Prime to Own Power minus 1 over Prime minus 1 being Prime
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Theorem
Let $n \in \Z_{>1}$ be an integer greater than $1$.
Then $\dfrac {n^n - 1} {n - 1}$ is a prime for $n$ equal to:
- $2, 3, 19, 31$
This sequence is A088790 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
This theorem requires a proof. In particular: Can easily be proved that $n$ must itself be prime for the expression to be prime. Then it's a matter of checking them all. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Note that if $4 p + 1$ is prime for prime $p$, then $\dfrac {p^p - 1} {p - 1}$ is divisible by $4 p + 1$:
Let $q = 4 p + 1$ be prime.
By First Supplement to Law of Quadratic Reciprocity:
- $\paren {\dfrac {-1} q} = 1$
that is, there exists some integer $I$ such that $I^2 \equiv -1 \pmod q$.
Then:
\(\ds \paren {1 + I}^4\) | \(=\) | \(\ds \paren {1 + 2 I + I^2}^2\) | Square of Sum | |||||||||||
\(\ds \) | \(\equiv\) | \(\ds \paren {2 I}^2\) | \(\ds \pmod q\) | Congruence of Powers | ||||||||||
\(\ds \) | \(\equiv\) | \(\ds -4\) | \(\ds \pmod q\) |
and thus:
\(\ds p^p\) | \(\equiv\) | \(\ds p^p \paren {1 + I}^{q - 1}\) | \(\ds \pmod q\) | Fermat's Little Theorem | ||||||||||
\(\ds \) | \(\equiv\) | \(\ds p^p \paren {1 + I}^{4 p}\) | \(\ds \pmod q\) | |||||||||||
\(\ds \) | \(\equiv\) | \(\ds p^p \paren {-4}^p\) | \(\ds \pmod q\) | |||||||||||
\(\ds \) | \(\equiv\) | \(\ds \paren {-4 p}^p\) | \(\ds \pmod q\) | |||||||||||
\(\ds \) | \(\equiv\) | \(\ds 1^p\) | \(\ds \pmod q\) | |||||||||||
\(\ds \) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod q\) |
Hence $q \divides \paren {p^p - 1}$.
Obviously $q > p - 1$.
Therefore $q \divides \dfrac {p^p - 1} {p - 1}$.
Sources
- 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.): $\mathbf A$: Prime Numbers: $\mathbf {A 3}$: Mersenne primes. Repunits. Fermat numbers. Primes of shape $k \cdot 2^n + 1$.
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $19$
- 2004: Richard K. Guy: Unsolved Problems in Number Theory (3rd ed.): $\mathbf A$: Prime Numbers: $\mathbf {A 3}$: Mersenne primes. Repunits. Fermat numbers. Primes of shape $k \cdot 2^n + 1$.