Kernel: Python 3 (Ubuntu Linux)

Quadratische Diophantische Gleichung in Sympy

Mittels SymPy in Python 3

In [1]:

from sympy import init_printing
init_printing()

In [2]:

from sympy import pprint, solve, S, Eq, symbols, simplify
x, y = symbols('x, y', integer=True)

In [3]:

from sympy.solvers.diophantine import diophantine

Allgemeine Lösung mittels Diophante Modul:

In [4]:

sol = diophantine(Eq(x**2 - 92 * y**2, 1))
sol

\left\{\left ( - 120 \sqrt{23} \left(- 240 \sqrt{23} + 1151\right)^{t} + \frac{1151}{2} \left(- 240 \sqrt{23} + 1151\right)^{t} + 120 \sqrt{23} \left(240 \sqrt{23} + 1151\right)^{t} + \frac{1151}{2} \left(240 \sqrt{23} + 1151\right)^{t}, \quad - 60 \left(- 240 \sqrt{23} + 1151\right)^{t} + \frac{1151 \sqrt{23}}{92} \left(- 240 \sqrt{23} + 1151\right)^{t} - \frac{1151 \sqrt{23}}{92} \left(240 \sqrt{23} + 1151\right)^{t} - 60 \left(240 \sqrt{23} + 1151\right)^{t}\right ), \left ( - \frac{1151}{2} \left(- 240 \sqrt{23} + 1151\right)^{t} + 120 \sqrt{23} \left(- 240 \sqrt{23} + 1151\right)^{t} - \frac{1151}{2} \left(240 \sqrt{23} + 1151\right)^{t} - 120 \sqrt{23} \left(240 \sqrt{23} + 1151\right)^{t}, \quad - \frac{1151 \sqrt{23}}{92} \left(- 240 \sqrt{23} + 1151\right)^{t} + 60 \left(- 240 \sqrt{23} + 1151\right)^{t} + 60 \left(240 \sqrt{23} + 1151\right)^{t} + \frac{1151 \sqrt{23}}{92} \left(240 \sqrt{23} + 1151\right)^{t}\right )\right\}

Herauspicken der ersten Lösung

In [20]:

xsol, ysol = list(sol)[0]
xsol

- 120 \sqrt{23} \left(- 240 \sqrt{23} + 1151\right)^{t} + \frac{1151}{2} \left(- 240 \sqrt{23} + 1151\right)^{t} + 120 \sqrt{23} \left(240 \sqrt{23} + 1151\right)^{t} + \frac{1151}{2} \left(240 \sqrt{23} + 1151\right)^{t}

In [21]:

ysol

- 60 \left(- 240 \sqrt{23} + 1151\right)^{t} + \frac{1151 \sqrt{23}}{92} \left(- 240 \sqrt{23} + 1151\right)^{t} - \frac{1151 \sqrt{23}}{92} \left(240 \sqrt{23} + 1151\right)^{t} - 60 \left(240 \sqrt{23} + 1151\right)^{t}

In [0]:

Ersetzen des freien Parameters mit 0

In [22]:

t = xsol.free_symbols.pop()

In [23]:

xsol.subs({t : 0})

1151

In [24]:

ysol.subs({t : 0})

-120

Probe

In [10]:

(x**2 - 92 * y**2).subs({x : -1151, y : 120})

1

In [0]:

In [11]:

x2 = xsol.subs({t : 1})
y2 = ysol.subs({t : 1})

In [12]:

x2

- 120 \sqrt{23} \left(- 240 \sqrt{23} + 1151\right) + 120 \sqrt{23} \left(240 \sqrt{23} + 1151\right) + 1324801

In [13]:

y2

- \frac{1151 \sqrt{23}}{92} \left(240 \sqrt{23} + 1151\right) - 138120 + \frac{1151 \sqrt{23}}{92} \left(- 240 \sqrt{23} + 1151\right)

In [18]:

lhs2 = (x**2 - 92 * y**2).subs({x : x2, y : y2})
lhs2

- 92 \left(- \frac{1151 \sqrt{23}}{92} \left(240 \sqrt{23} + 1151\right) - 138120 + \frac{1151 \sqrt{23}}{92} \left(- 240 \sqrt{23} + 1151\right)\right)^{2} + \left(- 120 \sqrt{23} \left(- 240 \sqrt{23} + 1151\right) + 120 \sqrt{23} \left(240 \sqrt{23} + 1151\right) + 1324801\right)^{2}

In [15]:

x2.simplify()

2649601

In [16]:

y2.simplify()

-276240

In [17]:

lhs2.simplify()

1

In [0]: