# Gardner's Conjecture on Divisibility of Sum of Even Amicable Pairs

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## Famous False Conjecture

The sum of every amicable pair of even integers is divisible by $9$.

For example:

\(\displaystyle 220 + 284\) | \(=\) | \(\displaystyle 504\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 9 \times 56\) |

\(\displaystyle 1184 + 1210\) | \(=\) | \(\displaystyle 2394\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 9 \times 266\) |

\(\displaystyle 2620 + 2924\) | \(=\) | \(\displaystyle 5544\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 9 \times 616\) |

... and so on.

## Refutation

Counterexamples are rare, but they exist:

For example:

\(\displaystyle 666 \, 030 \, 256 + 696 \, 630 \, 544\) | \(=\) | \(\displaystyle 1 \, 362 \, 660 \, 800\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2^6 \times 5^2 \times 31 \times 83 \times 331\) |

$\blacksquare$

## Source of Name

This entry was named for Martin Gardner.

## Historical Note

Martin Gardner made this conjecture in $1968$ after noticing that all amicable pairs of even integers that were known at that time had this property.

The counterexample given was provided by Elvin J. Lee, having originally been discovered by Paul Poulet.

## Sources

- 1969: Elvin J. Lee:
*On divisibility by nine of the sums of even amicable pairs*(*Math. Comp.***Vol. 23**: pp. 545 – 548) www.jstor.org/stable/2004382 - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $220$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $220$