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

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Theorem

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

$72, 78, 84, 90$
$216, 222, 228, 234$
$76 \, 326, 76 \, 332, 76 \, 338, 76 \, 344$


Proof

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

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


Now we show:

\(\displaystyle \map \phi {72}\) \(=\) \(\displaystyle 24\) $\phi$ of $72$
\(\displaystyle \map \phi {78}\) \(=\) \(\displaystyle 24\) $\phi$ of $78$
\(\displaystyle \map \phi {84}\) \(=\) \(\displaystyle 24\) $\phi$ of $84$
\(\displaystyle \map \phi {90}\) \(=\) \(\displaystyle 24\) $\phi$ of $90$

$\Box$



\(\displaystyle 222 - 216\) \(=\) \(\displaystyle 6\)
\(\displaystyle 228 - 222\) \(=\) \(\displaystyle 6\)
\(\displaystyle 234 - 228\) \(=\) \(\displaystyle 6\)

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


Now we show:

\(\displaystyle \map \phi {216}\) \(=\) \(\displaystyle 72\) $\phi$ of $216$
\(\displaystyle \map \phi {222}\) \(=\) \(\displaystyle 72\) $\phi$ of $222$
\(\displaystyle \map \phi {228}\) \(=\) \(\displaystyle 72\) $\phi$ of $228$
\(\displaystyle \map \phi {234}\) \(=\) \(\displaystyle 72\) $\phi$ of $234$

$\Box$


\(\displaystyle 76 \, 332 - 76 \, 326\) \(=\) \(\displaystyle 6\)
\(\displaystyle 76 \, 338 - 76 \, 332\) \(=\) \(\displaystyle 6\)
\(\displaystyle 76 \, 344 - 76 \, 338\) \(=\) \(\displaystyle 6\)

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


Now we show:

\(\displaystyle \map \phi {76 \, 326}\) \(=\) \(\displaystyle 25 \, 440\) $\phi$ of $76 \, 326$
\(\displaystyle \map \phi {76 \, 332}\) \(=\) \(\displaystyle 25 \, 440\) $\phi$ of $76 \, 332$
\(\displaystyle \map \phi {76 \, 338}\) \(=\) \(\displaystyle 25 \, 440\) $\phi$ of $76 \, 338$
\(\displaystyle \map \phi {76 \, 344}\) \(=\) \(\displaystyle 25 \, 440\) $\phi$ of $76 \, 344$

$\blacksquare$


Sources