Euler's Rule

The numbers  and  are an amicable pair if the three integers

			(1)
			(2)
			(3)

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