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In mathematics, a Descartes number is a number which is close to being a perfect number. They are named for René Descartes who observed that the number D = 3⋅7⋅11⋅13⋅22021 = 198585576189 would be an odd perfect number if only 22021 were a prime number, since the sum-of-divisors function for D satisfies

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

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

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

References

• Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip (2008). "Descartes numbers". In De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (eds.). Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006. CRM Proceedings and Lecture Notes. Vol. 46. Providence, RI: American Mathematical Society. pp. 167–173. ISBN 978-0-8218-4406-9. Zbl 1186.11004.
• Klee, Victor; Wagon, Stan (1991). Old and new unsolved problems in plane geometry and number theory. The Dolciani Mathematical Expositions. Vol. 11. Washington, DC: Mathematical Association of America. ISBN 0-88385-315-9. Zbl 0784.51002.

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