{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This document is intended to be used with my paper \"An Improvement on a Gap Principle Counting Technique and Some Consequences\" of which you can find a preprint here: https://arxiv.org/abs/2210.09631"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import sqrt,floor,log\n",
    "from collections import deque\n",
    "import decimal\n",
    "decimal.getcontext().prec = 10**4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Function Definitions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below corresponds to the quantity $$p_0(n) := \\begin{cases} 3 & \\text{ if } 5 \\leq n \\leq 8 \\\\ 2 & \\text{ if } n \\geq 9 \\end{cases}$$ defined immediately before equation (4)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def pZero(n):\n",
    "    if (n <= 8):\n",
    "        return 3.0\n",
    "    else:\n",
    "        return 2.0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below corresponds to the quantity $$K_d(n) = 2\\sqrt{\\frac{2n}{(n-1)(n-2)}} \\cdot \\left(2.032^{1/n} \\left(1 + \\sqrt{\\frac{2}{(n-2)p_0^n}}\\right)\\right)^d$$ defined in the statement of theorem 3.3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def K(d,n):\n",
    "    return 2*sqrt(float((2*n)/((n-1)*(n-2))))*(2.032**(1/n)*(1+sqrt(float(2/((n-2)* pZero(n)**n)))))**d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below corresponds to the quantity $$n^* = \\frac{n-2}{2}$$ defined in the statement of theorem 3.3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def star(n):\n",
    "    return (n-2) * 0.5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below corresponds to the quantity $$Q_1 := \\frac{p_0(n)^{n^* - d_0}}{K_{d_0}(n)}$$ defined shortly before equation (7)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def qOne(dZero,d,n):\n",
    "    return pZero(n)**(star(n)-dZero)/K(dZero,n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below checks to make sure that the choices of $d_0$ and $d$ are valid for a given value of $n$.  Namely, it checks the three conditions stated in equation (7):\n",
    "\n",
    "\\begin{align*}\n",
    "    &0 \\leq d_0 \\leq n^* - 1.4\\\\\n",
    "    &1 < d \\leq n^*\\\\\n",
    "    &Q_1^{d-1} > \\max(1,K_d(n))\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def validSmall(dZero,d,n):\n",
    "    return (0<=dZero) and (dZero <= star(n)-1.4) and (1 < d) and (d <= star(n)) and (qOne(dZero,d,n)>max(1,K(d,n)**(1/(d-1))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below checks to make sure that the values $a$ and $b$ are valid given a value of $n$.  Namely, it is checking the condition given in equation (8), i.e. $$0 < a < b < 1 - \\sqrt{2 \\cdot \\frac{n + a^2}{n^2}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def validLarge(a,b,n):\n",
    "    return (0<a) and (a < b) and (b < 1 - sqrt(float(2*(n + a**2)/n**2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below defines the quantity $L$ given shortly after equation (8), namely $$L = \\frac{\\sqrt{2(n + a^2)}}{1 - b}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def L(a,b,n):\n",
    "    return sqrt(float(2*(n+a**2)))/(1-b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below defines the quantity $D$ given shortly after equation (8), namely $$D = \\frac{L}{n - L}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def D(a,b,n):\n",
    "    return L(a,b,n)/(n - L(a,b,n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below defines the quantity $A$ given shortly after equation (8), namely $$A = \\frac{1}{a^2}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def A(a,b,n):\n",
