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,

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

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,Currently, the only known almost perfect numbers are the non-negative powers of 2, whence the only known odd almost perfect number is 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 $3$, then has over a million distinct prime divisors.

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.arXiv version
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See also

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

Notes

References

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Divisor function
Integer sequences

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