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 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,
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, 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, then has over a million distinct prime divisors.
