Dirichlet.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "平方剰余記号"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Leg(n, p):\n",
    "    if pow(n, (p-1)//2, p) == 1:\n",
    "        return 1\n",
    "    elif pow(n, (p-1)//2, p) == (p - 1):\n",
    "        return -1\n",
    "    else:\n",
    "        return 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\sin(\\dfrac{k\\pi}{p})$を計算"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "def sin(k,p):\n",
    "    return math.sin(math.pi*k/p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6180339887498948"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 5\n",
    "prod = 1\n",
    "for k in range(1, p//2 + 1):\n",
    "    prod = prod * (sin(k, p) ** Leg(k, p))\n",
    "prod"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6180339887498949"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(-1+math.sqrt(5))/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.3027756377319947"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 13\n",
    "prod = 1\n",
    "for k in range(1, p//2 + 1):\n",
    "    prod = prod * (sin(k, p) ** Leg(k, p))\n",
    "prod"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.30277563773199456"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(-3+math.sqrt(13))/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.12310562561766056"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 17\n",
    "prod = 1\n",
    "for k in range(1, p//2 + 1):\n",
    "    prod = prod * (sin(k, p) ** Leg(k, p))\n",
    "prod"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pell方程式の解を総当たりで求める\n",
    "$$\n",
    "x^2-py^2=\\pm4\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 1 -4\n",
      "3 1 4\n",
      "4 2 -4\n",
      "7 3 4\n"
     ]
    }
   ],
   "source": [
    "p=5\n",
    "for x in range(1,10):\n",
    "    for y in range(1,10):\n",
    "        if x**2 - p*y**2 == 4:\n",
    "            print(x,y,4)\n",
    "        elif x**2 - p*y**2 == -4:\n",
    "            print(x,y,-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3 1 -4\n",
      "11 3 4\n",
      "36 10 -4\n"
     ]
    }
   ],
   "source": [
    "p=13\n",
    "n=50\n",
    "for x in range(1,n):\n",
    "    for y in range(1,n):\n",
    "        if x**2 - p*y**2 == 4:\n",
    "            print(x,y,4)\n",
    "        elif x**2 - p*y**2 == -4:\n",
    "            print(x,y,-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "8 2 -4\n",
      "66 16 4\n"
     ]
    }
   ],
   "source": [
    "p=17\n",
    "n=100\n",
    "for x in range(1,n):\n",
    "    for y in range(1,n):\n",
    "        if x**2 - p*y**2 == 4:\n",
    "            print(x,y,4)\n",
    "        elif x**2 - p*y**2 == -4:\n",
    "            print(x,y,-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.12310562561766059"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(-8+2*math.sqrt(p))/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.19258240356725195"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 29\n",
    "prod = 1\n",
    "for k in range(1, p//2 + 1):\n",
    "    prod = prod * (sin(k, p) ** Leg(k, p))\n",
    "prod"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5 1 -4\n",
      "27 5 4\n"
     ]
    }
   ],
   "source": [
    "p=29\n",
    "n=100\n",
    "for x in range(1,n):\n",
    "    for y in range(1,n):\n",
    "        if x**2 - p*y**2 == 4:\n",
    "            print(x,y,4)\n",
    "        elif x**2 - p*y**2 == -4:\n",
    "            print(x,y,-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.19258240356725187"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(-5+1*math.sqrt(p))/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0827625302982197"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 37\n",
    "prod = 1\n",
    "for k in range(1, p//2 + 1):\n",
    "    prod = prod * (sin(k, p) ** Leg(k, p))\n",
    "prod"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "12 2 -4\n"
     ]
    }
   ],
   "source": [
    "p=37\n",
    "n=100\n",
    "for x in range(1,n):\n",
    "    for y in range(1,n):\n",
    "        if x**2 - p*y**2 == 4:\n",
    "            print(x,y,4)\n",
    "        elif x**2 - p*y**2 == -4:\n",
    "            print(x,y,-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(-12+2*math.sqrt(p))/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = 41\n",
    "prod = 1\n",
    "for k in range(1, p//2 + 1):\n",
    "    prod = prod * (sin(k, p) ** Leg(k, p))\n",
    "prod"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p=41\n",
    "n=100\n",
    "for x in range(1,n):\n",
    "    for y in range(1,n):\n",
    "        if x**2 - p*y**2 == 4:\n",
    "            print(x,y,4)\n",
    "        elif x**2 - p*y**2 == -4:\n",
    "            print(x,y,-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(-64+10*math.sqrt(p))/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = 53\n",
    "prod = 1\n",
    "for k in range(1, p//2 + 1):\n",
    "    prod = prod * (sin(k, p) ** Leg(k, p))\n",
    "prod"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p=53\n",
    "n=100\n",
    "for x in range(1,n):\n",
    "    for y in range(1,n):\n",
    "        if x**2 - p*y**2 == 4:\n",
    "            print(x,y,4)\n",
    "        elif x**2 - p*y**2 == -4:\n",
    "            print(x,y,-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(-7+1*math.sqrt(p))/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = 61\n",
    "prod = 1\n",
    "for k in range(1, p//2 + 1):\n",
    "    prod = prod * (sin(k, p) ** Leg(k, p))\n",
    "prod"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p=61\n",
    "n=100\n",
    "for x in range(1,n):\n",
    "    for y in range(1,n):\n",
    "        if x**2 - p*y**2 == 4:\n",
    "            print(x,y,4)\n",
    "        elif x**2 - p*y**2 == -4:\n",
    "            print(x,y,-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(-39+5*math.sqrt(p))/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00029239763581977255"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 229\n",
    "prod = 1\n",
    "for k in range(1, p//2 + 1):\n",
    "    prod = prod * (sin(k, p) ** Leg(k, p))\n",
    "prod"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "15 1 -4\n"
     ]
    }
   ],
   "source": [
    "p=229\n",
    "n=100\n",
    "for x in range(1,n):\n",
    "    for y in range(1,n):\n",
    "        if x**2 - p*y**2 == 4:\n",
    "            print(x,y,4)\n",
    "        elif x**2 - p*y**2 == -4:\n",
    "            print(x,y,-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.06637297521077823"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(-15+1*math.sqrt(p))/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0002923976358197762"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((-15+1*math.sqrt(p))/2)**3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = 257\n",
    "prod = 1\n",
    "for k in range(1, p//2 + 1):\n",
    "    prod = prod * (sin(k, p) ** Leg(k, p))\n",
    "prod"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p=257\n",
    "n=100\n",
    "for x in range(1,n):\n",
    "    for y in range(1,n):\n",
    "        if x**2 - p*y**2 == 4:\n",
    "            print(x,y,4)\n",
    "        elif x**2 - p*y**2 == -4:\n",
    "            print(x,y,-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(-32+2*math.sqrt(p))/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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