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

Jump to navigation
Jump to search

## 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

- Apr. 1972: M. Lal and P. Gillard:
*On the Equation $\map \phi n = \map \phi {n + k}$*(*Math. Comp.***Vol. 26**,*no. 118*: pp. 579 – 583) www.jstor.org/stable/2005186 - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $72$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $72$