TOPICS
Search

Euler's Rule

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

The numbers 2^npq and 2^nr are an amicable pair if the three integers

p=2^m(2^(n-m)+1)-1
(1)
q=2^n(2^(n-m)+1)-1
(2)
r=2^(n+m)(2^(n-m)+1)^2-1
(3)

are all prime numbers for some positive integer m satisfying 1<=m<=n-1 (Dickson 2005, p. 42). However, there are many amicable pairs which do not satisfy Euler's rule, so it is a sufficient but not necessary condition for amicability. Euler's rule is a generalization of Thâbit ibn Kurrah rule.

The first few (m,n) for which Euler's rule is satisfied are (m,n)=(1,2), (3,4), (6,7), (1,8), (29,40), ... (OEIS A094445 and A094446), with no others for n<2500, corresponding to the triples (p,q,r)=(5,11,71), (23, 47, 1151), (191, 383, 73727), ..., giving the amicable pairs (220, 284), (17296, 18416), (9363584, 9437056), ....

See also

Amicable Pair, Thâbit ibn Kurrah Rule

Explore with Wolfram|Alpha

References

Borho, W. "On Thabit ibn Kurrah's Formula for Amicable Numbers." Math. Comput. 26, 571-578, 1972.Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, 2005.Euler, L. "De Numeris Amicabilibus." In Opera Omnia, Series Prima, Vol. 2. Leipzig, Germany: Teubner, pp. 63-162, 1915.Sloane, N. J. A. Sequences A094445 and A094446 in "The On-Line Encyclopedia of Integer Sequences."te Riele, H. J. J. "Four Large Amicable Pairs." Math. Comput. 28, 309-312, 1974.

Referenced on Wolfram|Alpha

Euler's Rule

Cite this as:

Weisstein, Eric W. "Euler's Rule." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulersRule.html

Subject classifications