Euler,s Rule
المؤلف:
Dickson, L. E
المصدر:
History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, 2005.
الجزء والصفحة:
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24-11-2020
2139
Euler's Rule
The numbers 
and 
are an amicable pair if the three integers
are all prime numbers for some positive integer 
satisfying 
(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 
for which Euler's rule is satisfied are 
, 
, 
, 
, 
, ... (OEIS A094445 and A094446), with no others for 
, corresponding to the triples 
, (23, 47, 1151), (191, 383, 73727), ..., giving the amicable pairs (220, 284), (17296, 18416), (9363584, 9437056), ....
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.a
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