    "    return 1/a**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below gives the definition of $\\chi_n$ given in equation (9), namely $$\\chi_n = D(A + 1) + 1$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def chiN(a,b,n):\n",
    "    return D(a,b,n)*(A(a,b,n) + 1)+1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below gives the definition of $\\pi_n$ given in equation (10), namely $$\\pi_n = (D(4 + A) + 2)\\log(2) +  \\frac{(D + 1)\\log(n)}{2} + \\frac{nAD}{2}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def piN(a,b,n):\n",
    "    return (D(a,b,n)*(4 + A(a,b,n)) + 2)*float(log(2))+(D(a,b,n) + 1)*float(log(n))/2 + n*A(a,b,n)*D(a,b,n)/2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below gives the definition of $E$ given shortly after equation (8), namely $$E = \\frac{1}{2(b^2 - a^2)}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def E(a,b,n):\n",
    "    return 1/(2*(b**2-a**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below gives the definition of $Z$ given in equation (11), namely $$Z = \\left\\lfloor \\frac{\\log E + 2\\log n - \\log(L - 2)}{\\log(n-1)}\\right\\rfloor + 2$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Z(dZero,d,a,b,n):\n",
    "    return floor((float(log(E(a,b,n))) + 2*float(log(n)) - float(log(L(a,b,n) - 2)))/float(log(n-1)))+2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below gives the definition of $T$ given in equation (6), namely $$T = \\left\\lfloor\\max\\left(\\frac{\\log\\left(\\frac{\\chi_n}{\\frac{d_0(d - 1) + d}{n(d - 1)}} + 1\\right)}{\\log d}, \\frac{\\log\\left(\\frac{\\pi_n}{\\log K_d(n)^{-\\frac{1}{d-1}}Q_1} + 1\\right)}{\\log d}\\right)\\right\\rfloor + 2$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "def T(dZero,d,a,b,n):\n",
    "    firstQuantity = float(log(chiN(a,b,n) * n * (d - 1) / (dZero * (d-1) + d) + 1))/float(log(d))\n",
    "    secondQuantity = float(log(piN(a,b,n)/float(log(K(d,n)**(-1/(d-1))*qOne(dZero,d,n))) + 1))/float(log(d))\n",
    "    return floor(max(firstQuantity,secondQuantity))+2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function below takes in a range of degrees and a precision value.  It loops over values $n$ satisfying $n_{\\text{min}} \\leq n < n_{\\text{max}}$ and for each $n$, it returns optimal parameter choices to minimize the sum $T+Z$.\n",
    "\n",
    "Based on $n$, it computes an upper bound for $a$, namely the largest value of $a$ for which a valid choice of $b$ exists.  While this is not necessary, it does improve the efficiency slightly.  The value was computed by solving the equation $a = 1 - \\sqrt{2 \\cdot \\frac{n + a^2}{n^2}}$ for $a$.\n",
    "\n",
    "With this upper bound for $a$, the code loops over values $a \\in \\{\\text{prec},2\\cdot\\text{prec},3\\cdot\\text{prec},\\cdots,k\\cdot\\text{prec}\\}$ where $k\\cdot\\text{prec} \\leq a_{\\text{upper}} < (k + 1)\\cdot\\text{prec}$.\n",
    "\n",
    "After choosing a value of $a$, the code loops over values of $d_0 \\in \\{0,u,2u,3u,\\cdots,ku\\}$ where $u = \\text{prec} \\cdot (n^* - 1.4)$ and $ku \\leq n^* - 1.4 < (k + 1)u$.\n",
    "\n",
    "Next, the code loops over values of $b \\in \\{a + \\text{prec},a + 2\\cdot\\text{prec},\\dots,a + k\\cdot\\text{prec}\\}$ where $a + k\\cdot\\text{prec} < 1 - \\sqrt{2\\frac{n + a^2}{n^2}} < a + (k+1)\\cdot\\text{prec}$.\n",
    "\n",
    "Once the code has chosen each of the above parameters, it computes $T$ and $Z$ for the given parameters.  If $T + Z$ for the given parameters is smaller than what is stored in $\\text{tempMinT} + \\text{tempMinZ}$, the code sets $\\text{tempMinT} = T$ and $\\text{tempMinZ} = Z$, then checks to make sure that the parameters are valid and satisfy the necessary hypotheses in Proposition 3.2 before storing the parameters $a,b,$ and $d_0$.  After checking all values of $a,b,$ and $d_0$, the code stores the optimal parameter values along with the value of $n$ in an array.\n",
