In the $B^+\to D^+ D^-K^+$ decay, LHCb has reported observation of open-charm exotic states $X_{0,1}^0\equiv X_{0,1}(2900)^0$ with four different quark flavors~$(ud\bar s \bar c)$, where subscripts (0,1) denote spins. To confirm discovery, we propose $\Lambda_b\to \Sigma_c^{0(++)} X_{0,1}^{\prime\,0(--)}$ in final state interaction, $X_{0,1}^{\prime\,0(--)}$ $s u\bar d c$ ($ds\bar u c$) are new...

Fermat’s little theorem says for prime p that ap−1 ≡ 1 mod p for all a 6≡ 0 mod p. A naive extension of this to a composite modulus n ≥ 2 would be: for a 6≡ 0 mod n, an−1 ≡ 1 mod n. Let’s call this “Fermat’s little congruence.” It may or may not be true. When n is prime, it is true for all a 6≡ 0 mod n. But when n is composite it usually has many counterexamples. Example 1.1. When n = 15, the t...

A convex subnearlattice of a nearlattice S containing a fixed element n∈S is called an n-ideal. The n-ideal generated by a single element is called a principal n-ideal. The set of finitely generated principal nideals is denoted by Pn(S), which is a nearlattice. A distributive nearlattice S with 0 is called m-normal if its every prime ideal contains at most m number of minimal prime ideals. In t...

We consider the partition function bp(n), which counts the number of partitions of the integer n into distinct parts with no part divisible by the prime p. We prove the following: Let p be a prime greater than 3 and let r be an integer between 1 and p−1, inclusively, such that 24r + 1 is a quadratic nonresidue modulo p. Then, for all nonnegative integers n, bp(pn + r) ≡ 0 (mod 2).

We prove the existence of a subset, with positive natural density, squarefree integers n>0 such that 4–rank ideal class group ℚ(-n,n) is ω 3 (n)-1, where (n) number prime divisors n are modulo 4. Recall for groups associated to ℚ(n) or ℚ(-n) an analogous subset does not exist.

Let f be a Rademacher or Steinhaus random multiplicative function. ?>0 small. We prove that, as x?+?, we almost surely have | ? n?xP(n)>xf(n)|? x(loglogx)1?4+?, where P(n) stands for the largest prime factor of n. This gives an indication sure size fluctuations f.

We prove the generalized Hyers--Ulam stability  of $n$-th order linear differential equation of the form $$y^{(n)}+p_{1}(x)y^{(n-1)}+ cdots+p_{n-1}(x)y^{prime}+p_{n}(x)y=f(x),$$ with condition that there exists a non--zero solution of corresponding homogeneous equation. Our main results extend and improve the corresponding results obtained by many authors.

This is a quite faithful rendering of a Colloquio De Giorgi I had the honor to give at Scuola Normale Superiore on March 21, 2012. The idea was to explain some open problems in arithmetic algebraic geometry which are simple to state but which remain shrouded in mystery. 1. An interactive game: dimension zero Suppose I give you an integer N ≥ 2, and tell you that I am thinking of a monic integer...

For a 6= 0 we define {E n } by ∑ k=0 ( n 2k ) a2kE n−2k = (1− a)n (n = 0,1,2, . . .), where [n/2] = n/2 or (n−1)/2 according as 2 | n or 2 n. In the paper we establish many congruences for E n modulo prime powers, and show that there is a set X and a map T : X → X such that (−1)nE 2n is the number of fixed points of T n. MSC: Primary 11B68, Secondary 11A07

Let m and n > 0 be integers. Suppose that p is an odd prime dividing m− 4. We show that