Fix > 0, and let p1 = 2, p2 = 3, . . . be the sequence of all primes. We prove that if (q, a) = 1, then there are infinitely many pairs pr, pr+1 such that pr ≡ pr+1 ≡ a mod q and pr+1 − pr < log pr. The proof combines the ideas of Shiu and of Goldston–Pintz–Yıldırım.

One of the basic principles behind conditioning is that “conditioning on a high-probability event keeps things reasonably unchanged; however, if we condition on a low-probability event, then all bets could be off”. More precisely, suppose we wish to estimate Pr[A | B]. Note, in general, that this could be less than, equal to, or greater than Pr[A]. We now prove that if Pr[B] is “high”, then Pr[...

Introduction: Different etiologies of pleural effusion are diagnosed based on serum and plural fluid characteristics. The aim of this study was to assess and compare the serum and pleural fluid protein to lactate dehyrogenase (Pr/LDH) ratio in exudative pleural effusions. Materials and Methods: This study was conducted on 60 patients with exudative pleural effusion including: 20 cases with para...

PURPOSE To investigate whether there is a difference in patient and tumor characteristics in cases with single receptor positive (SRP) (ER-/PR+ and ER+/PR-) breast carcinoma in comparison with the double receptor positive (DRP) (ER+/PR+) and double receptor negative (DRN) (ER-/PR-) tumors. METHODS A total of 255 breast cancer patients were categorized on the basis of their tumor hormonal rece...

High mortality in pregnant women (PR) is a characteristic of hepatitis E in developing countries. To understand the pathogenesis of HEV infection in self-limiting disease during pregnancy, we compared clinical (PR-patients) and subclinical-HEV-infections in pregnant women in the first (SC-PR-1) and later (2nd and 3rd, SC-PR-2+3) trimesters with the respective healthy controls and acute non-PR p...