Euler numbers modulo powers of two seems to be a difficult task. In this paper we achieve this and apply the explicit congruence to give a new proof of a classical result due to M.A. Stern.     

Quartic, octic residues and Lucas sequences

Let be a prime and a,b∈Z with a²+b²≠p. Suppose p=x²+(a²+b²)y² for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1). We also pose many conjectures concerning , or .     

Pseudo Frobenius numbers

For a prime $p$, we call a positive integer $n$ a Frobenius $p$-number if there exists a finite group with exactly $n$ subgroups of order $p^a$ for some $a\ge 0$. Extending previous results on Sylow’s theorem, we prove in this paper that every Frobenius $p$-number $n\equiv 1(mod{p}^2)$ is a Sylow $p$-number, i. e., the number of Sylow $p$-subgroups of some finite group. As a consequence, we verify that 46 is a pseudo Frobenius 3-number, that is, no finite group has exactly 46 subgroups of order $3^a$ for any $a\ge 0$.     

Translation invariance in groups of prime order

We prove that there is an absolute constant c>0 with the following property: if Z/pZ denotes the group of prime order p, and a subset A⊂Z/pZ satisfies 1<|A|<p/2, then for any positive integer there are at most 2m non-zero elements b∈Z/pZ with |(A+b)?A|?m. This (partially) extends onto prime-order groups the result, established earlier by S. Konyagin and the present author for the group of integers. We notice that if A⊂Z/pZ is an arithmetic progression and m<|A|<p/2, then there are exactly 2m non-zero elements b∈Z/pZ with |(A+b)?A|?m. Furthermore, the bound c|A|/ln|A| is best possible up to the value of the constant c. On the other hand, it is likely that the assumption can be dropped or substantially relaxed.