Squares from products of integers

This is a preprint of an article published in the Gazette of the Australian Mathematical Society, 31 (2004) no.1, pp.40-42.Notice that 1_2_3_4+1 = 52 , 2_3_4_5+1 = 112 , 3_4_5_6+1 = 192 , . . . . Indeed, it is well known that the product of any four consecutive integers always differs by one from a perfect square. However, a little experimentation readily leads one to guess that there is no integer n, other than four, so that the product of any n consecutive integers always differs from a perfect square by some fixed integer c = c(n) depending only on n. The two issues that are present here can be readily dealt with. The apparently special status of the number four arises from the fact that any quadratic polynomial can be completed by a constant to become the square of a polynomial. Second, [5] provides an elegant proof that there is in fact no integer n larger than four with the property stated above. In [5] one finds a reminder that a polynomial taking too many square values must be the square of a polynomial (see [4, Chapter VIII.114 and .190], and [2]). One might therefore ask whether there are polynomials other than integer multiples of x(x + 1)(x + 2)(x + 3) and 4x(x + 1), with integer zeros and differing by a nonzero constant from the square of a polynomial. We will show that this is quite a good question in that it has a nontrivial answer, inter alia giving new insight into the results of [5]

NT-proBNP changes in patients with ascites during large volume paracentesis

N-terminal probrain natriuretic peptide (NT-proBNP) is a hormone involved in the regulation of cardiovascular homeostasis. Changes in serum NT-proBNP during large volume paracentesis (LVP) in patients with ascites have never before been examined. Aims. To determine if significant changes in serum NT-proBNP occur in patients undergoing LVP and the associated clinical correlates in patients with cirrhosis. Method. A total of 45 patients with ascites were prospectively recruited. Serum NTproBNP, biochemistry, and haemodynamics were determined at baseline and at key time points during and after paracentesis. Results. 34 patients were analysed; 19 had ascites due to cirrhosis and 15 from malignancy. In those with cirrhosis, NT-proBNP decreased by 77.3 pg/mL at 2 L of drainage and 94.3 pg/mL at the end of paracentesis, compared with an increase of 10.5 pg/mL and 77.2 pg/mL in cancer patients at the same time points ( = 0.05 and = 0.03). Only congestive cardiac failure (CCF) was an independent predictor of significant NT-proBNP changes at the end of drainage in cirrhotic patients ( < 0.01). There were no significant changes in haemodynamics or renal biochemistry for either group. Conclusion. Significant reductions in serum NTproBNP during LVP occur in patients with cirrhosis but notmalignancy, and only comorbid CCF appeared to predict such changes

Hepatitis C Virus Induces the Cannabinoid Receptor 1

BACKGROUND: Activation of hepatic CB(1) receptors (CB(1)) is associated with steatosis and fibrosis in experimental forms of liver disease. However, CB(1) expression has not been assessed in patients with chronic hepatitis C (CHC), a disease associated with insulin resistance, steatosis and metabolic disturbance. We aimed to determine the importance and explore the associations of CB(1) expression in CHC. METHODS: CB(1) receptor mRNA was measured by real time quantitative PCR on extracted liver tissue from 88 patients with CHC (genotypes 1 and 3), 12 controls and 10 patients with chronic hepatitis B (CHB). The Huh7/JFH1 Hepatitis C virus (HCV) cell culture model was used to validate results. PRINCIPAL FINDINGS: CB(1) was expressed in all patients with CHC and levels were 6-fold higher than in controls (P<0.001). CB(1) expression increased with fibrosis stage, with cirrhotics having up to a 2 fold up-regulation compared to those with low fibrosis stage (p<0.05). Even in mild CHC with no steatosis (F0-1), CB(1) levels remained substantially greater than in controls (p<0.001) and in those with mild CHB (F0-1; p<0.001). Huh7 cells infected with JFH-1 HCV showed an 8-fold upregulation of CB(1), and CB(1) expression directly correlated with the percentage of cells infected over time, suggesting that CB(1) is an HCV inducible gene. While HCV structural proteins appear essential for CB(1) induction, there was no core genotype specific difference in CB(1) expression. CB(1) significantly increased with steatosis grade, primarily driven by patients with genotype 3 CHC. In genotype 3 patients, CB(1) correlated with SREBP-1c and its downstream target FASN (SREBP-1c; R=0.37, FASN; R=0.39, p<0.05 for both). CONCLUSIONS/SIGNIFICANCE: CB(1) is up-regulated in CHC and is associated with increased steatosis in genotype 3. It is induced by the hepatitis C virus

MBOAT7 rs641738 increases risk of liver inflammation and transition to fibrosis in chronic hepatitis C

Cirrhosis likely shares common pathophysiological pathways despite arising from a variety of liver diseases. A recent GWAS identified rs641738, a polymorphism in the MBOAT7 locus, as being associated with the development of alcoholic cirrhosis. Here we explore the role of this variant on liver inflammation and fibrosis in two cohorts of patients with chronic hepatitis C. In 2,051 patients, rs641738 associated with severe hepatic inflammation and increased risk of fibrosis, as well as fast fibrosis progression. At functional level, rs641738 associated with MBOAT7 transcript and protein levels in liver and blood, and with serum inflammatory, oxidative stress and macrophage activation markers. MBOAT7 was expressed in immune cell subsets, implying a role in hepatic inflammation. We conclude that the MBOAT7 rs641738 polymorphism is a novel risk variant for liver inflammation in hepatitis C, and thereby for liver fibrosis

