Problem 399

Squarefree Fibonacci Numbers

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 and 610.

The 200th 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 000th squarefree fibonacci number.
Give as your answer 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 200th 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 p² (this is part of Wall's conjecture). This has been verified for primes ≤ 3·10¹⁵, 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 000th squarefree fibonacci number, rather it represents only a lower bound for that number.

无平方因子斐波那契数

前15个斐波那契数是：
1，1，2，3，5，8，13，21，34，55，89，144，233，377，610。
可以看出8和144都不是无平方因子数：8能被4整除，而144能被4和9整除。
所以前13个无平方因子斐波那契数为：
1，1，2，3，5，13，21，34，55，89，233，377和610。

第200个无平方因子斐波那契数为：
971183874599339129547649988289594072811608739584170445.
这个数的最后十六位数字是：1608739584170445；用科学计数法表示，这个数可以写成9.7e53。

求第100,000,000个无平方因子斐波那契数。
你的答案应当为这个数的最后十六位数字和这个数的科学计数法表示（四舍五入到小数点后一位小数），用半角逗号隔开。
对于第200个无平方因子斐波那契数，你的答案应当写成：1608739584170445,9.7e53

注意：在这个问题中，我们假定对于任意素数p，第一个能够被p整除的斐波那契数不能够被p²整除（这是沃尔猜想的一部分）。目前，对于≤ 3·10¹⁵的素数，这个假定已经得到验证，然而并没有完全证明。如果最终这个假定被证伪，那么这个问题目前给出的答案并不保证是第100,000,000个无平方因子斐波那契数，而只能作为其下界。