Computational experiences on norm form equations with solutions forming arithmetic progressions

In the present paper we solve the equation NK/Q(x0 + x1α + x2α2 + ... + xn -1αn -1) = 1 in x0, ... , xn -1 ∈ Z, such that x0, ... , xn -1 is an arithmetic progression, where α is a root of the polynomial xn - a, for all integers 2 ≤ a ≤ 100 and n ≥ 3

On the diophantine equation xp−x=yq−yx^p-x=y^q-y

We consider the diophantine equation x^p-x=y^q-y \tag"$(*)$" in integers $(x,p,y,q)$. We prove that for given $p$ and $q$ with 2\le p < q $(*)$ has only finitely many solutions. Assuming the abc-conjecture we can prove that $p$ and $q$ are bounded. In the special case $p=2$ and $y$ a prime power we are able to solve $(*)$ completely

Subcutaneous tocilizumab in rheumatoid arthritis: findings from the common-framework phase 4 study programme TOZURA conducted in 22 countries

Objectives The aim of this pooled analysis of the TOZURA study programme was to evaluate the efficacy and safety of subcutaneous tocilizumab (TCZ-SC) as monotherapy or in combination with conventional synthetic DMARDs (csDMARDs) in patients with moderate to severe RA who had an inadequate response to csDMARD or anti-TNF agent therapy or who were MTX naïve. Methods TOZURA is a multinational, open-label, single-arm, common-framework, phase 4 study programme (11 protocols, 22 countries). Patients received TCZ-SC 162 mg each week for ⩾24 weeks, administered at the investigator’s discretion, as monotherapy or in combination with a csDMARD. Efficacy, safety and immunogenicity were evaluated; propensity score–based matching was used for between-group comparisons. Results Of 1804 patients, 353 (19.6%) received monotherapy and 1451 (80.4%) received combination therapy. The 28-joint DAS using ESR (DAS28-ESR) in both groups decreased significantly from baseline to week 24 (mean change: monotherapy −3.40, combination therapy −3.46), with no significant difference between groups (P = 0.46). The proportion of patients who achieved DAS28-ESR or Clinical Disease Activity Index remission or ACR 20/50/70/90 responses was similar between groups. Overall, 13.9% of patients withdrew—6.2% for safety reasons and 1.6% for insufficient therapeutic response; 5.8% of patients experienced one or more serious adverse events [14.6/100 patient-years (PY)]; six deaths occurred (0.64/100 PY). Conclusion In a common framework of 11 studies in 22 countries, this phase 4 study programme confirmed TCZ-SC’s known efficacy and safety profile with comparable effects as monotherapy and in combination with csDMARDs

Effects of concomitant glucocorticoids in TOZURA, a common-framework study programme of subcutaneous tocilizumab in rheumatoid arthritis

Objectives This post hoc analysis of the TOZURA study programme evaluated the efficacy and safety of subcutaneous tocilizumab (TCZ-SC) as monotherapy or with concomitant conventional synthetic DMARDs (csDMARDs) in patients with RA categorized by baseline glucocorticoid (GC) use. Methods TOZURA was a multinational, open-label, single-arm, common-framework study programme (11 protocols, 22 countries) in patients with moderate to severe RA in whom csDMARDs or biologic therapies had failed or who were MTX naïve. Patients received once-weekly TCZ-SC 162 mg for ⩾24 weeks as monotherapy or in combination with csDMARDs and/or oral GC use (⩽10 mg/day prednisone or equivalent), which was to be continued unchanged for 24 weeks. Treatment subgroups were defined by baseline GC use and analysed for efficacy and safety. Results Of 1804 patients who received TCZ-SC, 145 received monotherapy + GC, 208 received monotherapy without GC, 730 received combination therapy + GC and 721 received combination therapy without GC. The median GC dose in both GC subgroups was 5 mg/day. The proportion of patients who achieved clinical remission, defined as DAS in 28 joints using ESR <2.6, increased similarly from baseline to week 24 in all subgroups. Improvements in patient-reported outcomes were similar in all subgroups. Overall adverse event profiles were generally similar between subgroups, with some slight numerical differences between GC and non-GC subgroups. Conclusion The incremental efficacy benefits of TCZ-SC as monotherapy and in combination with csDMARDs were similar between patients with and without previous and continued oral GC treatment, with generally similar safety profiles