코딩 연습

The positive integers, \(x, \;y,\) and \(z\), are consecutive terms of an arithmetic progression. Given that \(n\) is a positive integer, the equation, \(x^2 - y^2 - z^2 =n\), has exactly one solution when \(n=20\):\[13^2-10^2-7^2=20\] In fact there are twenty-five values of \(n\) below one hundred for which the equation has a unique solution. How many values of \(n\) less than fifty million hav..

Given the positive integers \(x,\; y\), and \(z\), are consecutive terms of an arithmetic progression, the least value of the positive integers \(n\), for which the equation, \(x^2 - y^2 - z^2 =n\), has exactly two solutions is \(n=27\):\[34^2-27^2-20^2=12^2-9^2-6^2=27\] It turns out that \(n=1155\) is the least value which has exactly ten solutions. How many values of \(n\) less than one millio..

Consider the consecutive primes \(p_1=19\) and \(p_2=23\). It can be verified that \(1219\) is the smallest number such that the last digits are formed by \(p_1\) whilst also being divisible by \(p_2\). In fact, with the exception of \(p_1-3\) and \(p_2=5\), for every pair of consecutive primes, \(p_2 > p_1\), there exist values of \(n\) for which the last digits are formed by \(p_1\) and \(n\) ..