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a formula for amicable pairs
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(Definition)
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The following formula is due to Thabit ibn Qurra (836-901), a mathematician who worked in Baghdad's ``House of Wisdom'' translating Greek and Syrian works (such as Apollonius's ``Conics'' or works of Euclid and Archimedes). As he translated the texts, ibn Qurra produced a mathematical body of his own.
Theorem 1 Let $n\geq 1$ be a natural number and suppose that the numbers $$3\cdot 2^n -1, \quad 3\cdot 2^{n-1}-1 \quad \text{ and } \quad 9\cdot 2^{2n-1}-1$$ are all prime. Then the numbers: $$2^n(3\cdot 2^n-1)(3\cdot 2^{n-1}-1)\quad \text{ and } \quad 2^n(9\cdot 2^{2n-1}-1)$$ are amicable numbers.
Example 1 When $n=2$ one has: $$3\cdot 2^2 -1=11, \quad 3\cdot 2^{2-1}-1=5 \quad \text{ and } \quad 9\cdot 2^{4-1}-1=71$$ which are all primes. Thus, the numbers: $$2^2(3\cdot 2^2-1)(3\cdot 2^{2-1}-1)=220\quad \text{ and } \quad 2^2(9\cdot 2^{4-1}-1)=284$$ form an amicable pair. In fact, this is the smallest amicable pair. For $n=4$ one obtains the amicable pair $17296$ and $18416$ .
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"a formula for amicable pairs" is owned by alozano.
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Cross-references: amicable numbers, prime, numbers, natural number, body, IBN, Thabit ibn Qurra, formula
There are 2 references to this entry.
This is version 3 of a formula for amicable pairs, born on 2006-04-27, modified 2006-04-28.
Object id is 7876, canonical name is AFormulaForAmicablePairs.
Accessed 1866 times total.
Classification:
| AMS MSC: | 11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors) |
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Pending Errata and Addenda
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