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 < ...

In this study, we considered heterogeneous and homogeneous chemical reactions in Eyring-Powell ferrofluid past a stretchable surface. addition, thermal radiation magnetic dipole impacts are also considered. The direct numerical solution of the governing model is complex. For this, simplified into coupled ordinary differential equations (ODEs) with dimensionless variables. Then, determined compu...

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...