Problem 304: Primonacci

(see projecteuler.net/problem=304)

For any positive integer n the function next\_prime(n) returns the smallest prime p such that p > n.

The sequence a(n) is defined by:
a(1)=next\_prime(10^14) and a(n)=next\_prime(a(n-1)) for n>1.

The fibonacci sequence f(n) is defined by: f(0)=0, f(1)=1 and f(n)=f(n-1)+f(n-2) for n>1.

The sequence b(n) is defined as f(a(n)).

Find sum{b(n)} for 1<=n<=100 000. Give your answer mod 1234567891011.

My Algorithm

I need two thing for the problem: a fast Fibonacci generator and a prime test for large numbers.
My toolbox already has a Miller-Rabin test, that means I only have to find a fast Fibonacci generator.
Last week I solved problem 137 with the matrix form found on en.wikipedia.org/wiki/Fibonacci_number

Using those two tools I can easily compute F(10^14) and test each consecutive Fibonacci number until I have 100000 primes.

Alternative Approaches

When I read the Wikipedia a second time I noted a simplification of the matrix form called "fast doubling" on other websites:
F_{2n-1} = F^2_n + F^2_{n-1}
F_{2n} = F_n (2 F_{n-1} + F_n)

My function fibonacci implements that algorithm. It's about 2 to 3x faster than fibonacciMatrix (from problem 137).
Unfortunately, looking for the initial Fibonacci numbers poses a tiny fraction of the overall computation time.
Therefore it doesn't make a difference - but hopefully my new fibonacci function can be used in other problems, too.

I played around with my trusted powmod function, too. It's pretty fast when compiled with GCC but noticeable slower
with Visual C++ because of its lack of 128-bit integer arithmetic.
Based on the discussion apps.topcoder.com/forums/?module=Thread&threadID=670443&start=0&mc=10 I wrote a second
powmod which is about 30% faster on Visual C++ and doesn't need a mulmod function.
The old powmod still remains my fastest implementation of powmod for GCC.

All these little optimizations caused a little bit of code bloat - that's one of my longest solutions to a Project Euler problem ...
(actually it became the new number 1 spot on my list).

Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Input data (separated by spaces or newlines):
Note: Enter the the lower limit for the first prime, the number of primes to be searched and the modulo.

This is equivalent to
echo "10000 10000 10000000000" | ./304

Output:

(please click 'Go !')

Note: the original problem's input 100000000000000 100000 1234567891011 cannot be entered
because just copying results is a soft skill reserved for idiots.

(this interactive test is still under development, computations will be aborted after one second)

My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository. 

       
#include <iostream>

       
 

       
// ---------- mulmod, powmod and Miller-Rabin test from my toolbox ----------

       
 

       
// return (a*b) % modulo

       
unsigned long long mulmod(unsigned long long a, unsigned long long b, unsigned long long modulo)

       
{

       
  // (a * b) % modulo = (a % modulo) * (b % modulo) % modulo

       
  a %= modulo;

       
  b %= modulo;

       
 

       
  // fast path

       
  if (a <= 0xFFFFFFF && b <= 0xFFFFFFF)

       
    return (a * b) % modulo;

       
 

       
  // we might encounter overflows (slow path)

       
  // the number of loops depends on b, therefore try to minimize b

       
  if (b > a)

       
    std::swap(a, b);

       
 

       
#ifndef __GNUC__

       
  // bitwise multiplication

       
  unsigned long long result = 0;

       
  while (a > 0 && b > 0)

       
  {

       
    // b is odd ? a*b = a + a*(b-1)

       
    if (b & 1)

       
    {

       
      result += a;

       
      if (result >= modulo)

       
        result -= modulo;

       
      // skip b-- because the bit-shift at the end will remove the lowest bit anyway

       
    }

       
 

       
    // b is even ? a*b = (2*a)*(b/2)

       
    a <<= 1;

       
    if (a >= modulo)

       
      a -= modulo;

       
 

       
    // next bit

       
    b >>= 1;

       
  }

       
 

       
  return result;

       
 

       
#else

       
 

       
  // based on GCC's 128 bit implementation

       
  return ((unsigned __int128)a * b) % modulo;

       
 

       
#endif

       
}

       
 

       
#ifdef __GNUC__

       
// return (base^exponent) % modulo => simple implementation

       
unsigned long long powmod(unsigned long long base, unsigned long long exponent, unsigned long long modulo)

       
{

       
  unsigned long long result = 1;

       
  while (exponent > 0)

       
  {

       
    // fast exponentation:

       
    // odd exponent ? a^b = a*a^(b-1)

       
    if (exponent & 1)

       
      result = mulmod(result, base, modulo);

