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ipynb/Beal.ipynb
274
ipynb/Beal.ipynb
@ -33,7 +33,7 @@
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"\n",
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"# Online Beal Counterexample Checker\n",
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"\n",
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"Got a counterexample? Check it with my [Online Beal Counterexample Checker](http://norvig.com/bealcheck.html).\n",
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"Got a counterexample? Verify it with my [Beal Counterexample Checker](http://norvig.com/bealcheck.html).\n",
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"\n",
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"# How to Not Win A Million Dollars\n",
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"\n",
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@ -53,7 +53,7 @@
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"* Another claimed that $2^3+2^3=2^4$ was a counterexample, because all the bases are 2, which is prime, and prime numbers have no prime factors. But that's not true; a prime number has itself as a factor.\n",
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"\n",
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"\n",
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"* A creative person offered $1359072^4 - 940896^4 = 137998080^3$, which fails both because $3^3 2^5 11^2$ is a common factor, and because it has a subtraction rather than an addition (although, as Julius Jacobsen pointed out, it could be rewritten as $137998080^3 + 940896^4 = 1359072^4$).\n",
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"* A creative person offered $ 1359072^4 - 940896^4 = 137998080^3$, which fails both because $ 3^3 2^5 11^2 $ is a common factor, and because it has a subtraction rather than an addition (although, as Julius Jacobsen pointed out, it could be rewritten as $ 137998080^3 + 940896^4 = 1359072^4 $).\n",
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"\n",
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"\n",
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"* Mustafa Pehlivan came up with an example involving 76-million-digit numbers, which took some work to prove wrong (using modulo arithmetic). \n",
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@ -88,46 +88,13 @@
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"output_type": "execute_result"
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}
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],
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"source": [
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"A, B, C = 60000000000000000000, 70000000000000000000, 82376613842809255677\n",
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"x = y = z = 3.\n",
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"A ** x + B ** y == C ** z"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"That's interesting; the equality is true. But to show that this is a valid counterexample, we have to show that `A, B, C` have no common factor; that is, that their greatest common divisor (`gcd`) is 1:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {
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"button": false,
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"collapsed": false,
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"new_sheet": false,
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"run_control": {
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"read_only": false
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}
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"True"
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"from math import gcd #### In Python versions < 3.5, use \"from fractions import gcd\"\n",
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"\n",
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"gcd(gcd(A, B), C) == 1"
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"A, B, C = 60000000000000000000, 70000000000000000000, 82376613842809255677\n",
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"x = y = z = 3.\n",
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"\n",
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"A ** x + B ** y == C ** z and gcd(gcd(A, B), C) == 1"
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]
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},
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{
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@ -140,14 +107,14 @@
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}
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},
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"source": [
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"**WOW! The result is `True`!** Is this a real counterexample to Beal? And also a disproof of Fermat's Last Theorem?\n",
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"**WOW! The result is `True`!** The two sides of the equation are equal, and the greatest common divisor is 1. Is this a real counterexample to Beal? And also a disproof of Fermat's Last Theorem?\n",
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"\n",
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"Alas, it is not. The decimal point in \"`x = y = z = 3.`\" indicates a floating point number, with inexact, limited precision. Change the inexact \"`3.`\" to an exact \"`3`\" and the two sides of the equation are no longer equal. Below we see they are the same for the first 18 digits, but differ starting with the 19th: "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"execution_count": 2,
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"metadata": {
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"button": false,
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"collapsed": false,
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@ -164,7 +131,7 @@
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" 559000000000000000063037470301555182935702892172500189973733)"
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]
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},
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"execution_count": 3,
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -196,131 +163,92 @@
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"$3987^{12} + 4365^{12} = 4472^{12}$ and $1782^{12} + 1841^{12} = 1922^{12}$. These were designed by *Simpsons* writer David X. Cohen to be correct up to the precision found in most handheld calculators. Cohen found the equations with a program that must have been something like this:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"1782**12 + 1841**12 == 1922**12 has relative error of 3e-10\n",
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"3987**12 + 4365**12 == 4472**12 has relative error of 2e-11\n"
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]
