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

## 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

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

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

Now we show:

 $\displaystyle \map \phi {72}$ $=$ $\displaystyle 24$ Euler Phi Function of 72 $\displaystyle \map \phi {78}$ $=$ $\displaystyle 24$ Euler Phi Function of 78 $\displaystyle \map \phi {84}$ $=$ $\displaystyle 24$ Euler Phi Function of 84 $\displaystyle \map \phi {90}$ $=$ $\displaystyle 24$ Euler Phi Function of 90