# Numbers n whose Euler Phi value Divides n + 1/Mistake/Second Edition

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## Source Work

1994: Richard K. Guy: *Unsolved Problems in Number Theory* (2nd ed.):

- $\mathbf B$: Divisibility
- $\mathbf {B 37}$: Does $\map \phi n$ properly divide $n - 1$?

## Mistake

*Victor Meally notes that $\map \phi n$ sometimes divides $n + 1$, e.g. for $n = n_1 = 3 \cdot 5 \cdot 17 \cdot 353 \cdot 929$ and $n = n_1 \cdot 83623937$. [Note that $353 = 11 \cdot 2^5 + 1, 929 = 29 \cdot 2^5 + 1, 83623937 = 11 \cdot 29 \cdot 2^{18} + 1$ and $\paren {353 - 2^8} \paren {929 - 2^8} = 2^{16} - 2^8 + 1$.]*

## Correction

This was in fact published by Derrick Henry Lehmer in an article in $1932$.

As this article was itself cited by Guy in the references of this actual chapter, it is an oversight of his not to have noticed it.

In his *Unsolved Problems in Number Theory, 3rd ed.* of $2004$, Guy correctly attributes the result to Lehmer, although still misses his final observation in an endnote of the above article:

*In the same way if $6992962672132097$ is a prime**$3 \cdot 5 \cdot 17 \cdot 353 \cdot 929 \cdot 83623937 \cdot 6992962672132097 = 48901526933832864378258473353215$*

*is a solution of $(2)$.*

Guy continues to attribute the above to Meally.

## Also see

## Sources

- 1932: D.H. Lehmer:
*On Euler's Totient Function*(*Bull. Amer. Math. Soc.***Vol. 38**: pp. 745 – 751)

- 1994: Richard K. Guy:
*Unsolved Problems in Number Theory*(2nd ed.): $\mathbf B$: Divisibility: $\mathbf {B 37}$: Does $\map \phi n$ properly divide $n - 1$?