On a variant of a Diophantine equation of Cassels

Recently, Yuan and Li considered a variant y2=px(Ax2-2) of Cassels\u27 equation y2=3x(x2+2). They proved that the equation has at most five solutions in positive integers (x, y). In this note, we improve Yuan-Li\u27s result by showing that for any prime p and any odd positive integer A, the Diophantine equation y2=px(Ax2-2) has at most three solutions in positive integers (x, y)

An exponential Diophantine equation related to the difference between powers of two consecutive Balancing numbers

In this paper, we find all solutions of the exponential Diophantine equation $B_{n+1}^x-B_n^x=B_m$ in positive integer variables $(m, n, x)$, where $B_k$ is the $k$-th term of the Balancing sequence.Comment: Comments are welcom

On Diophantine, pronic and triangular triples of balancing numbers

In this paper, we search for some Diophantine triples of balancing numbers. We prove that, if $(6pm2)B_nB_k+1$ and $(6pm2)B_{n+2}B_k+1$ are both perfect squares then $k=n+1$, for any positive integer $n geq 1$. In addition, we define pronic $m$-tuples, triangular $m$-tuples and prove some results related to pronic and triangular triples of balancing numbers

Complete solution of the exponential Diophantine equation (P_n^x+P_{n+1}^x=P_m^y)

In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (m, n, x,y), where (P_k) is the kth term of the Pell sequence

The Dual Targeting of FcRn and FcγRs via Monomeric Fc Fragments Results in Strong Inhibition of IgG-Dependent Autoimmune Pathologies

Novel molecules that directly target the neonatal Fc receptor (FcRn) and/or Fc gamma receptors (FcγRs) are emerging as promising treatments for immunoglobulin G (IgG)-dependent autoimmune pathologies. Mutated Fc regions and monoclonal antibodies that target FcRn are currently in clinical development and hold promise for reducing the levels of circulating IgG. Additionally, engineered structures containing multimeric Fc regions allow the dual targeting of FcRn and FcγRs; however, their tolerance needs to first be validated in phase I clinical studies. Here, for the first time, we have developed a modified monomeric recombinant Fc optimized for binding to all FcRns and FcγRs without the drawback of possible tolerance associated with FcγR cross-linking. A rational approach using Fc engineering allowed the selection of LFBD192, an Fc with a combination of six mutations that exhibits improved binding to human FcRn and FcγR as well as mouse FcRn and FcγRIV. The potency of LFBD192 was compared with that of intravenous immunoglobulin (IVIg), an FcRn blocker (Fc-MST-HN), and a trimeric Fc that blocks FcRn and/or immune complex-mediated cell activation through FcγR without triggering an immune reaction in several in vitro tests and validated in three mouse models of autoimmune disease

B lymphocytes trigger monocyte mobilization and impair heart function after acute myocardial infarction.

Acute myocardial infarction is a severe ischemic disease responsible for heart failure and sudden death. Here, we show that after acute myocardial infarction in mice, mature B lymphocytes selectively produce Ccl7 and induce Ly6C(hi) monocyte mobilization and recruitment to the heart, leading to enhanced tissue injury and deterioration of myocardial function. Genetic (Baff receptor deficiency) or antibody-mediated (CD20- or Baff-specific antibody) depletion of mature B lymphocytes impeded Ccl7 production and monocyte mobilization, limited myocardial injury and improved heart function. These effects were recapitulated in mice with B cell-selective Ccl7 deficiency. We also show that high circulating concentrations of CCL7 and BAFF in patients with acute myocardial infarction predict increased risk of death or recurrent myocardial infarction. This work identifies a crucial interaction between mature B lymphocytes and monocytes after acute myocardial ischemia and identifies new therapeutic targets for acute myocardial infarction.This work was supported by Inserm, British Heart Foundation (Z.M.), European Research Council (Z.M.), Fondation Coeur et Recherche (Z.M., T.S., N.D.), Fondation pour la Recherche Medicale (J.S.S.), European Union Seven Framework programme TOLERAGE (Z.M.), Fondation Leducq transatlantic network (C.J.B., D.T., A.T., J.S.S., Z.M.), National Institutes of Health grants AI56363 and AI057157, and a grant from The Lymphoma Research Foundation (T.F.T).This is the author accepted manuscript. The final version is available from Nature Publishing Group at http://dx.doi.org/10.1038/nm.3284

A note of three prime representation problems

In this note, we consider a three prime representation problem asked by Sárközy. We give a negative answer to the upper density case of the problem and obtain a conclusion that the Three Primes Theorem still holds for a general thin subset of primes

On the solutions of the exponential diophantine equation ax + by = (m2 + 1)z

Click on the link to view the abstract.Keywords: Exponential Diophantine equation, Terai conjecture, positive integer solution, linear forms in two logarithms, lower bound, primitive divisorQuaestiones Mathematicae 36(2013), 119–13