       
 

       
    // even exponent ? a^b = (a*a)^(b/2)

       
    base = mulmod(base, base, modulo);

       
    exponent >>= 1;

       
  }

       
  return result;

       
}

       
 

       
#else

       
 

       
// return (base^exponent) % modulo => faster implementation

       
unsigned long long powmod(unsigned long long base, unsigned long long exponent, unsigned long long modulo)

       
{

       
  // based on https://apps.topcoder.com/forums/?module=Thread&threadID=670443&start=0&mc=10

       
  // fastest generic code (but slower than G++ optimized code)

       
  unsigned long long result = 1;

       
  base %= modulo;

       
 

       
  while (exponent > 0)

       
  {

       
    // odd exponent ? a^b = a*a^(b-1)

       
    if (exponent & 1)

       
    {

       
      // "unrolled" code of:

       
      // result = mulmod(result, base, modulo);

       
      // base   = mulmod(base,   base, modulo);

       
      auto x = result;

       
      auto y = base;

       
      auto b = base;

       
 

       
      result = 0;

       
      base   = 0;

       
      while (b > 0)

       
      {

       
        if (b & 1)

       
        {

       
          result += x; if (result >= modulo) result -= modulo;

       
          base   += y; if (base   >= modulo) base   -= modulo;

       
        }

       
 

       
        x += x; if (x >= modulo) x -= modulo;

       
        y += y; if (y >= modulo) y -= modulo;

       
 

       
        b >>= 1;

       
      }

       
    }

       
    else

       
    {

       
      // even exponent ? a^b = (a*a)^(b/2)

       
      auto y = base;

       
      auto b = base;

       
 

       
      base = 0;

       
      while (b > 0)

       
      {

       
        if (b & 1)

       
        {

       
          base += y; if (base >= modulo) base -= modulo;

       
        }

       
 

       
        y += y; if (y >= modulo) y -= modulo;

       
        b >>=1;

       
      }

       
    }

       
 

       
    exponent >>=1;

       
  }

       
 

       
  return result;

       
}

       
#endif

       
 

       
// Miller-Rabin-test

       
bool isPrime(unsigned long long p)

       
{

       
  // IMPORTANT: requires mulmod(a, b, modulo) and powmod(base, exponent, modulo)

       
 

       
  // some code from             https://ronzii.wordpress.com/2012/03/04/miller-rabin-primality-test/

       
  // with optimizations from    http://ceur-ws.org/Vol-1326/020-Forisek.pdf

       
  // good bases can be found at http://miller-rabin.appspot.com/

       
 

       
  // trivial cases

       
  const unsigned int bitmaskPrimes2to31 = (1 <<  2) | (1 <<  3) | (1 <<  5) | (1 <<  7) |

       
                                          (1 << 11) | (1 << 13) | (1 << 17) | (1 << 19) |

       
                                          (1 << 23) | (1 << 29); // = 0x208A28Ac

       
  if (p < 31)

       
    return (bitmaskPrimes2to31 & (1 << p)) != 0;

       
 

       
  if (p %  2 == 0 || p %  3 == 0 || p %  5 == 0 || p % 7 == 0 || // divisible by a small prime

       
      p % 11 == 0 || p % 13 == 0 || p % 17 == 0)

       
    return false;

       
 

       
  if (p < 17*19) // we filtered all composite numbers < 17*19, all others below 17*19 must be prime

       
    return true;

       
 

       
  // fine-tuning for the problem:

       
  // fast check of more small primes (up to 100)

       
  if (p % 19 == 0 || p % 23 == 0 || p % 29 == 0 || p % 31 == 0 || p % 37 == 0 ||

       
      p % 41 == 0 || p % 43 == 0 || p % 47 == 0 || p % 53 == 0 || p % 59 == 0 ||

       
      p % 61 == 0 || p % 67 == 0 || p % 71 == 0 || p % 73 == 0 || p % 79 == 0 ||

       
      p % 83 == 0 || p % 89 == 0 || p % 97 == 0)

       
    return false;

       
 

       
  // test p against those numbers ("witnesses")

       
  // good bases can be found at http://miller-rabin.appspot.com/

       
  const unsigned int STOP = 0;

       
  const unsigned int TestAgainst1[] = { 377687, STOP };

       
  const unsigned int TestAgainst2[] = { 31, 73, STOP };

       
  const unsigned int TestAgainst3[] = { 2, 7, 61, STOP };

       
  // first three sequences are good up to 2^32

       
  const unsigned int TestAgainst4[] = { 2, 13, 23, 1662803, STOP };

       
  const unsigned int TestAgainst7[] = { 2, 325, 9375, 28178, 450775, 9780504, 1795265022, STOP };

