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id	title	challengeType	forumTopicId	dashedName
5900f4fc1000cf542c51000e	Problem 399: Squarefree Fibonacci Numbers	5	302064	problem-399-squarefree-fibonacci-numbers

--description--

The first 15 fibonacci numbers are:

1,1,2,3,5,8,13,21,34,55,89,144,233,377,610.

It can be seen that 8 and 144 are not squarefree: 8 is divisible by 4 and 144 is divisible by 4 and by 9.

So the first 13 squarefree fibonacci numbers are:

1,1,2,3,5,13,21,34,55,89,233,377 \text{ and } 610.

The $200$th squarefree fibonacci number is: 971183874599339129547649988289594072811608739584170445. The last sixteen digits of this number are: 1608739584170445 and in scientific notation this number can be written as 9.7e53.

Find the $100\,000\,000$th squarefree fibonacci number. Give as your answer as a string with its last sixteen digits followed by a comma followed by the number in scientific notation (rounded to one digit after the decimal point). For the $200$th squarefree number the answer would have been: 1608739584170445,9.7e53

Note: For this problem, assume that for every prime p, the first fibonacci number divisible by p is not divisible by p^2 (this is part of Wall's conjecture). This has been verified for primes ≤ 3 \times {10}^{15}, but has not been proven in general.

If it happens that the conjecture is false, then the accepted answer to this problem isn't guaranteed to be the $100\,000\,000$th squarefree fibonacci number, rather it represents only a lower bound for that number.

--hints--

squarefreeFibonacciNumbers() should return a string.

assert(typeof squarefreeFibonacciNumbers() === 'string');

squarefreeFibonacciNumbers() should return the string 1508395636674243,6.5e27330467.

assert.strictEqual(squarefreeFibonacciNumbers(), '1508395636674243,6.5e27330467');

--seed--

--seed-contents--

function squarefreeFibonacciNumbers() {

  return true;
}

squarefreeFibonacciNumbers();

--solutions--

// solution required

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