On divisibility by nine of the sums of even amicable pairs

Author:
Elvin Lee

Journal:
Math. Comp. **23** (1969), 545-548

MSC:
Primary 10.07

MathSciNet review:
0248074

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Abstract: Most known even amicable pairs have sums divisible by nine [9]. The general form of the exceptions to the rule of divisibility by nine (Gardner's rule) is deduced and the results expressed in the form of a theorem. A computer search based on a corollary to the theorem is described and six new exceptions to Gardner's rule are found.

**[1]**Oystein Ore,*Number Theory and Its History*, McGraw-Hill Book Company, Inc., New York, 1948. MR**0026059****[2]**Edward Brind Escott,*Amicable numbers*, Scripta Math.**12**(1946), 61–72. MR**0017293****[3]**P. Poulet, "43 new couples of amicable numbers,"*Scripta Math.*, v. 14, 1948, p. 77.**[4]**Mariano García,*New amicable pairs*, Scripta Math.**23**(1957), 167–171. MR**0098703****[5]**H. L. Rolf, "Friendly numbers,"*Amer. Math. Monthly*, v. 72, 1965, p. 455.**[6]**J. Alanen, O. Ore, and J. Stemple,*Systematic computations on amicable numbers*, Math. Comp.**21**(1967), 242–245. MR**0222006**, 10.1090/S0025-5718-1967-0222006-7**[7]**Elvin J. Lee,*Amicable numbers and the bilinear diophantine equation*, Math. Comp.**22**(1968), 181–187. MR**0224543**, 10.1090/S0025-5718-1968-0224543-9**[8]**Paul Bratley and John McKay,*More amicable numbers*, Math. Comp.**22**(1968), 677–678. MR**0225706**, 10.1090/S0025-5718-1968-0225706-9**[9]**Martin Gardner,*The celebrated four-color map problem of topology*, Sci. Amer.**203**(1960), no. 3, 218–226. MR**0114216****[10]**M. Gardner, Private communication.

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1969-0248074-6

Article copyright:
© Copyright 1969
American Mathematical Society