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}
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],
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"source": [
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"from itertools import combinations\n",
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"\n",
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"def Fermat(A, B, n):\n",
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" \"Return a tuple of the form (error, A, B, C, n)\"\n",
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" C = int(round((A ** n + B ** n) ** (1/n)))\n",
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" lhs, rhs = A ** n + B ** n, C ** n\n",
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" error = abs(lhs - rhs) / rhs\n",
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" return (error, A, B, C, n)\n",
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"\n",
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"def simpsons(bases, powers):\n",
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" \"\"\"Find the Fermat equation that minimizes the error.\"\"\"\n",
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" return min(Fermat(A, B, n) \n",
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" for A, B in combinations(bases, 2) \n",
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" for n in powers)\n",
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"\n",
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"def show(fermat):\n",
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" (error, A, B, C, n) = fermat\n",
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" print('{}**{} + {}**{} == {}**{} has relative error of {:.0g}'\n",
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" .format(A, n, B, n, C, n, error))\n",
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" \n",
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"show(simpsons(range(1000, 2000), [12]))\n",
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"show(simpsons(range(3000, 5000), [12]))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"These are the same two equations that David X. Cohen found. The sides of the equations are nearly equal, up to 10 or 11 decimal places, respectively.\n",
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"\n",
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"Can we do better? The following search will take about 8 minutes:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"from collections import namedtuple\n",
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"from itertools import combinations\n",
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"\n",
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"Fermat = namedtuple('Fermat', 'A, B, C, n')\n",
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"\n",
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"def err(equation):\n",
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" \"The relative error in a Fermat-style equation\"\n",
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" (A, B, C, n) = equation\n",
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" return abs(A ** n + B ** n - C ** n) / C ** n\n",
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"\n",
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"def simpsons(bases, powers):\n",
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" \"\"\"Find the Fermat equation that minimizes the error (the\n",
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" difference between A ** n + B ** n and C ** n). \n",
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" Let A, B range over all pairs of bases and n over all powers.\"\"\"\n",
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" return min((Fermat(A, B, int((A ** n + B ** n) ** (1/n)), n) \n",
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" for A, B in combinations(bases, 2) for n in powers),\n",
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" key=err)"
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"2196**3 + 5984**3 == 6081**3 has relative error of 4e-12\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {
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"button": false,
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"collapsed": false,
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"new_sheet": false,
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"run_control": {
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"read_only": false
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}
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"Fermat(A=3987, B=4365, C=4472, n=12)"
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"(224866629440, 224866629441)"
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]
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},
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"execution_count": 5,
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"simpsons(range(3000, 5000), range(11, 14))"
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"(_, A, B, C, n) = f = simpsons(range(1000, 6000), range(3, 13))\n",
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"show(f)\n",
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"A ** n + B ** n, C ** n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {
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"button": false,
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"collapsed": false,
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"new_sheet": false,
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"run_control": {
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"read_only": false
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}
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"Fermat(A=1010, B=1897, C=1988, n=3)"
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]
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},
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"execution_count": 6,
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"cell_type": "markdown",
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"simpsons(range(1000, 2000), range(3, 16))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(63976656349698612616236230953154487896987106,\n",
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" 63976656348486725806862358322168575784124416)"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"3987 ** 12 + 4365 ** 12, 4472 ** 12"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(2541210258614589176288669958142428526657,\n",
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" 2541210259314801410819278649643651567616)"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"1782 ** 12 + 1841 ** 12, 1922 ** 12"
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"This is an equation that differs by just 1 in the 12th and last decmal place."
