Descartes number

{{Short description|Number which would have been an odd perfect number if one of its composite factors were prime}}

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 {{math|D {{=}} 3² ⋅ 7² ⋅ 11² ⋅ 13² ⋅ 22021 {{=}} (3⋅1001)² ⋅ (22⋅1001 − 1) {{=}} 198585576189}} would be an odd perfect number if only {{math|22021}} were a prime number, since the sum-of-divisors function for {{math| D }} would satisfy, if 22021 were prime,

: $\begin{align}
\sigma(D)
&= (3^2+3+1)\cdot(7^2+7+1)\cdot(11^2+11+1)\cdot(13^2+13+1)\cdot(22021+1) \\
&= (13)\cdot(3\cdot19)\cdot(7\cdot19)\cdot(3\cdot61)\cdot(22\cdot1001) \\
&= 3^2\cdot7\cdot13\cdot19^2\cdot61\cdot(22\cdot7\cdot11\cdot13) \\ &= 2 \cdot (3^2\cdot7^2\cdot11^2\cdot13^2) \cdot (19^2\cdot61) \\ &= 2 \cdot (3^2\cdot7^2\cdot11^2\cdot13^2) \cdot 22021 = 2D,
\end{align}$

where we ignore the fact that 22021 is composite ({{math|22021 {{=}} 19² ⋅ 61}}).

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

If {{math|m}} is an odd almost perfect number,Currently, the only known almost perfect numbers are the non-negative powers of 2, whence the only known odd almost perfect number is {{math|2⁰ {{=}} 1.}} that is, {{math|σ(m) {{=}} 2m − 1}} and {{math| 2m − 1 }} is taken as a 'spoof' prime, then {{math|n {{=}} m ⋅ (2m − 1)}} is a Descartes number, since {{math|σ(n) {{=}} σ(m ⋅ (2m − 1)) {{=}} σ(m) ⋅ 2m {{=}} (2m − 1) ⋅ 2m {{=}} 2n}}. If {{math|2m − 1}} were prime, {{math|n}} would be an odd perfect number.

Properties

If {{math|n}} is a cube-free Descartes number not divisible by {{math|3}}, then {{math|n}} has over one million distinct prime divisors.{{Citation |last1=Banks |first1=William D. |title=Descartes numbers |date=2008 |work=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006 |pages=167–173 |url=https://zbmath.org/?format=complete&q=an:1186.11004 |access-date=2024-05-13 |place=Providence, RI |publisher=American Mathematical Society (AMS) |language=English |isbn=978-0-8218-4406-9 |last2=Güloğlu |first2=Ahmet M. |last3=Nevans |first3=C. Wesley |last4=Saidak |first4=Filip|zbl=1186.11004 }} If $D=pq$ is a Descartes number other than Descartes' example, with spoof-prime factor $p$ , then $q > 10^{12}$ .{{harvtxt|Tóth|2021}}

Generalizations

John Voight generalized Descartes numbers to allow negative bases. He found the example $3^4 7^2 11^2 19^2 (-127)^1$ .{{cite news |last1=Nadis |first1=Steve |title=Mathematicians Open a New Front on an Ancient Number Problem |url=https://www.quantamagazine.org/mathematicians-open-a-new-front-on-an-ancient-number-problem-20200910/ |access-date=3 October 2021 |work=Quanta Magazine |date=September 10, 2020}} Subsequent work by a group at Brigham Young University found more examples similar to Voight's example, 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.{{cite journal |author1=Andersen, Nickolas|author2= Durham, Spencer|author3= Griffin, Michael J.|author4= Hales, Jonathan|author5= Jenkins, Paul|author6= Keck, Ryan|author7= Ko, Hankun|author8= Molnar, Grant|author9= Moss, Eric|author10= Nielsen, Pace P.|author11= Niendorf, Kyle|author12= Tombs, Vandy|author13= Warnick, Merrill|author14= Wu, Dongsheng |title=Odd, spoof perfect factorizations |journal=J. Number Theory |year=2020 |issue=234 |pages=31–47|arxiv=2006.10697 }} A generalization of Descartes numbers to multiperfect numbers has also been constructed. ({{harvtxt|Tóth|2025}}).

Notes

References

{{cite book | last1=Banks | first1=William D. | last2=Güloğlu | first2=Ahmet M. | last3=Nevans | first3=C. Wesley | last4=Saidak | first4=Filip | chapter=Descartes numbers | pages=167–173 | editor1-last=De Koninck | editor1-first=Jean-Marie | editor1-link=Jean-Marie De Koninck | editor2-last=Granville | editor2-first=Andrew | editor2-link=Andrew Granville | editor3-last=Luca | editor3-first=Florian | title=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006 | location=Providence, RI | publisher=American Mathematical Society | series=CRM Proceedings and Lecture Notes | volume=46 | year=2008 | isbn=978-0-8218-4406-9 | zbl=1186.11004 }}
{{cite book | last1=Klee | first1=Victor | author-link1=Victor Klee | last2=Wagon | first2=Stan | author-link2=Stan Wagon | title=Old and new unsolved problems in plane geometry and number theory | series=The Dolciani Mathematical Expositions | volume=11 | location=Washington, DC | publisher=Mathematical Association of America | year=1991 | isbn=0-88385-315-9 | zbl=0784.51002 | url-access=registration | url=https://archive.org/details/oldnewunsolvedpr0000klee }}
{{cite journal|last=Tóth|first=László|title=On the Density of Spoof Odd Perfect Numbers|journal=Comput. Methods Sci. Technol.|volume=27|year=2021|issue=1 |arxiv=2101.09718|url=https://cmst.eu/wp-content/uploads/files/10.12921_cmst.2021.0000005_TOTH.pdf}}
{{cite journal|last=Tóth|first=László|title=Odd Spoof Multiperfect Numbers|journal=Integers|volume=25|year=2025|issue=A19 |arxiv=2502.16954|url=https://math.colgate.edu/~integers/z19/z19.pdf}}

Category:Divisor function

Category:Integer sequences

Descartes number

Properties

Generalizations

See also

Notes

References