       
 

       
  // good up to 2^64

       
  const unsigned int* testAgainst = TestAgainst7;

       
  // use less tests if feasible

       
  if (p < 5329)

       
    testAgainst = TestAgainst1;

       
  else if (p < 9080191)

       
    testAgainst = TestAgainst2;

       
  else if (p < 4759123141ULL)

       
    testAgainst = TestAgainst3;

       
  else if (p < 1122004669633ULL)

       
    testAgainst = TestAgainst4;

       
 

       
  // find p - 1 = d * 2^j

       
  auto d = p - 1;

       
  d >>= 1;

       
  unsigned int shift = 0;

       
  while ((d & 1) == 0)

       
  {

       
    shift++;

       
    d >>= 1;

       
  }

       
 

       
  // test p against all bases

       
  do

       
  {

       
    auto x = powmod(*testAgainst++, d, p);

       
    // is test^d % p == 1 or -1 ?

       
    if (x == 1 || x == p - 1)

       
      continue;

       
 

       
    // now either prime or a strong pseudo-prime

       
    // check test^(d*2^r) for 0 <= r < shift

       
    bool maybePrime = false;

       
    for (unsigned int r = 0; r < shift; r++)

       
    {

       
      // x = x^2 % p

       
      // (initial x was test^d)

       
      x = mulmod(x, x, p);

       
      // x % p == 1 => not prime

       
      if (x == 1)

       
        return false;

       
 

       
      // x % p == -1 => prime or an even stronger pseudo-prime

       
      if (x == p - 1)

       
      {

       
        // next iteration

       
        maybePrime = true;

       
        break;

       
      }

       
    }

       
 

       
    // not prime

       
    if (!maybePrime)

       
      return false;

       
  } while (*testAgainst != STOP);

       
 

       
  // prime

       
  return true;

       
}

       
 

       
// ---------- compute Fibonacci % modulo ----------

       
 

       
// version 1: matrix algorithm

       
unsigned long long fibonacciMatrix(unsigned long long n, unsigned long long modulo)

       
{

       
  // fast exponentiation: same idea as powmod from my toolbox

       
 

       
  // matrix values from https://en.wikipedia.org/wiki/Fibonacci_number#Matrix_form

       
  unsigned long long fibo  [2][2]= { { 1, 1 },

       
                                     { 1, 0 } };

       
  // initially identity matrix

       
  unsigned long long result[2][2]= { { 1, 0 },   // { { F(n+1), F(n)   },

       
                                     { 0, 1 } }; //   { F(n),   F(n-1) } }

       
 

       
  while (n > 0)

       
  {

       
    // fast exponentation:

       
    // odd exponent ? a^n = a*a^(n-1)

       
    if (n & 1)

       
    {

       
      // compute new values, store them in temporaries

       
      auto t00 = mulmod(result[0][0], fibo[0][0], modulo) + mulmod(result[0][1], fibo[1][0], modulo);

       
      auto t01 = mulmod(result[0][0], fibo[0][1], modulo) + mulmod(result[0][1], fibo[1][1], modulo);

       
      auto t10 = mulmod(result[1][0], fibo[0][0], modulo) + mulmod(result[1][1], fibo[1][0], modulo);

       
      auto t11 = mulmod(result[1][0], fibo[0][1], modulo) + mulmod(result[1][1], fibo[1][1], modulo);

       
 

       
      if (t00 >= modulo) t00 -= modulo;

       
      if (t01 >= modulo) t01 -= modulo;

       
      if (t10 >= modulo) t10 -= modulo;

       
      if (t11 >= modulo) t11 -= modulo;

       
 

       
      // copy back to matrix

       
      result[0][0] = t00; result[0][1] = t01;

       
      result[1][0] = t10; result[1][1] = t11;

       
    }

       
 

       
    // even exponent ? a^n = (a*a)^(n/2)

       
 

       
    // compute new values, store them in temporaries

       
    auto t00 = mulmod(fibo[0][0], fibo[0][0], modulo) + mulmod(fibo[0][1], fibo[1][0], modulo);

       
    auto t01 = mulmod(fibo[0][0], fibo[0][1], modulo) + mulmod(fibo[0][1], fibo[1][1], modulo);

       
    auto t10 = mulmod(fibo[1][0], fibo[0][0], modulo) + mulmod(fibo[1][1], fibo[1][0], modulo);

       
    auto t11 = mulmod(fibo[1][0], fibo[0][1], modulo) + mulmod(fibo[1][1], fibo[1][1], modulo);

       
 

       
    if (t00 >= modulo) t00 -= modulo;

       
    if (t01 >= modulo) t01 -= modulo;

       
    if (t10 >= modulo) t10 -= modulo;

       
    if (t11 >= modulo) t11 -= modulo;