    "\n",
    "The code returns an array of the form \\[\\[$n_{\\text{min}},d_0,d,a,b,T,Z$\\],\\[$n_{\\text{min}}+1,d_0,d,a,b,T,Z$\\],...,\\[$n_{\\text{max}}-1,d_0,d,a,b,T,Z$\\]\\] where $d$ is always chosen to be $n^*$ and for any given $n$ satisfying $n_{\\text{min}} \\leq n < n_{\\text{max}}$, the printed values of $d_0,a,b$ minimize the sum $T + Z$.\n",
    "\n",
    "A few miscellaneous comments:\n",
    "- The minimal possible values of $T$ and $Z$ are $2$.  So if $T + Z = 4$ at any point in the code, we can assume that is the minimal sum and there is no need to check for further values of $a,b,d_0$\n",
    "- It seems like the value $d = n^*$ always allows for $T + Z$ to be minimal.  Hence, we do not bother looping over different values of $d$ for efficiency.  That said, one can easily modify the code to loop over different values of $d$ too in exchange for longer run time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def minTPlusZ(nMin,nMax,prec):\n",
    "    toReturn = []\n",
    "    for n in range(nMin,nMax):\n",
    "        nStar = star(n)\n",
    "        aUpper = (2*n**2 - sqrt(4*n**4 - 4*(n**2 - 2*n)*(n**2 - 2)))/(2*(n**2 - 2))\n",
    "        a = prec\n",
    "        tempMinA=prec\n",
    "        dZero = 0\n",
    "        tempMinDZero = 0\n",
    "        b = a + prec\n",
    "        tempMinB = a + prec\n",
    "        tempMinT = T(dZero,nStar,a,b,n)\n",
    "        tempMinZ = Z(dZero,nStar,a,b,n)\n",
    "        tempMinSum = tempMinT + tempMinZ\n",
    "        while ((a <= aUpper) and (tempMinSum > 4)):\n",
    "            while ((dZero <= nStar - 1.4) and (tempMinSum > 4)):\n",
    "                while ((b < 1-sqrt(2*(n + a**2)/n**2)) and (tempMinSum > 4)):\n",
    "                    tempT = T(dZero,nStar,a,b,n)\n",
    "                    tempZ = Z(dZero,nStar,a,b,n)\n",
    "                    if (tempT + tempZ < tempMinSum):\n",
    "                        tempMinA = a\n",
    "                        tempMinB = b\n",
    "                        tempMinDZero = dZero\n",
    "                        tempMinT = tempT\n",
    "                        tempMinZ = tempZ\n",
    "                        tempMinSum = tempT + tempZ\n",
    "                    b += prec\n",
    "                dZero += prec * (nStar - 1.4)\n",
    "                b = a + prec\n",
    "            a += prec\n",
    "            dZero = 0\n",
    "            b = a + prec\n",
    "        assert validSmall(tempMinDZero,nStar,n), \"d0,d,n are invalid\"\n",
    "        assert validLarge(tempMinA,tempMinB,n), \"a,b,n are invalid\"\n",
    "        assert chiN(tempMinA,tempMinB,n) >= 2, \"chiN is too small\"\n",
    "        assert piN(tempMinA,tempMinB,n) >= 5*log(2) + 2*log(n), \"piN is too small\"\n",
    "        toReturn.append([n,tempMinDZero,nStar,tempMinA,tempMinB,tempMinT,tempMinZ])\n",
    "    return toReturn"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The idea of the program below is to take a maximal value of $n$ as input, check to see if there exist choices of $a,b,d_0$ so that $T + Z = 4$.  If so, decrement $n$ by $1$ and then try again.  Continue until it is no longer possible to find choices of $a,b,d_0$ so that $T + Z = 4$.\n",
    "\n",
    "In more detail, the program loops over values of $n \\leq n_{\\text{max}}$ for which $T+Z = 4$.  Given $n$, the program defines the same upper bound on $a$ as above (see the commentary on the previous function).  From there, it loops over values of $a \\in \\{a_{\\text{upper}} - \\text{prec},a_{\\text{upper}} - 2\\cdot\\text{prec},\\dots,a_{\\text{upper}} - k\\cdot\\text{prec}\\}$ where $a_{\\text{upper}} - k\\cdot\\text{prec} > 0 \\geq a_{\\text{upper}} - (k+1)\\cdot\\text{prec}$.\n",
