# 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

\(\displaystyle 78 = 72\) | \(=\) | \(\displaystyle 6\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle 84 - 78\) | \(=\) | \(\displaystyle 6\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle 90 - 84\) | \(=\) | \(\displaystyle 6\) | $\quad$ | $\quad$ |

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

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $72$