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We completely solve the Diophantine equations $x²+2^a{q}^b=yⁿ$ (for q = 17, 29, 41). We also determine all $C=p{₁}^{a₁}\cdots p_k^{a_k}$ and $C=2^{a₀}p{₁}^{a₁}\cdots p_k^{a_k}$, where $p₁,...,p_k$ are fixed primes satisfying certain conditions. The corresponding Diophantine equations x² + C = yⁿ may be studied by the method used by Abu Muriefah et al. (2008) and Luca and Togbé (2009).

Let $p_i$ denote the ith prime. We conjecture that there are precisely 28 solutions to the equation $n²-1=p{₁}^{\alpha ₁}\cdots p_k^{{\alpha }_k}$ in positive integers n and α₁,..., ${\alpha }_k$. This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).

We construct $p$-adic $L$-functions (in general case unbounded) attached to “motivic" primitive Hilbert cusp forms as a non-archimedean Mellin transform of the corresponding admissible measure. In order to prove the growth conditions of the appropriate complex-valued distributions we represent them as Rankin type representation and use Atkin–Lehner theory and explicit form of Fourier coefficients of Eisenstein series.

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We consider the Diophantine equation $(x+y)(x²+Bxy+y²)=D{z}^p$, where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D’s, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).