Thabit's Rule
Theorem
Let $n$ be a positive integer such that:
\(\ds a\) | \(=\) | \(\ds 3 \times 2^n - 1\) | ||||||||||||
\(\ds b\) | \(=\) | \(\ds 3 \times 2^{n - 1} - 1\) | ||||||||||||
\(\ds c\) | \(=\) | \(\ds 9 \times 2^{2 n - 1} - 1\) |
are all prime.
Then:
- $\tuple {2^n a b, 2^n c}$
forms an amicable pair.
Proof
Let $r = 2^n a b, s = 2^n c$.
Let $\map {\sigma_1} k$ denote the divisor sum of an integer $k$.
From Divisor Sum of Power of 2:
- $\map {\sigma_1} {2^n} = 2^{n + 1} - 1$
From Divisor Sum of Prime Number:
\(\ds \map {\sigma_1} a\) | \(=\) | \(\ds 3 \times 2^n\) | ||||||||||||
\(\ds \map {\sigma_1} b\) | \(=\) | \(\ds 3 \times 2^{n - 1}\) | ||||||||||||
\(\ds \map {\sigma_1} c\) | \(=\) | \(\ds 9 \times 2^{2 n - 1}\) |
From Divisor Sum Function is Multiplicative:
\(\ds \map {\sigma_1} r\) | \(=\) | \(\ds \map {\sigma_1} {2^n} \map {\sigma_1} a \map {\sigma_1} b\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2^{n + 1} - 1} \paren {3 \times 2^n} \paren {3 \times 2^{n - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2^{n + 1} - 1} \paren {9 \times 2^{2 n - 1} }\) |
and:
\(\ds \map {\sigma_1} s\) | \(=\) | \(\ds \map {\sigma_1} {2^n} \map {\sigma_1} c\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2^{n + 1} - 1} \paren {9 \times 2^{2 n - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} r\) |
Thus it is seen that:
- $\map {\sigma_1} r = \map {\sigma_1} s$
Now we have:
\(\ds r + s\) | \(=\) | \(\ds 2^n a b + 2^n c\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^n \paren {\paren {3 \times 2^n - 1} \paren {3 \times 2^{n - 1} - 1} + 9 \times 2^{2 n - 1} - 1 }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^n \paren {9 \times 2^{2 n - 1} - 3 \times 2^n - 3 \times 2^{n - 1} + 1 + 9 \times 2^{2 n - 1} - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^n \paren {2 \times 9 \times 2^{2 n - 1} - 3 \times 2 \times 2^{n - 1} - 3 \times 2^{n - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^{n + 1} \paren {9 \times 2^{2 n - 1} - 9 \times 2^{n - 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^{n + 1} \paren {9 \times 2^{2 n - 1} } - 2^{n + 1} \paren {9 \times 2^{n - 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^{n + 1} \paren {9 \times 2^{2 n - 1} } - \paren {9 \times 2^{2 n - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2^{n + 1} - 1} \paren {9 \times 2^{2 n - 1} }\) |
and so it is seen that:
- $r + s = \map {\sigma_1} r = \map {\sigma_1} s$
Hence the result, by definition of amicable pair.
$\blacksquare$
Also see
Source of Name
This entry was named for Thabit ibn Qurra.
Historical Note
Thabit's Rule has been discovered and rediscovered several times during the course of history.
The first person on record to discuss it was Thabit ibn Qurra, in his Book on the Determination of Amicable Numbers.
The Thabit pair $\tuple {17 \,296, 18 \, 416}$ was discovered by Ibn al-Banna' al-Marrakushi, who may well have also rediscovered this rule.
In $1636$ it was rediscovered by Pierre de Fermat, who also rediscovered the Thabit pair $\tuple {17 \,296, 18 \, 416}$.
At around the same time, René Descartes also rediscovered it, and also rediscovered the Thabit pair $\tuple {9 \, 363 \, 584, 9 \, 437 \, 056}$.
These three remain the only known Thabit pairs.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $220$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $220$