Descartes number explained

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

\begin{align} \sigma(D) & = ( 3 ^{2+3+1) ⋅ (7} ^{2+7+1) ⋅ (11} ^{2+11+1) ⋅ (13} ^{2+13+1) ⋅ (22021+1)} \\ & = ( 13 ) ⋅ ( 3 ⋅ 19 ) ⋅ ( 7 ⋅ 19 ) ⋅ ( 3 ⋅ 61 ) ⋅ ( 22 ⋅ 1001 ) \\ & = 3 ^{2 ⋅ 7 ⋅ 13 ⋅ 19} ^{2 ⋅ 61 ⋅ (22 ⋅ 7 ⋅ 11 ⋅ 13)} \ & = 2 ⋅ ( 3 ^{2 ⋅ 7} ^{2 ⋅ 11} ^{2 ⋅ 13} ^{2)} ⋅ ( 19 ^{2 ⋅ 61)} \ & = 2 ⋅ ( 3 ^{2 ⋅ 7} ^{2 ⋅ 11} ^{2 ⋅ 13} ^{2)} ⋅ 22021 = 2 D, \end{align}

where we ignore the fact that 22021 is composite .

A Descartes number is defined as an odd number where and are coprime and, whence is taken as a 'spoof' prime. The example given is the only one currently known.

If is an odd almost perfect number,^[1] that is, and is taken as a 'spoof' prime, then is a Descartes number, since . If were prime, would be an odd perfect number.

Properties

Banks et al. showed in 2008 that if is a cube-free Descartes number not divisible by

3 , then has over one million distinct prime divisors.

Tóth showed in 2021 that if

D = pq denotes a Descartes number (other than Descartes’ example), with pseudo-prime factor p , then q > 10 ^{12} .

Generalizations

John Voight generalized Descartes numbers to allow negative bases. He found the example

3 ^{4} 7 ^{2} 11 ^{2} 19 ^{2} ( -127 ) ^{1} .^[2] Subsequent work by a group at Brigham Young University found more examples similar to Voight's example,^[2] 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.^[3]

See also

• Erdős–Nicolas number, another type of almost-perfect number

References

• Book: Banks . William D. . Güloğlu . Ahmet M. . Nevans . C. Wesley . Saidak . Filip . Descartes numbers . 167–173 . De Koninck . Jean-Marie . Jean-Marie De Koninck . Granville . Andrew . Andrew Granville . Luca . Florian . Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006 . Providence, RI . . CRM Proceedings and Lecture Notes . 46 . 2008 . 978-0-8218-4406-9 . 1186.11004 .
• Book: Klee . Victor . Victor Klee . Wagon . Stan . Stan Wagon . Old and new unsolved problems in plane geometry and number theory . The Dolciani Mathematical Expositions . 11 . Washington, DC . . 1991 . 0-88385-315-9 . 0784.51002 . registration .
• Tóth. László. On the Density of Spoof Odd Perfect Numbers. Comput. Methods Sci. Technol.. 27. 2021. 1 . 2101.09718. .

Notes and References

1. Currently, the only known almost perfect numbers are the non-negative powers of 2, whence the only known odd almost perfect number is
2. News: Nadis . Steve . Mathematicians Open a New Front on an Ancient Number Problem . 3 October 2021 . Quanta Magazine . September 10, 2020.
3. 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 . Odd, spoof perfect factorizations . J. Number Theory . 2020 . 234 . 31–47. 2006.10697 . arXiv version