## Connectedness of a network

### Problem 186

Published on Saturday, 15th March 2008, 05:00 am; Solved by 1908; Difficulty rating: 60%Here are the records from a busy telephone system with one million users:

RecNr | Caller | Called |

1 | 200007 | 100053 |

2 | 600183 | 500439 |

3 | 600863 | 701497 |

... | ... | ... |

The telephone number of the caller and the called number in record n are Caller(n) = S_{2n-1} and Called(n) = S_{2n} where S_{1,2,3,...} come from the "Lagged Fibonacci Generator":

For 1 ≤ k ≤ 55, S_{k} = [100003 - 200003k + 300007k^{3}] (modulo 1000000)

For 56 ≤ k, S_{k} = [S_{k-24} + S_{k-55}] (modulo 1000000)

If Caller(n) = Called(n) then the user is assumed to have misdialled and the call fails; otherwise the call is successful.

From the start of the records, we say that any pair of users X and Y are friends if X calls Y or vice-versa. Similarly, X is a friend of a friend of Z if X is a friend of Y and Y is a friend of Z; and so on for longer chains.

The Prime Minister's phone number is 524287. After how many successful calls, not counting misdials, will 99% of the users (including the PM) be a friend, or a friend of a friend etc., of the Prime Minister?