    "\n",
    "Next, the program loops over $b \\in \\{a_{\\text{upper}},a_{\\text{upper}} - \\text{prec},a_{\\text{upper}} - 2\\cdot\\text{prec},\\dots,a_{\\text{upper}} - k\\cdot\\text{prec}\\}$ where $a_{\\text{upper}} - k\\cdot\\text{prec} > a \\geq a_{\\text{upper}} - (k+1)\\cdot\\text{prec}$.\n",
    "\n",
    "Next, the program loops over values of $d_0 \\in \\{n^* - 1.4, n^* - 1.4 - u, n^* - 1.4 - 2u,\\dots,n^* - 1.4 - ku\\}$ where $u = \\text{prec}\\cdot(n^* - 1.4)$ and $n^* - 1.4 - ku \\geq 0 > n^* - 1.4 - (k+1)u$.\n",
    "\n",
    "Upon choosing values for $a,b,d_0$, the program uses $d = n^*$ and computes the values of $T$ and $Z$ at these parameters.  If $T$ and $Z$ are minimal (equivalently, if $T + Z = 4$), the program checks to make sure that the parameters are valid and that the hypotheses of Proposition 3.2 are met before saving the values of $a,b,d_0,d$, then decrementing $n$ by 1 and trying again.  If the program cannot find values of $a,b,d_0$ to obtain $T + Z = 4$, it terminates.\n",
    "\n",
    "The program returns a double ended array (deque) of the form \\[\\[$n_{\\text{min}},d_0,d,a,b,T,Z$\\],\\[$n_{\\text{min}}+1,d_0,d,a,b,T,Z$\\],...,\\[$n_{\\text{max}},d_0,d,a,b,T,Z$\\]\\] where $n_{\\text{min}}$ is the smallest value of $n$ for which the program could find parameters to get $T = Z = 2$ and for each $n$ satisfying $n_{\\text{min}} \\leq n \\leq n_{\\text{max}}$, the values of $d_0,d,a,b$ yield $T = Z = 2$.\n",
    "\n",
    "Additional comments:\n",
    "- Most of the comments that apply to the above program also apply here.\n",
    "- This program initializes $b = a_{\\text{upper}}$ because by coincidence, this happens to find additional values of $n$ for which $T = Z = 2$.  The next program (\"minNWithMinTPlusZAlt\") starts with a slightly larger value of $b$.\n",
    "- This program begins with maximal possible values of $a,b,d_0$ because these larger values tend to produce smaller values of $T$ and $Z$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def minNWithMinTPlusZ(nMax,prec):\n",
    "    n = nMax\n",
    "    nStar = star(n)\n",
    "    aUpper = (2*n**2 - sqrt(4*n**4 - 4*(n**2 - 2*n)*(n**2 - 2)))/(2*(n**2 - 2))\n",
    "    a = aUpper - prec\n",
    "    b = aUpper\n",
    "    dZero = nStar - 1.4\n",
    "    tempTPlusZ = 4\n",
    "    listOfParams = deque([])\n",
    "    while ((n >= 6) and (tempTPlusZ == 4)):\n",
    "        tempT = T(dZero,nStar,a,b,n)\n",
    "        tempZ = Z(dZero,nStar,a,b,n)\n",
    "        tempTPlusZ = tempT + tempZ\n",
    "        while ((a > 0) and (tempTPlusZ > 4)):\n",
    "            while ((b > a) and (tempTPlusZ > 4)):\n",
    "                while ((dZero >= 0) and (tempTPlusZ > 4)):\n",
    "                    tempT = T(dZero,nStar,a,b,n)\n",
    "                    tempZ = Z(dZero,nStar,a,b,n)\n",
    "                    tempTPlusZ = tempT + tempZ\n",
    "                    if (tempTPlusZ == 4):\n",
    "                        assert validSmall(dZero,nStar,n), \"d0,d,n are invalid\"\n",
    "                        assert validLarge(a,b,n), \"a,b,n are invalid\"\n",
    "                        assert chiN(a,b,n) >= 2, \"chiN is too small\"\n",
    "                        assert piN(a,b,n) >= 5*log(2) + 2*log(n), \"piN is too small\"\n",
    "                        listOfParams.appendleft([n,dZero,nStar,a,b,tempT,tempZ])\n",
    "                    dZero -= prec*(nStar - 1.4)\n",
    "                b -= prec\n",
    "                dZero = nStar - 1.4\n",
    "            a -= prec\n",
    "            b = aUpper\n",
    "        n -= 1\n",
