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

## 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

 $\ds 78 - 72$ $=$ $\ds 6$ $\ds 84 - 78$ $=$ $\ds 6$ $\ds 90 - 84$ $=$ $\ds 6$

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

Now we show:

 $\ds \map \phi {72}$ $=$ $\ds 24$ $\phi$ of $72$ $\ds \map \phi {78}$ $=$ $\ds 24$ $\phi$ of $78$ $\ds \map \phi {84}$ $=$ $\ds 24$ $\phi$ of $84$ $\ds \map \phi {90}$ $=$ $\ds 24$ $\phi$ of $90$

$\Box$

 $\ds 222 - 216$ $=$ $\ds 6$ $\ds 228 - 222$ $=$ $\ds 6$ $\ds 234 - 228$ $=$ $\ds 6$

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

Now we show:

 $\ds \map \phi {216}$ $=$ $\ds 72$ $\phi$ of $216$ $\ds \map \phi {222}$ $=$ $\ds 72$ $\phi$ of $222$ $\ds \map \phi {228}$ $=$ $\ds 72$ $\phi$ of $228$ $\ds \map \phi {234}$ $=$ $\ds 72$ $\phi$ of $234$

$\Box$

 $\ds 76 \, 332 - 76 \, 326$ $=$ $\ds 6$ $\ds 76 \, 338 - 76 \, 332$ $=$ $\ds 6$ $\ds 76 \, 344 - 76 \, 338$ $=$ $\ds 6$

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

Now we show:

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

$\blacksquare$