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]
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},
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{
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@ -348,7 +276,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"execution_count": 5,
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"metadata": {
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"button": false,
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"collapsed": true,
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@ -414,7 +342,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"execution_count": 6,
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"metadata": {
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"button": false,
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"collapsed": false,
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@ -428,8 +356,8 @@
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"CPU times: user 362 ms, sys: 3.33 ms, total: 366 ms\n",
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"Wall time: 406 ms\n"
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"CPU times: user 354 ms, sys: 4.32 ms, total: 358 ms\n",
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"Wall time: 376 ms\n"
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]
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}
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],
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@ -447,12 +375,12 @@
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}
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},
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"source": [
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"The execution time goes up roughly with the square of `max_A`, so the following should take about 100 times longer:"
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"The execution time goes up roughly with the square of `max_A`, so the following should take about 25 times longer:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"execution_count": 7,
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"metadata": {
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"button": false,
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"collapsed": false,
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@ -466,13 +394,13 @@
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"CPU times: user 43.3 s, sys: 335 ms, total: 43.7 s\n",
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"Wall time: 46.5 s\n"
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"CPU times: user 11.1 s, sys: 152 ms, total: 11.3 s\n",
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"Wall time: 12.3 s\n"
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]
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}
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],
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"source": [
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"%time beal(1000, 100)"
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"%time beal(500, 100)"
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]
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},
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{
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@ -497,7 +425,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"execution_count": 8,
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"metadata": {
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"button": false,
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"collapsed": false,
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@ -518,7 +446,7 @@
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" 6: [216, 1296, 7776, 279936]}"
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]
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},
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"execution_count": 12,
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -543,7 +471,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"execution_count": 9,
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"metadata": {
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"button": false,
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"collapsed": false,
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@ -579,7 +507,7 @@
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" 279936: 6}"
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]
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},
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"execution_count": 13,
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"execution_count": 9,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -591,7 +519,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"execution_count": 10,
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"metadata": {
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"collapsed": false
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},
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@ -602,7 +530,7 @@
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"3"
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]
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},
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"execution_count": 14,
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"execution_count": 10,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -631,7 +559,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"execution_count": 11,
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"metadata": {
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"button": false,
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"collapsed": false,
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@ -680,7 +608,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"execution_count": 12,
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"metadata": {
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"button": false,
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"collapsed": false,
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@ -696,7 +624,7 @@
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"{True}"
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]
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},
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"execution_count": 16,
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"execution_count": 12,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -730,7 +658,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"execution_count": 13,
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"metadata": {
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"button": false,
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"collapsed": true,
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@ -761,7 +689,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 18,
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"execution_count": 14,
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"metadata": {
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"button": false,
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"collapsed": false,
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@ -777,7 +705,7 @@
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"'tests pass'"
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]
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},
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"execution_count": 18,
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"execution_count": 14,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -882,7 +810,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 19,
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"execution_count": 15,
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"metadata": {
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"button": false,
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"collapsed": true,
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@ -947,7 +875,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 20,
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"execution_count": 16,
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"metadata": {
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"collapsed": false
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},
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@ -963,7 +891,7 @@
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" 6: [(216, 3), (296, 4), (776, 5), (936, 7)]}"
|
||||
]
|
||||
},
|
||||
"execution_count": 20,
|
||||
"execution_count": 16,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
@ -989,7 +917,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 21,
|
||||
"execution_count": 17,
|
||||
"metadata": {
|
||||
"button": false,
|
||||
"collapsed": false,
|
||||
@ -1024,7 +952,7 @@
|
||||
" 936: [(6, 7)]})"
|
||||
]
|
||||
},
|
||||
"execution_count": 21,
|
||||
"execution_count": 17,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
@ -1048,7 +976,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 22,
|
||||
"execution_count": 18,
|
||||
"metadata": {
|
||||
"button": false,
|
||||
"collapsed": false,
|
||||
@ -1062,8 +990,8 @@
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"CPU times: user 1min 2s, sys: 646 ms, total: 1min 3s\n",
|
||||
"Wall time: 1min 8s\n"
|
||||
"CPU times: user 55.9 s, sys: 530 ms, total: 56.4 s\n",
|
||||
"Wall time: 59.6 s\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
|
||||
Loading…
Reference in New Issue
Block a user