Paraneoplastic hepatic dysfunction in metastatic prostate cancer : the role of cytokine dysregulation

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Determinants in the Study of Thue&apos;s Method and Curves with Prescribed Singularities

CONTENTS 1. Introduction 2. Approximation of Algebraic Numbers by Algebraic Numbers 3. Confluent Vandermonde Determinants 4. Questions, and Some Answers 5. Some Extraordinary Determinants References Van der Poorten was supported in part by grants from the Australian Research Council and by a research agreement with Digital Equipment Corporation. 1991 Mathematics subject classification: 11J68, 11C20 Our investigations in the 1980&apos;s of Thue&apos;s method yielded determinants that we were only able to analyse successfully in part. We explain the context of our work, recount our experiences, mention our conjectures, and allude to a number of open questions. 1. INTRODUCTION During 1983, and again in 1986, we attempted the explicit construction of the auxiliary polynomial required in Thue&apos;s method, as then recently refined by the first author. That led to our battling, and partly taming, some interesting determinants. Of course, the moment we finished w

Small bowel varices secondary to chronic superior mesenteric vein thrombosis in a patient with heterozygous Factor V Leiden mutation : a case report

Introduction: Bleeding ectopic small bowel varices pose a clinical dilemma for the physician, given their diagnostic obscurity and the lack of evidence-based medicine to guide therapy. They often occur in the context of portal hypertension, secondary to either liver disease or extrahepatic causes. Rarely is their presence associated with chronic superior mesenteric vein thrombosis and hereditary coagulopathies. Case presentation: A 74-year-old white woman, with a heterozygous Factor V Leiden mutation and no underlying liver disease or portal hypertension, presented over the course of 13 months for recurrent episodes of melena and per rectal bleeding. An initial endoscopy showed a clean-based chronic gastric ulcer, while colonoscopies showed multiple, non-bleeding angioectasias which were treated with argon plasma coagulation. Subsequent video capsule endoscopy and double balloon enteroscopy revealed red wale marks overlying engorged submucosal veins in her distal ileum, consistent with ectopic varices. A chronic superior mesenteric vein thrombus, found via computed tomography venogram, was the cause of the ileal varices. She underwent curative surgical resection of the affected bowel, with no re-bleeding episodes 17 months post-surgery, despite needing lifelong anticoagulation for recurrent venous thromboembolisms. Conclusions: Clinicians should consider ectopic varices in patients who present with obscure gastrointestinal bleeding, even in the absence of portal hypertension or liver disease. In those with a known thrombophilia, patients should be screened for splanchnic thrombosis, which may precipitate ectopic varices

Compounded levofloxacin triple therapy is safe and effective for refractory Helicobacter pylori

Failure of first line and subsequent Helicobacter pylori therapy is a significant problem, as alternate treatments are cumbersome and difficult to access. The purpose of this study was to evaluate the efficacy and safety of a compounded levofloxacin triple therapy in clinical practice as a second or third-line salvage regimen for Helicobacter pylori. Patients referred after first or subsequent treatment failures were prescribed compounded levofloxacin 500 mg, amoxicillin 1 g, and esomeprazole 40 mg, all twice daily for 10 days. Eradication success was determined by 14C-urea breath test or histology at least 4 weeks after completion of therapy. The study included 93 patients, the majority of whom were female (57%) with a mean age of 44. The most common indication for treatment was dyspepsia/risk reduction (84%). Median number of previous treatments was 1 (range: 1 through 6) with treatment used as second line in 83%. Helicobacter pylori eradication was achieved in 89.2% (74/83) per protocol and 79.6% (74/93) on an intention-to-treat basis. Outcome was independent of gender, ethnicity, treatment indication, or number. Treatment was well tolerated, with minor adverse events in 8.4% and only one patient discontinuing therapy. Compounded levofloxacin triple therapy is an effective and safe second line treatment for Helicobacter pylori, with eradication rates comparable to standard levofloxacin-based regimens

Midkine Increases Diagnostic Yield in AFP Negative and NASH-Related Hepatocellular Carcinoma

<div><p>Robust biomarkers for population-level hepatocellular carcinoma (HCC) surveillance are lacking. We compared serum midkine (MDK), dickkopf-1 (DKK1), osteopontin (OPN) and AFP for HCC diagnosis in 86 HCC patients matched to 86 cirrhotics, 86 with chronic liver disease (CLD) and 86 healthy controls (HC). Based on the performance of each biomarker, we assessed a separate longitudinal cohort of 28 HCC patients, at and before cancer diagnosis. Serum levels of MDK and OPN were higher in HCC patients compared to cirrhosis, CLD and HC groups. DKK1 was not different between cases and controls. More than half of HCC patients had normal AFP. In this AFP-negative HCC cohort, 59.18% (n = 29/49) had elevated MDK, applying the optimal cut-off of 0.44 ng/ml. Using AFP ≥ 20 IU/ml or MDK ≥ 0.44 ng/ml, a significantly greater number (76.7%; n = 66/86) of HCC cases were detected. The area under the receiver operating curve for MDK was superior to AFP and OPN in NASH-HCC diagnosis. In the longitudinal cohort, MDK was elevated in 15/28 (54%) of HCC patients at diagnosis, of whom 67% had elevated MDK 6 months prior. <b>Conclusion:</b> AFP and MDK have a complementary role in HCC detection. MDK increases the diagnostic yield in AFP-negative HCC and has greater diagnostic performance than AFP, OPN and DKK-1 in the diagnosis of NASH-HCC. Additionally, MDK has a promising role in the pre-clinical diagnosis of HCC.</p></div