1 pi − 2 j+1 > 7pr − 4pr > pr . If r is odd, set nk = ∏r1 pi − 2k+1. Then nk ≡ 3(4) (since 32 ≡ 1(4)). But no pi divides nk for 1 ≤ i ≤ r , so the integer nk has some prime factor qk ≡ 3(4) with qk > pr . If j = k, say j > k, the assumption that qk also divides n j leads to the same contradiction as earlier: since nk − n j = 2 j+1 − 2k+1 = 2k+1(2 j−k − 1), we have qk | 2 j−k − 1 and hence qk < ...

Electron beam irradiated sheep wool dosed (0–410) kGy showing good adsorption properties was tested for Cr(III) and Co(II) of higher concentrations. Fitting to ten isotherm models examined as follows: Langmuir, Freundlich, Dubinin-Radushkevich, Temkin, Flory-Huggins, Halsey, Harkins-Jura, Jovanovic, Elovich Redlich-Peterson. Both cations being Lewis acids generate complex salts, such carboxylat...

Pneumorrhachis (PR), which involves the entrapment of air or gas within the spinal canal, is a rare clinical entity. The pathogenesis and etiology of this uncommon entity vary and may present a diagnostic challenge. Air in the spinal canal can be divided into primary and secondary PR, extra- or intradural PR and etiologically classified as iatrogenic, traumatic and non-traumatic. PR is typicall...

background and objectives: breast cancer is the most common malignancy among women in the world. the aim of this study was to assess estrogen receptor (er), progesterone receptor (pr) and her-2/neu of infiltrating ductal carcinoma (idc) with tumor size, histologic grade, lymph node metastasis and age. methods: this study was carried out on 300 tissue blocks ofpatients with idc who underwent mas...