# Anti-indicator

In Theory of the numbers, one says that a whole positive N is a anti-indicator if the equation $\ varphi \left(X\right) = N \,$, of unknown factor X , does not have a solution, the function $\ varphi \,$ indicating the Indicatrice of Euler. All the odd entireties are anti-indicators, except 1, since, in this case, X = 1 and X = 2 are solutions of the preceding equation.

The first even anti-indicators are:

14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, 158, 170, 174, 182, 186, 188, 194, 202, 206, 214, 218, 230, 234, 236, 242, 244, 246, 248, 254, 258, 266, 274, 278, 284, 286, 290, 298, 302, 304, 308, 314, 318

An even anti-indicator can be form $p + 1 \,$, where $p \,$ is a Prime number, but never of the form $p - 1 \,$, since $p - 1 = \ varphi \left(p\right) \,$ when p is first (the positive entireties lower than a given prime number are very first with him). Same manner, a oblong Number $n \left(N - 1\right) \,$ cannot be an anti-indicator when N is first since $\ varphi \left(p^2\right) = p \left(p - 1\right) \,$ for any prime number $p \,$.

## See too

• Anticoïndicateur

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