They are named after René Descartes who observed that the number D = 32⋅72⋅112⋅132⋅22021 = (3⋅1001)2 ⋅ (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, where we ignore the fact that 22021 is composite (22021 = 192 ⋅ 61).
A Descartes number is defined as an odd number n = m ⋅ p where m and p are coprime and 2n = σ(m) ⋅ (p + 1), whence p is taken as a 'spoof' prime.
If 2m − 1 were prime, n would be an odd perfect number.
Banks et al. showed in 2008 that if n is a cube-free Descartes number not divisible by
, then n has over one million distinct prime divisors.
John Voight generalized Descartes numbers to allow negative bases.
[3] Subsequent work by a group at Brigham Young University found more examples similar to Voight's example,[3] 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.
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