Descartes number

2020 Mathematics Subject Classification: Primary: 11A [MSN][ZBL]

A number which is close to being a perfect number. They are named for René Descartes who observed that the number

$$D= 198585576189 = 3^2⋅7^2⋅11^2⋅13^2⋅22021$$

would be an odd perfect number if only 22021 were a prime number, since the sum-of-divisors function for $D$ satisfies

$$\sigma(D) = (3^2+3+1)\cdot(7^2+7+1)\cdot(11^2+11+1)\cdot(13^3+13+1)\cdot(22021+1) \ .$$

A Descartes number is defined as an odd number $n = m p$ where $m$ and $p$ are coprime and $2n = \sigma(m)\cdot(p+1)$. The example given is the only one currently known.

If $m$ is an odd almost perfect number, that is, $\sigma(m) = 2m-1$, then $m(2m−1)$ is a Descartes number.

References

• Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip. "Descartes numbers". In De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (edd). Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006. CRM Proceedings and Lecture Notes 46 Providence, RI: American Mathematical Society (2008) pp. 167–173. ISBN 978-0-8218-4406-9. Zbl 1186.11004.