Descartes number

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

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

where we ignore the fact that 22021 is composite (Template:Math).

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

If Template:Math is an odd almost perfect number, that is, Template:Math and Template:Math is taken as a 'spoof' prime, then Template:Math is a Descartes number, since Template:Math. If Template:Math were prime, Template:Math would be an odd perfect number.

Properties

Banks et al. showed in 2008 that if Template:Math is a cube-free Descartes number not divisible by $3$, then Template:Math 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 $q>10^{12}$ (Template:Harvtxt).

Generalizations

John Voight generalized Descartes numbers to allow negative bases. He found the example $3^4{7}^{2}11^{2}19^2(-127{)}^1$. 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.

Tóth generalized the definition of Descartes numbers to [[Multiply perfect number |multiperfect numbers]] and found several new examples, such as $8999757$, which would be an odd multiperfect number, if only one of its prime factors, $61$, was a square (Template:Harvtxt).

See also

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

References

• Template:Cite book
• Template:Cite book
• Template:Cite journal
• Template:Cite journal

Template:Numtheory-stub