Book:Richard K. Guy/Unsolved Problems in Number Theory/Third Edition/Errata

Errata for 2004: Richard K. Guy: Unsolved Problems in Number Theory (3rd ed.)

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

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

Lehmer gives $8$ solutions to $\map \phi n \mid n + 1$, namely $n = 2$, $n = 2^{2^k} - 1$ for $1 \le k \le 5$, $n = n_1 = 3 \cdot 5 \cdot 17 \cdot 353 \cdot 929$ 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 $\paren {353 - 2^8} \paren {929 - 2^8} = 2^{16} - 2^8 + 1$.] This exhausts the solutions with less than seven factors. Victor Meally notes that $n = n_1 \cdot 83623937 \cdot 699296672132097$ would be a solution were the largest factor a prime, put Peter Borwein notes that this is divisible by $73$.

Triples with Same Sum and Same Product

$\mathbf D$: Diophantine Equations: $\mathbf {D 16}$: Triples with Same Sum and Same Product

It may be of interest to ask for the smallest sums or products with each multiplicity. For example, for $4$ triples, J. G. Mauldon finds the smallest common sum to be $118$ ... and the smallest common product to be $25200$ ...