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A conjecture of Bratley and McKay, according to which odd amicable numbers should be divisible by three, is disproved by some counterexamples. 1. Even and Odd Amicable Numbers. Two natural numbers A and B are amicable if each of them is the sum of all proper divisors of the other. If A = B, they are called perfect numbers, otherwise they form an amicable pair. The first perfect numbers 6, 28, 496, 8128, and the smallest amicable pair 220, 284, were known to the Greek mathematicians. Two further amicable pairs were discovered by medieval Islamic mathematicians, and rediscovered by Fermat and Descartes. All of these were even numbers. In fact, they were found by the famous rules given by Euclid for perfect, resp. by Thabit ibn Kurrah for amicable numbers (see, e.g., [11, [51 for a survey of this subject), and so were even by construction. L. Euler was the first to study systematically the question whether or not, also odd numbers with these properties may be found. The existence of odd perfect numbers has remained a famous open problem in number theory, while the existence of odd amicable numbers was established by Euler. He described several methods to construct numerical examples, one of which is, for example, A =32 *7*13*5*17=69615, B =32 *7*13*107=87633. Since Euler's time, many more even and odd amicable pairs have been found and published: Hundreds of them before, and thousands after the employment of electronic computers in number theory. A superficial glance at the list of hitherto known odd amicable pairs illustrates the fact that the lack of two as a common factor has to be compensated by a sufficient amount of divisibility by the other small prime factors, like three, five, seven. In fact, all odd amicable pairs that we know [21, [61, [71, [81 actually contain some power of three as a common factor. With some familiarity with the various known methods to find odd amicable pairs, it soon becomes clear, that it is actually very hard to avoid three as a common factor. Paul Bratley and John McKay even conjectured that all odd amicable numbers must be divisible by three, see [31, and also R. Guy's book on open problems in number theory [4]. On the other hand, to avoid three is a priori not impossible, but it only leads to very large numbers in all calculations, which are difficult to deal with. Therefore, we made a systematic attempt to decide the question posed in the title of this note by a constructive search, or, in other words, to disprove the conjecture of Bratley and McKay by a counterexample. Received January 12, 1987; revised July 27, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 11A25. ? 1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page

Are there odd amicable numbers not divisible by three?