NOTES ON THE DIVISIBILITY BY NINE OF THE SUMS OF EVEN AMICABLE PAIRS

This article presents a perspective on the development of the theory of amicable numbers, focusing particularly on the contributions of Poulet, Gardner and Elvin Lee on the divisibility by nine of the sums of even amicable pairs.

In this way, this manuscript, after evaluating the contributions brought by these authors, retrieves Elvin Lee's indication that not all sums of even amicable pairs is divisible by nine, highlighting the eleven examples that refuted Gardner, and which will be called, in this article, exceptional even amicable pairs.

Finally, based on the eleven counterproofs mentioned, the article will propose two conjectures: 1) The final digits of the exceptional even amicable pairs follows a pattern; and 2) They will never end with the digits 2-2; 2-4; 2-6; 4-2; 4-4; 4-8; 6-2; 6-6; 8-4; 8-6; 8-8.