Paul Poulet

Paul Poulet (1887–1946) was a self-taught Belgian mathematician who made several important contributions to number theory, including the discovery of sociable numbers in 1918. He is also remembered for calculating the pseudoprimes to base two, first up to 50 million in 1926, then up to 100 million in 1938. These are now often called Poulet numbers in his honour (they are also known as Fermatians or Sarrus numbers). In 1925, he published forty-three new multiperfect numbers, including the first two known octo-perfect numbers. His achievements are particularly remarkable given that he worked without the aid of modern computers and calculators.

Poulet published at least two books about his mathematical work, Parfaits, amiables et extensions (1918) (Perfect and Amicable Numbers and Their Extensions) and La chasse aux nombres (1929) (The Hunt for Numbers). He wrote the latter in the French village of Lambres-lez-Aire in the Pas-de-Calais, a short distance across the border with Belgium. Both were published by éditions Stevens of Brussels.

In a sociable chain, or aliquot cycle, a sequence of divisor-sums returns to the initial number. These are the two chains Poulet described in 1918:

12496 → 14288 → 15472 → 14536 → 14264 → 12496 (5 links)

14316 → 19116 → 31704 → 47616 → 83328 → 177792 → 295488 → 629072 → 589786 → 294896 → 358336 → 418904 → 366556 → 274924 → 275444 → 243760 → 376736 → 381028 → 285778 → 152990 → 122410 → 97946 → 48976 → 45946 → 22976 → 22744 → 19916 → 17716 → 14316 (28 links)

The second chain remains by far the longest known, despite the exhaustive computer searches begun by the French mathematician Henri Cohen in 1969. Poulet introduced sociable chains in a paper in the journal L'Intermédiaire des Mathématiciens #25 (1918). The paper ran like this:

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