PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (230)
Orphanage
Unclass'd
Unproven (519)
Corrections (17)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: Very high
[parent] a formula for amicable pairs (Definition)

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
$\displaystyle 3\cdot 2^n -1, \quad 3\cdot 2^{n-1}-1$    and $\displaystyle \quad 9\cdot 2^{2n-1}-1$
are all prime. Then the numbers:
$\displaystyle 2^n(3\cdot 2^n-1)(3\cdot 2^{n-1}-1)$    and $\displaystyle \quad 2^n(9\cdot 2^{2n-1}-1)$
are amicable numbers.
Example 1   When $ n=2$ one has:
$\displaystyle 3\cdot 2^2 -1=11, \quad 3\cdot 2^{2-1}-1=5$    and $\displaystyle \quad 9\cdot 2^{4-1}-1=71$
which are all primes. Thus, the numbers:
$\displaystyle 2^2(3\cdot 2^2-1)(3\cdot 2^{2-1}-1)=220$    and $\displaystyle \quad 2^2(9\cdot 2^{4-1}-1)=284$
form an amicable pair. In fact, this is the smallest amicable pair. For $ n=4$ one obtains the amicable pair $ 17296$ and $ 18416$.



"a formula for amicable pairs" is owned by alozano.
(view preamble)

View style:

See Also: Thabit number


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: amicable numbers, prime, numbers, natural number, body, IBN, Thabit ibn Qurra
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 1440 times total.

Classification:
AMS MSC: 11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)