Descartes number

In number theory, a Descartes number is an odd number which would have been an odd perfect number if one of its composite factors were prime. They are named after René Descartes who observed that the number $D = 3 2 \cdot 7 2 \cdot 11 2 \cdot 13 2 \cdot 22021 = (3\cdot1001) 2 \cdot (22\cdot1001 - 1) = 198585576189$ would be an odd perfect number if only $22021$ were a prime number, since in that case the sum-of-divisors function for $D$ would satisfy ${\begin{aligned}\sigma (D)&=(3^{2}+3+1)\cdot (7^{2}+7+1)\cdot (11^{2}+11+1)\cdot (13^{2}+13+1)\cdot (22021+1)\\&=(13)\cdot (3\cdot 19)\cdot (7\cdot 19)\cdot (3\cdot 61)\cdot (22\cdot 1001)\\&=3^{2}\cdot 7\cdot 13\cdot 19^{2}\cdot 61\cdot (22\cdot 7\cdot 11\cdot 13)\\&=2\cdot (3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2})\cdot (19^{2}\cdot 61)\\&=2\cdot (3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2})\cdot 22021=2D.\end{aligned}}$ In reality, 22021 is composite ( $22021 = 19 2 \cdot 61$ ), and $\sigma (D)=426027470778={\frac {23622}{11011}}D$ .

A Descartes number is defined as an odd number $n = m \cdot p$ where $m$ and $p$ are coprime and $2 n = σ (m) \cdot (p + 1)$ , whence $p$ is taken as a 'spoof' prime. The example given is the only one currently known.

If $m$ is an odd almost perfect number,^[1] that is, $σ(m) = 2 m - 1$ and $2 m - 1$ is taken as a 'spoof' prime, then $n = m \cdot (2 m - 1)$ is a Descartes number, since $σ(n) = σ(m \cdot (2 m - 1)) = σ(m) \cdot 2 m = (2 m - 1) \cdot 2 m = 2 n$ . If $2 m - 1$ were prime, $n$ would be an odd perfect number.

Properties

If $n$ is a cube-free Descartes number not divisible by $3$ , then $n$ has over one million distinct prime divisors.^[2] If $D=pq$ is a Descartes number other than Descartes' example, with spoof-prime factor $p$ , then $q>10^{12}$ .^[3]

Generalizations

John Voight generalized Descartes numbers to allow negative bases. He found the example $3^{4}7^{2}11^{2}19^{2}(-127)^{1}$ .^[4] Subsequent work by a group at Brigham Young University found more examples similar to Voight's example,^[4] and also allowed a new class of spoofs where one is allowed to also not notice that a prime is the same as another prime in the factorization.^[5] A generalization of Descartes numbers to multiperfect numbers has also been constructed. (Tóth (2025)).

Notes

^ Currently, the only known almost perfect numbers are the non-negative powers of 2, whence the only known odd almost perfect number is $20 = 1.$
^ Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip (2008), "Descartes numbers", Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006, Providence, RI: American Mathematical Society (AMS), pp. 167–173, ISBN 978-0-8218-4406-9, Zbl 1186.11004, retrieved 2024-05-13
^ Tóth (2021)
^ ^a ^b Nadis, Steve (September 10, 2020). "Mathematicians Open a New Front on an Ancient Number Problem". Quanta Magazine. Retrieved 3 October 2021.
^ Andersen, Nickolas; Durham, Spencer; Griffin, Michael J.; Hales, Jonathan; Jenkins, Paul; Keck, Ryan; Ko, Hankun; Molnar, Grant; Moss, Eric; Nielsen, Pace P.; Niendorf, Kyle; Tombs, Vandy; Warnick, Merrill; Wu, Dongsheng (2020). "Odd, spoof perfect factorizations". J. Number Theory (234): 31–47. arXiv:2006.10697.