In this article, we study the non-trivial solutions of the Diophantine equations x p + y p = 2 r z p and x p + y p = z 2 of prime exponent p with r ∈ N over a totally real field K. We show that there exists a constant BK,r > 0, depending on K, r, such that for all primes p > BK,r, the equation x p + y p = 2 r z p (resp., x p + y p = z 2 ) of exponent p has no non-trivial solution in WK(resp., in W ′ K ). Then, for r = 2, 3, we show that there exists a constant BK > 0 such that for all primes p > BK , the equation x p + y p = 2 r z p of exponent p has no non-trivial solution over K.

Let f be a primitive Hilbert modular form over F of weight k with coefficient field E f , generated by the Fourier coefficients C(p, f ) for p ∈ Spec(O F ). Under certain assumptions on the image of the residual Galois representations attached to f , we calculate the Dirichlet density of {p ∈ Spec(OFor k = 2, we show those assumptions are satisfied when [E f : Q] = [F : Q] is an odd prime. We also study analogous results for F f , the fixed field of E f by the set of all inner twists of f . Then, we provide some examples of f to support our results. Finally, we compute the density of {p ∈ Spec(O F )|C(p, f ) ∈ K} for fields K with F f ⊆ K ⊆ E f .

Groundwater hydrochemistry in some parts of Krushanprasad block was determined to assess its quality. Krushnaprasad block is not a single Unit. Some part of it is situated in the main land and some part in form of island within Chilika lagoon. Groundwater samples from 30 different island locations were collected in the post monsoon season. It was found that the groundwater is rich in sodium, magnesium and chloride ions as compared to calcium, potassium, bicarbonate and sulphate ions. The analysis was carried out as per international standards. The groundwater is high in TDS. The parameterschloride / bicarbonate, magnesium/calcium were used to identify the intrusion of salt water in the coastal aquifers. It was concluded that groundwater of Krushnaprasad block in the island regions is in danger under the saltwater intrusion condition. The main groundwater type is sodium chloride type followed by Magnesium chloride waters. The groundwater in most part of the study area is not fit for drinking and for agriculture. The anthropogenic factor may be taken as one of the factors for seawater intrusion. The results were observed and compared with standards of WHO, ICMR and ISI so as to determine its suitability for different use.

In this article, we calculate the density of primes p for which the p-th Fourier coefficient C * (p, f ) (resp., C(p, f )) of a primitive Hilbert modular form f generates the coefficient field F f (resp., E f ), under certain conditions on the images of λ-adic residual Galois representations attached to f . Then, we produce some examples of primitive forms f satisfying these conditions. Our work can be thought of as a generalization of [KSW08] to primitive Hilbert modular forms.