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A number consisting entirely of ones is called a repunit. We shall define \(R(k)\) to be a repunit of length \(k\); for example, \(R(6)=111111\). Let us consider repunits of the form \(R \left (10^n \right )\). Although \(R(10),\; R(100)\), or \(R(1000)\) are not divisible by \(17\), \(R(10000)\) is divisible by \(17\). Yet there is no value of \(n\) for which \(R \left ( 10^n \right )\) will di..

A number consisting entirely of ones is called a repunit. We shall define \(R(k)\) to be a repunit of length \(k\). For example \(R(10)=1111111111=11\times 41 \times 271 \times 9091\), and the sum of these prime factors is \(9414\). Find the sum of the first fort prime factors of \(R \left ( 10^9 \right ) \). 숫자 \(1\) 로만 이루어진 수를 repunit 이라고 한다. \(R(k)\) 를 \(k\) 개의 \(1\) 로 이루어진 수라고 하자. 예를 들어, \(R..

There are some prime values, \(p\), for which there exists a positive integer, \(n\), such that the expression \(n^3 + n^2 p\) is a perfect cube. For example, when \(p=19, \; 8^3 + 8^2 \times 19 = 12^3\). What is perhaps most surprising is that for each prime with this property the value of \(n\) is unique, and there are only four such primes below one-hundred. How many primes below one million ..