Noncototient

Article summary

In number theory, a noncototient is a positive integer that cannot be expressed as the difference between a positive integer and the number of coprime integers below it. That is, , where stands for Euler's totient function, has no solution for . The cototient of is defined as , so a noncototient is a number that is never a cototient. It is conjectured that all noncototients are even. This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number can be represented as a sum of two distinct primes and , then \begin{align} pq - \varphi(pq) &= pq - (p-1)(q-1) \\ &= p + q - 1 \\ &= n - 1. \end{align} It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations , , and . For even numbers, it can be shown \begin{align} 2pq - \varphi(2pq) &= 2pq - (p-1)(q-1) \\ &= pq + p + q - 1 \\ &= (p+1)(q+1) - 2 \end{align} Thus, all even numbers such that can be written as with primes are cototients. The first few noncototients are :10, 26, 34, 50, 52, 58, 86, 100, 116, 122, 130, 134, 146, 154, 170, 172, 186, 202, 206, 218, 222, 232, 244, 260, 266, 268, 274, 290, 292, 298, 310, 326, 340, 344, 346, 362, 366, 372, 386, 394, 404, 412, 436, 466, 470, 474, 482, 490, ...…

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