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stevenjohnstone's solution

to Nth Prime in the Lua Track

Published at Jul 26 2020 · 0 comments
Instructions
Test suite
Solution

Given a number n, determine what the nth prime is.

By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.

If your language provides methods in the standard library to deal with prime numbers, pretend they don't exist and implement them yourself.

Running the tests

To run the tests, run the command busted from within the exercise directory.

Further information

For more detailed information about the Lua track, including how to get help if you're having trouble, please visit the exercism.io Lua language page.

Source

A variation on Problem 7 at Project Euler http://projecteuler.net/problem=7

Submitting Incomplete Solutions

It's possible to submit an incomplete solution so you can see how others have completed the exercise.

nth-prime_spec.lua

local nth = require('nth-prime')

describe('nth-prime', function()
  local function benchmark(f)
    local start = os.clock()
    f()
    return os.clock() - start
  end

  it('should give 2 as the first prime', function()
    assert.equal(2, nth(1))
  end)

  it('should give 3 as the second prime', function()
    assert.equal(3, nth(2))
  end)

  it('should be able to calculate the nth prime for small n', function()
    assert.equal(13, nth(6))
  end)

  it('should be able to calculate the nth prime for large n', function()
    assert.equal(104743, nth(10001))
  end)

  it('should be efficient for large n', function()
    local execution_time = benchmark(function()
      nth(10001)
    end)

    assert(execution_time < 1, 'should take less than a second to execute')
  end)

  it('should raise an error for n <= 0', function()
    assert.has_error(function()
      nth(0)
    end)

    assert.has_error(function()
      nth(-1)
    end)
  end)
end)
local function mulmod(a, e, m)
    local result = 1
    while e > 0 do
        if e % 2 == 1 then
            result = result * a % m
            e = e - 1
        end
        e = e / 2
        a = a * a % m
    end
    return result
end

-- express n - 1 = 2^s * q for some odd q
local function sq(n)
    local q = n - 1
    local s = 0
    while q % 2 == 0 do
        s = s + 1
        q = q // 2
    end
    return s, q
end

local function is_prime(p)
    for f = 3, math.sqrt(p) do
        if p % f == 0 then
            return false
        end
    end
    return true
end


local function miller_rabin(n)
    local check = function()
        local a = math.random(2, n -1)
        local s, q = sq(n)
        local m = mulmod(a, q, n)
        if m == 1 or m == n - 1 then
            return true
        end

        for _ = 1, s - 1 do
            m = mulmod(m, 2, n)
            if m == n - 1 then
                return true
            end
        end
        return false
    end

    for _ = 1, 4 do
        if not check() then
            return false
        end
    end
    return is_prime(n)
end

local first_primes = {2, 3, 5, 7, 11, 13}
return function(n)
    assert(n > 0)
    local small_prime = first_primes[n]
    if small_prime ~= nil then
        return small_prime
    end
    local p, i = first_primes[#first_primes], #first_primes + 1
    while true do
        p = p + 2
        if miller_rabin(p) then
            if i == n then
                return p
            end
            i = i + 1
        end
    end
end

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