4 Positive Integers in Arithmetic Progression which have Same Euler Phi Value

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Theorem

The following sets of $4$ positive integers which form an arithmetic progression are the smallest which all have the same Euler $\phi$ value:

$72, 78, 84, 90$
$216, 222, 228, 234$
$76 \, 236, 76 \, 240, 76 \, 246, 76 \, 252$


Proof

\(\displaystyle 78 = 72\) \(=\) \(\displaystyle 6\) $\quad$ $\quad$
\(\displaystyle 84 - 78\) \(=\) \(\displaystyle 6\) $\quad$ $\quad$
\(\displaystyle 90 - 84\) \(=\) \(\displaystyle 6\) $\quad$ $\quad$

demonstrating that this is indeed an arithmetic progression, with a common difference of $6$.


Now we show:

\(\displaystyle \phi \left({72}\right)\) \(=\) \(\displaystyle 24\) $\quad$ Euler Phi Function of 72 $\quad$
\(\displaystyle \phi \left({78}\right)\) \(=\) \(\displaystyle 24\) $\quad$ Euler Phi Function of 78 $\quad$
\(\displaystyle \phi \left({84}\right)\) \(=\) \(\displaystyle 24\) $\quad$ Euler Phi Function of 84 $\quad$
\(\displaystyle \phi \left({90}\right)\) \(=\) \(\displaystyle 24\) $\quad$ Euler Phi Function of 90 $\quad$



Sources