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

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

Prime to Own Power minus 1 over Prime minus 1 being Prime

$\mathbf A$: Prime Numbers: $\mathbf {A 3}$: Mersenne primes. Repunits. Fermat numbers. Primes of shape $k \cdot 2^n + 1$.

Wagstaff observes that the only primes $< 180$ for which $\paren {p^p - 1} / \paren {p - 1}$ is prime are $2$, $3$, $7$, $19$ and $31$; ...

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

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

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 $\paren {353 - 2^8} \paren {929 - 2^8} = 2^{16} - 2^8 + 1$.]

Sums of Squares

$\mathbf C$: Additive Number Theory: $\mathbf {C 20}$: Sums of Squares

Apart from $256$ examples, the largest of which is $1167$, every number can be expressed as the sum of at most five composite numbers.