a formula for amicable pairs

The following formulaMathworldPlanetmathPlanetmath 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.


Let n1 be a natural numberMathworldPlanetmath and suppose that the numbers

32n-1,32n-1-1 and 922n-1-1

are all prime. Then the numbers:

2n(32n-1)(32n-1-1) and 2n(922n-1-1)

are amicable numbers.


When n=2 one has:

322-1=11,322-1-1=5 and 924-1-1=71

which are all primes. Thus, the numbers:

22(322-1)(322-1-1)=220 and 22(924-1-1)=284

form an amicable pair. In fact, this is the smallest amicable pair. For n=4 one obtains the amicable pair 17296 and 18416.

Title a formula for amicable pairs
Canonical name AFormulaForAmicablePairs
Date of creation 2013-03-22 15:52:45
Last modified on 2013-03-22 15:52:45
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Definition
Classification msc 11A05
Related topic ThabitNumber