Breeding amicable numbers in abundance

Abstract

We give some new methods for the constructive search for amicable number pairs. Our numerical experiments using these methods produced a total of 3501 new amicable pairs of a very special form. They provide some experimental evidence for the infinity of such pairs.

New amicable pairs of type $(2,2)$ and type $(3,2)$

“…Sometimes e 1 contained one more prime than e 2 . In this case, an amicable pair produced is of type (3,2). At first, the choices for e 1 and e 2 were often determined by looking at known pairs of other types.…”

“…The pair (17296, 18416) has often been attributed to Fermat, but it was actually first found in the early 14th century by Ibn al-Banna in Marakesh and also by Kamaladdin Farisi in Bagdad. The pair (9363584, 9437056) has often been attributed to Descartes, but it was actually first discovered by Muhammad Baqir Yazdi in Iran [3]. Euler found 59 more pairs by noticing that each member of a known pair had a common factor and then a product of different primes on the end.…”

The first known type (7,1) amicable pair

“…Type (i, 1) amicable pairs have been the object of special investigation and search [1], [2], [3], [5]. Often it is possible to construct other amicable pairs from them [6].…”

“…Often it is possible to construct other amicable pairs from them [6]. Also, until Euler's time [4], the only three known amicable pairs were of type (2,1). One of these is the smallest existing amicable pair (2 2 · 5 · 11, 2 2 · 71), which was known to the Pythagoreans.…”

Computation of all the amicable pairs below 10¹⁰

“…In [8] a substantial part of the new APs found there was constructed from such mother pairs. In [1], Borho and Hoffmann have partially generalized the methods from [8] by introducing the concept of a breeder: a breeder is a pair of positive integers (ax, a2) such that the equations ax + a2x = a(ax) = a(a2)(x + 1) In our list of 1427 APs we found a few APs, e.g., #647 and #955, which suggested that the condition gcà(a, u) = 1 in Theorem 1 may be dropped. In fact, we have It is known [5] that most even APs have a pair sum which is = 0 (mod 9).…”