On the Diophantine equations z^2 = f(x)^2 ± f(x)f(y) + f(y)^2
نویسندگان
چکیده
Using the theory of Pell equation, we study non-trivial positive integer solutions Diophantine equations $z^2=f(x)^2\pm f(x)f(y)+f(y)^2$ for certain polynomials f(x), which mean to construct integral triangles with two sides given by values f(x) and f(y) intersection angle $120^\circ$ or $60^\circ$.
منابع مشابه
The Diophantine Equation 8x + py = z2
Let p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p ≡ ± 3(mod 8), then the equation 8 (x) + p (y) = z (2) has no positive integer solutions (x, y, z); (ii) if p ≡ 7(mod 8), then the equation has only the solutions (p, x, y, z) = (2 (q) - 1, (1/3)(q + 2), 2, 2 (q) + 1), where q is an odd prime with q ≡ 1(mod 3); (iii) if p ≡ 1(mod 8)...
متن کاملDiophantine approximation and Diophantine equations
The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...
متن کاملOn Some Diophantine Equations (i)
In this paper we study the equation m−n = py,where p is a prime natural number, p≥ 3. Using the above result, we study the equations x + 6pxy + py = z and the equations ck(x 4 + 6pxy + py) + 4pdk(x y + pxy) = z, where the prime number p ∈ {3, 7, 11, 19} and (ck, dk) is a solution of the Pell equation, either of the form c −pd = 1 or of the form c − pd = −1. I. Preliminaries. We recall some nece...
متن کاملOn Some Diophantine Equations (iii)
In this paper we study the Diophantine equations ck(f +42fg+49g) + 28dk(f g + 7fg) = m, where (ck, dk) are solutions of the Pell equation c 2−7d2= 1.
متن کاملOn Some Diophantine Equations (ii)
In [7] we have studied the equation m − n = py, where p is a prime natural number p ≥ 3. Using the above result, in this paper, we study the equations ck(x 4 + 6px y +py) + 4pdk(x y + pxy) = 32z with p ∈ {5, 13, 29, 37}, where (ck, dk) is a solution of the Pell equation ∣∣c2 − pd2∣∣ = 1.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Notes on Number Theory and Discrete Mathematics
سال: 2021
ISSN: ['1310-5132', '2367-8275']
DOI: https://doi.org/10.7546/nntdm.2021.27.2.88-100