Rational Amicable Pair

A rational amicable pair consists of two integers a and b for which the divisor functions are equal and are of the form


where P(a,b) and Q(a,b) are bivariate polynomials, and for which the following properties hold (Y. Kohmoto):

1. All the degrees of terms of the numerator of the right fraction are the same.

2. All the degrees of terms of the denominator of the right fraction are the same.

3. The degree of P is one greater than the degree of Q.

If a=b and P(a,b) is of the form ma^r, then (◇) reduces to the special case


so if m/n is an integer, then a is a multiperfect number.

Consider polynomials of the form


For n=1, (◇) reduces to


of which no examples are known. For n=2, (◇) reduces to


so (a,b) form an amicable pair. For n=3, (◇) becomes


Kohmoto has found three classes of solutions of this type. The first is

 2^(m-1)M_m·3·5^2·13·31·139·277·3877[11·19; 239],

where M_m is a Mersenne prime with m!=2!=5, giving (26403469440047700, 30193441130006700), (7664549986025275200, 8764724625167659200), ... (OEIS A038362 and A038363). The second set of solutions is

 2^(m-1)·M_m·3·7·11^2·17^2·19^2·23·127·307·359·3739·22433·68209[83·1931; 162287]

where m!=2!=3!=7, giving the solution


The third type is the unique solution

 2^(11)·3^7·13·17·19^2·23·41·127·227·271·541·2269·124429[29·569; 17099],

Considering polynomials of the more general form


Kohmoto has found the (k,n)=(2,4) solution

 2^(m-1)·M_m·3·5·7·23^2·59·79·137·547·2477·158527·173428537·8671426849·[83·1931; 162287]

for m the index of a Mersenne prime with the exceptions of m=2 and 3.

Kohmoto (pers. comm., Feb. 2004) also found the (6,6) solution

 2^(m-1)·M_m·3^(10)·5·11·13·17·23^3·41·43·53^2·59·89·103·107·229·409·823·1031·1801·1831·3851·4271·19751·9322471·[83·1931; 162287]

for m the index of a Mersenne prime with the exceptions of m=2.

Considering polynomials of the form


for r/s=3/2, Kohmoto has found the solution

 2^8·3^2·13·17·41·53·73^2·1801·11971[5·11; 71].

Considering polynomials of the form


or equivalently,


Kohmoto has found the solutions listed in the following table.

6(1537536, 2269696)
8(22405565952, 21500290560)
9(8509664043532288000, 5783455883132928000)

See also

Amicable Pair

This entry contributed by Yasutoshi Kohmoto

Explore with Wolfram|Alpha


Sloane, N. J. A. Sequences A038362 and A038363 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Rational Amicable Pair

Cite this as:

Kohmoto, Yasutoshi. "Rational Amicable Pair." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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