Numbers n whose Euler Phi value Divides n + 1/Historical Note
Historical Note on Numbers $n$ whose Euler Phi value Divides $n + 1$
This result appears in an article from $1932$ by Derrick Henry Lehmer, where he performs a complete analysis of the situation where $k \map \phi n = n + 1$ where $n$ has fewer than $7$ distinct prime factors.
It appears that Victor Meally may have made the same observation, as Richard K. Guy attributes it to him (without providing a citation) in a note in his Unsolved Problems in Number Theory, 2nd ed. of $1994$.
Interestingly, while citing Lehmer's $1932$ article in the context of a conjecture about $k \map \phi n = n - 1$, Guy appears to completely fail to notice his analysis of $k \map \phi n = n + 1$.
In his Unsolved Problems in Number Theory, 3rd ed. of $2004$, he does now report on Lehmer's $1932$ article, but continues to credit Victor Meally with the observation that $83 \, 623 \, 935 \times 83 \, 623 \, 937 \times \paren {83 \, 623 \, 935 \times 83 \, 623 \, 937 + 2}$ would also be a solution if $83 \, 623 \, 935 \times 83 \, 623 \, 937 + 2$ were prime.
However, again, the latter result also appears in Lehmer's $1932$ article.
In Unsolved Problems in Number Theory, 3rd ed., Guy then reports that Peter Borwein established that $83 \, 623 \, 935 \times 83 \, 623 \, 937 + 2 = 6 \, 992 \, 962 \, 672 \, 132 \, 097$ is not prime, as it has $73$ as a prime factor.
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 $n - 1$?
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $83,623,935$
- 2004: Richard K. Guy: Unsolved Problems in Number Theory (3rd ed.): $\mathbf B$: Divisibility: $\mathbf {B 37}$: Does $\map \phi n$ properly divide $n - 1$?