задание 2

1 files changed, 208 insertions, 0 deletions

diff --git a/t4mk.ipynb b/t4mk.ipynb
index dcf2c42..b6d3fd1 100644
--- a/t4mk.ipynb
+++ b/t4mk.ipynb
@@ -218,6 +218,214 @@
    ]
   },
   {
+   "cell_type": "code",
+   "execution_count": 187,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\text{Свойство 1:}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle a \\equiv b \\pmod{p} \\implies \\left(\\frac{a}{p}\\right) = \\left(\\frac{b}{p}\\right)$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\text{Свойство 6:}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left(\\frac{2}{p}\\right) = \\begin{cases}\n",
+       " 1, &\\text{если } p \\equiv \\pm 1 \\pmod{8} \\\\\n",
+       "-1, &\\text{если } p \\equiv \\pm 3 \\pmod{8}\n",
+       "\\end{cases}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\text{Свойство 7 (Квадратичный закон взаимности Гаусса):}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left(\\frac{p}{q}\\right) = \\left(\\frac{q}{p}\\right) \\cdot (-1)^{\\frac{p - 1}{2} \\cdot \\frac{q - 1}{2}}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\text{Решение:}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left(\\frac{1016}{2945}\\right) = $"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle = (1) \\cdot \\left(\\frac{1016}{2945}\\right) = (1) \\cdot \\left(\\frac{2^3}{2945}\\right) \\cdot \\left(\\frac{127}{2945}\\right) = \\text{[по св. 6]} = (1) \\cdot \\left(\\frac{127}{2945}\\right) =\\text{[по св. 7 Гаусса]} = (1) \\cdot \\left(\\frac{2945}{127}\\right) \\cdot (-1)^{\\frac{2945 - 1}{2} \\cdot \\frac{127 - 1}{2}} =(1) \\cdot \\left(\\frac{2945}{127}\\right) = \\text{[по св. 1]} = (1) \\cdot \\left(\\frac{24}{127}\\right) = $"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle = (1) \\cdot \\left(\\frac{24}{127}\\right) = (1) \\cdot \\left(\\frac{2^3}{127}\\right) \\cdot \\left(\\frac{3}{127}\\right) = \\text{[по св. 6]} = (1) \\cdot \\left(\\frac{3}{127}\\right) =\\text{[по св. 7 Гаусса]} = (1) \\cdot \\left(\\frac{127}{3}\\right) \\cdot (-1)^{\\frac{127 - 1}{2} \\cdot \\frac{3 - 1}{2}} =(-1) \\cdot \\left(\\frac{127}{3}\\right) = \\text{[по св. 1]} = (-1) \\cdot \\left(\\frac{1}{3}\\right) = $"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle = (-1) \\cdot \\left(\\frac{1}{3}\\right) = \\text{[по св. 7 Гаусса]} = (-1) \\cdot \\left(\\frac{3}{1}\\right) \\cdot (-1)^{\\frac{3 - 1}{2} \\cdot \\frac{1 - 1}{2}} =(-1) \\cdot \\left(\\frac{3}{1}\\right) = \\text{[по св. 1]} = (-1) \\cdot \\left(\\frac{0}{1}\\right) = $"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle  = -1$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "a, n = 1016, 2945\n",
+    "\n",
+    "def jacobi(a, n):\n",
+    "  ans = 1\n",
+    "  if a == 1:\n",
+    "    return ans\n",
+    "  while a != 0:\n",
+    "    ltx = []\n",
+    "    ltx.append(fr'''= ({ans}) \\cdot \\left(\\frac{{{a}}}{{{n}}}\\right) = ''')\n",
+    "\n",
+    "    count = 0\n",
+    "    while a % 2 == 0:\n",
+    "      a //= 2\n",
+    "      count += 1\n",
+    "      if n % 8 in {3, 5}:\n",
+    "        ans = -ans\n",
+    "\n",
+    "    if count != 0:\n",
+    "      ltx.append(fr\"({ans}) \\cdot \\left(\\frac{{2^{count}}}{{{n}}}\\right) \\cdot \\left(\\frac{{{a}}}{{{n}}}\\right) = \")\n",
+    "      ltx.append(fr\"\\text{{[по св. 6]}} = ({ans}) \\cdot \\left(\\frac{{{a}}}{{{n}}}\\right) =\")\n",
+    "\n",
+    "    a, n = n, a\n",
+    "    ltx.append(fr\"\\text{{[по св. 7 Гаусса]}} = ({ans}) \\cdot \\left(\\frac{{{a}}}{{{n}}}\\right) \\cdot (-1)^{{\\frac{{{a} - 1}}{{2}} \\cdot \\frac{{{n} - 1}}{{2}}}} =\")\n",
+    "    if a % 4 == n % 4 == 3:\n",
+    "      ans = -ans\n",
+    "    ltx.append(fr\"({ans}) \\cdot \\left(\\frac{{{a}}}{{{n}}}\\right) = \")\n",
+    "    a %= n\n",
+    "    ltx.append(fr\"\\text{{[по св. 1]}} = ({ans}) \\cdot \\left(\\frac{{{a}}}{{{n}}}\\right) = \")\n",
+    "\n",
+    "    display(Math(''.join(ltx)))\n",
+    "\n",
+    "  if n == 1:\n",
+    "    return ans\n",
+    "  else:\n",
+    "    return 0\n",
+    "\n",
+    "display(Math(r'\\text{Свойство 1:}'))\n",
+    "display(Math(r'a \\equiv b \\pmod{p} \\implies \\left(\\frac{a}{p}\\right) = \\left(\\frac{b}{p}\\right)'))\n",
+    "\n",
+    "display(Math(r'\\text{Свойство 6:}'))\n",
+    "display(Math(r'''\\left(\\frac{2}{p}\\right) = \\begin{cases}\n",
+    " 1, &\\text{если } p \\equiv \\pm 1 \\pmod{8} \\\\\n",
+    "-1, &\\text{если } p \\equiv \\pm 3 \\pmod{8}\n",
+    "\\end{cases}'''))\n",
+    "\n",
+    "display(Math(r'\\text{Свойство 7 (Квадратичный закон взаимности Гаусса):}'))\n",
+    "display(Math(r'\\left(\\frac{p}{q}\\right) = \\left(\\frac{q}{p}\\right) \\cdot (-1)^{\\frac{p - 1}{2} \\cdot \\frac{q - 1}{2}}'))\n",
+    "\n",
+    "display(Math(r'\\text{Решение:}'))\n",
+    "display(Math(fr\"\\left(\\frac{{{a}}}{{{n}}}\\right) = \"))\n",
+    "res = jacobi(a, n)\n",
+    "display(Math(f\" = {res}\"))"
+   ]
+  },
+  {
    "cell_type": "markdown",
    "metadata": {},
    "source": [