On equal values of power sums of arithmetic progressions

In this paper we consider the Diophantine equation \begin{align*}b^k +\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l + \left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l, \end{align*} where $a,b,c,d,k,l$ are given integers. We prove that, under some reasonable assumptions, this equation has only finitely many integer solutions.Comment: This version differs slightly from the published version in its expositio

On conjectures and problems of Ruzsa concerning difference graphs of S-units

Given a finite nonempty set of primes S, we build a graph $\mathcal{G}$ with vertex set $\mathbb{Q}$ by connecting x and y if the prime divisors of both the numerator and denominator of x-y are from S. In this paper we resolve two conjectures posed by Ruzsa concerning the possible sizes of induced nondegenerate cycles of $\mathcal{G}$, and also a problem of Ruzsa concerning the existence of subgraphs of $\mathcal{G}$ which are not induced subgraphs.Comment: 15 page

ID1 and ID3 Regulate the Self-Renewal Capacity of Human Colon Cancer-Initiating Cells through p21

SummaryThere is increasing evidence that some cancers are hierarchically organized, sustained by a relatively rare population of cancer-initiating cells (C-ICs). Although the capacity to initiate tumors upon serial transplantation is a hallmark of all C-ICs, little is known about the genes that control this process. Here, we establish that ID1 and ID3 function together to govern colon cancer-initiating cell (CC-IC) self-renewal through cell-cycle restriction driven by the cell-cycle inhibitor p21. Regulation of p21 by ID1 and ID3 is a central mechanism preventing the accumulation of excess DNA damage and subsequent functional exhaustion of CC-ICs. Additionally, silencing of ID1 and ID3 increases sensitivity of CC-ICs to the chemotherapeutic agent oxaliplatin, linking tumor initiation function with chemotherapy resistance

Differential clonal evolution in oesophageal cancers in response to neo-adjuvant chemotherapy

How chemotherapy affects carcinoma genomes is largely unknown. Here we report whole-exome and deep sequencing of 30 paired oesophageal adenocarcinomas sampled before and after neo-adjuvant chemotherapy. Most, but not all, good responders pass through genetic bottlenecks, a feature associated with higher mutation burden pre-treatment. Some poor responders pass through bottlenecks, but re-grow by the time of surgical resection, suggesting a missed therapeutic opportunity. Cancers often show major changes in driver mutation presence or frequency after treatment, owing to outgrowth persistence or loss of sub-clones, copy number changes, polyclonality and/or spatial genetic heterogeneity. Post-therapy mutation spectrum shifts are also common, particularly C>A and TT>CT changes in good responders or bottleneckers. Post-treatment samples may also acquire mutations in known cancer driver genes (for example, SF3B1, TAF1 and CCND2) that are absent from the paired pre-treatment sample. Neo-adjuvant chemotherapy can rapidly and profoundly affect the oesophageal adenocarcinoma genome. Monitoring molecular changes during treatment may be clinically useful

Tumour Cell Heterogeneity.

The population of cells that make up a cancer are manifestly heterogeneous at the genetic, epigenetic, and phenotypic levels. In this mini-review, we summarise the extent of intra-tumour heterogeneity (ITH) across human malignancies, review the mechanisms that are responsible for generating and maintaining ITH, and discuss the ramifications and opportunities that ITH presents for cancer prognostication and treatment

30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)