    "        aUpper = (2*n**2 - sqrt(4*n**4 - 4*(n**2 - 2*n)*(n**2 - 2)))/(2*(n**2 - 2))\n",
    "        nStar = star(n)\n",
    "        a = aUpper - prec\n",
    "        b = aUpper\n",
    "        dZero = nStar - 1.4\n",
    "    return listOfParams"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This program is identical to the above program with a slightly different initialization of $b$.  See the program immediately above this for comments."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "def minNWithMinTPlusZAlt(nMax,prec):\n",
    "    n = nMax\n",
    "    nStar = star(n)\n",
    "    aUpper = (2*n**2 - sqrt(4*n**4 - 4*(n**2 - 2*n)*(n**2 - 2)))/(2*(n**2 - 2))\n",
    "    a = aUpper - prec\n",
    "    b = 1 - sqrt(2*(n+a**2)/n**2)-prec\n",
    "    dZero = nStar - 1.4\n",
    "    tempTPlusZ = 4\n",
    "    listOfParams = deque([])\n",
    "    while ((n >= 6) and (tempTPlusZ == 4)):\n",
    "        tempT = T(dZero,nStar,a,b,n)\n",
    "        tempZ = Z(dZero,nStar,a,b,n)\n",
    "        tempTPlusZ = tempT + tempZ\n",
    "        while ((a > 0) and (tempTPlusZ > 4)):\n",
    "            while ((b > a) and (tempTPlusZ > 4)):\n",
    "                while ((dZero >= 0) and (tempTPlusZ > 4)):\n",
    "                    tempT = T(dZero,nStar,a,b,n)\n",
    "                    tempZ = Z(dZero,nStar,a,b,n)\n",
    "                    tempTPlusZ = tempT + tempZ\n",
    "                    if (tempTPlusZ == 4):\n",
    "                        assert validSmall(dZero,nStar,n), \"d0,d,n are invalid\"\n",
    "                        assert validLarge(a,b,n), \"a,b,n are invalid\"\n",
    "                        assert chiN(a,b,n) >= 2, \"chiN is too small\"\n",
    "                        assert piN(a,b,n) >= 5*log(2) + 2*log(n), \"piN is too small\"\n",
    "                        listOfParams.appendleft([n,dZero,nStar,a,b,tempT,tempZ])\n",
    "                    dZero -= prec*(nStar - 1.4)\n",
    "                b -= prec\n",
    "                dZero = nStar - 1.4\n",
    "            a -= prec\n",
    "            b = 1 - sqrt(2*(n+a**2)/n**2)-prec\n",
    "        n -= 1\n",
    "        aUpper = (2*n**2 - sqrt(4*n**4 - 4*(n**2 - 2*n)*(n**2 - 2)))/(2*(n**2 - 2))\n",
    "        nStar = star(n)\n",
    "        a = aUpper - prec\n",
    "        b = 1 - sqrt(2*(n+a**2)/n**2)-prec\n",
    "        dZero = nStar - 1.4\n",
    "    return listOfParams"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Computations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These computations verify Lemma 3.2 for $n \\leq 506$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[6, 0, 2.0, 0.18000000000000002, 0.2900000000000001, 10, 4],\n",
       " [7, 0.5500000000000003, 2.5, 0.20000000000000004, 0.2800000000000001, 7, 4],\n",
       " [8, 0.9920000000000008, 3.0, 0.16, 0.4100000000000002, 7, 3],\n",
       " [9, 0.8820000000000005, 3.5, 0.17, 0.4000000000000002, 6, 3],\n",
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       " [218, 57.56400000000005, 108.0, 0.03, 0.8700000000000006, 3, 2]]"
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     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "minTPlusZ(6,219,0.01)"
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  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minNWithMinTPlusZ(507,0.01)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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       "        110.0,\n",
       "        0.4179093101235729,\n",
       "        0.8859093101235733,\n",
       "        2,\n",
       "        2]])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minNWithMinTPlusZ(222,0.001)"
   ]
  }
 ],
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