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 $D = 3^{2}⋅7^{2}⋅11^{2}⋅13^{2}⋅22021 = (3⋅1001)^{2}&thinsp;⋅&thinsp;(22⋅1001 − 1) = 198585576189$ would be an odd perfect number if only $22021$ were a prime number, since the sum-of-divisors function for $D$ 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 ($22021 = 19^{2}&thinsp;⋅&thinsp;61$).

A Descartes number is defined as an odd number $n = m&thinsp;⋅&thinsp;p$ where $m$ and $p$ are coprime and $2n = σ(m)&thinsp;⋅&thinsp;(p + 1)$, 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, that is, $2^{0} = 1.$ and $σ(m) = 2m − 1$ is taken as a 'spoof' prime, then $2m − 1$ is a Descartes number, since $n = m&thinsp;⋅&thinsp;(2m − 1)$. If $σ(n) = σ(m&thinsp;⋅&thinsp;(2m − 1)) = σ(m)&thinsp;⋅&thinsp;2m = (2m − 1)&thinsp;⋅&thinsp;2m = 2n$ were prime, $2m − 1$ would be an odd perfect number.

Properties
Banks et al. showed in 2008 that if $n$ is a cube-free Descartes number not divisible by $$3$$, then $n$ has over a 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 $$3^4 7^2 11^2 19^2 (-127)^1 $$. 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.