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

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

• Numbers $n$ whose Euler Phi value Divides $n + 1$

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 $n - 1$?