       
 

       
    // copy back to matrix

       
    fibo[0][0] = t00; fibo[0][1] = t01;

       
    fibo[1][0] = t10; fibo[1][1] = t11;

       
 

       
    n >>= 1;

       
  }

       
 

       
  return result[0][1]; // same as result[1][0]

       
}

       
 

       
// version 2: fast doubling algorithm, modulo < 2^62

       
unsigned long long fibonacci(unsigned long long n, unsigned long long modulo)

       
{

       
  // extract highest bit

       
  auto bit = n;

       
  while (bit & (bit - 1))

       
    bit &= bit - 1;

       
 

       
  // F(0) and F(1)

       
  auto a = 0ULL;

       
  auto b = 1ULL;

       
 

       
  // from highest to lowest bit

       
  while (bit != 0)

       
  {

       
    // F(2n)   = F(n) * (2 * F(n+1) - F(n))

       
    auto nextA = mulmod(a, 2*b + modulo - a, modulo); // plus modulo to avoid negative results

       
    // F(2n+1) = F(n)^2 + F(n+1)^2

       
    b = mulmod(a, a, modulo) + mulmod(b, b, modulo);

       
    if (b >= modulo)

       
      b -= modulo;

       
 

       
    a = nextA;

       
 

       
    // odd ?

       
    if (n & bit)

       
    {

       
      // one step further, need F(2n+1) and F(2n+2)

       
      auto next = a + b;

       
      if (next >= modulo)

       
        next -= modulo;

       
 

       
      a = b;

       
      b = next;

       
    }

       
 

       
    bit >>= 1;

       
  }

       
 

       
  return a;

       
}

       
 

       
int main()

       
{

       
  auto n         = 100000000000000ULL;

       
  auto numPrimes =          100000;

       
  auto Modulo    =   1234567891011ULL;

       
  std::cin >> n >> numPrimes >> Modulo;

       
 

       
  // F(n) = F(n-1) + F(n-2)

       
  // my "seed" values to find F(10^14 + x)

       
  auto last    = fibonacci(n - 1, Modulo); // replace by fibonacciMatrix for the other algorithm

       
  auto current = fibonacci(n    , Modulo);

       
  // actually it's already in result[0][0] of matrix algorithm

       
  // and the second result of the fast doubling algorithm

       
 

       
  auto sum = 0ULL;

       
  for (auto i = 1; i <= numPrimes; i++)

       
  {

       
    do

       
    {

       
      n++;

       
 

       
      // next Fibonacci number

       
      auto next = (last + current) % Modulo;

       
      last      = current;

       
      current   = next;

       
    }

       
    while (!isPrime(n));

       
 

       
    // if n is prime then add F(n)

       
    sum += current;

       
    // don't overflow

       
    sum %= Modulo;

       
  }

       
 

       
  // display result

       
  std::cout << sum << std::endl;

       
  return 0;

       
}

This solution contains 63 empty lines, 77 comments and 7 preprocessor commands.

Benchmark

The correct solution to the original Project Euler problem was found in 2.4 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

Changelog

August 7, 2017 submitted solution
August 7, 2017 added comments

Difficulty

35% Project Euler ranks this problem at 35% (out of 100%).

Links

projecteuler.net/thread=304 - the best forum on the subject (note: you have to submit the correct solution first)

Code in various languages:

Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p304.py (written by Nayuki)
C github.com/LaurentMazare/ProjectEuler/blob/master/e304.c (written by Laurent Mazare)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p304.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem304.java (written by Magnus Solheim Thrap)
Mathematica github.com/steve98654/ProjectEuler/blob/master/304.nb
Perl github.com/shlomif/project-euler/blob/master/project-euler/407/euler-304-v2.pl (written by Shlomi Fish)
Sage github.com/roosephu/project-euler/blob/master/304.sage (written by Yuping Luo)

Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own.
Maybe not all linked resources produce the correct result and/or exceed time/memory limits.

Heatmap

Please click on a problem's number to open my solution to that problem:

green   solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too
yellow solutions score less than 100% at Hackerrank (but still solve the original problem easily)
gray problems are already solved but I haven't published my solution yet
blue solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much
orange problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte
red problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too
black problems are solved but access to the solution is blocked for a few days until the next problem is published
[new] the flashing problem is the one I solved most recently

I stopped working on Project Euler problems around the time they released 617.
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The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Look at my progress and performance pages to get more details.

Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.
All of my solutions can be used for any purpose and I am in no way liable for any damages caused.
You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler.
Thanks for all their endless effort !!!

more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !