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On prime values of some polynomials. (English) Zbl 1082.11018
The author applies the deformation method of quadratic matrix equations to discuss the solubility and the number of solutions of some inhomogeneous quadratic Diophantine equations involving prime numbers. For example, let $$q \equiv 1$$ mod 8 be a prime number such that $$q^s + 2$$ is also a prime number for some odd integer $$s$$. Then the equation $$2x^2 + 2y^2 - 2q^{\frac{s+1}{2}}x - q = 0$$ has exactly 8 integer solutions.
##### MSC:
 11D09 Quadratic and bilinear Diophantine equations 11D45 Counting solutions of Diophantine equations