In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, a Descartes number is an
odd number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because
\begin
-2 \cdot 2 &= -4 \\
0 \cdot 2 &= 0 \\
41 ...
which would have been an
odd perfect number, if one of its
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
...
factors
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, suc ...
were
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. They are named after
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
who observed that the number would be an odd perfect number if only were a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, since the
sum-of-divisors function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...
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
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
and , whence is taken as a 'spoof' prime. The example given is the only one currently known.
If is an odd
almost perfect number
In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number ''n'' such that the sum of all divisors of ''n'' (the sum-of-divisors function ''σ''(''n'')) is equal to 2''n'' ∠...
,
[Currently, the only known almost perfect numbers are the non-negative ]powers of 2
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negative ...
, 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
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
by
, then has over a million distinct prime divisors.
Generalizations
John Voight generalized Descartes numbers to allow negative bases. He found the example
.
Subsequent work by a group at
Brigham Young University
Brigham Young University (BYU, sometimes referred to colloquially as The Y) is a private research university in Provo, Utah. It was founded in 1875 by religious leader Brigham Young and is sponsored by the Church of Jesus Christ of Latter-day ...
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]
/ref>
See also
*Erdős–Nicolas number
In number theory, an Erdős–Nicolas number is a number that is not perfect, but that equals one of the partial sums of its divisors.
That is, a number is Erdős–Nicolas number when there exists another number such that
: \sum_d=n.
The fir ...
, another type of almost-perfect number
Notes
References
*
*
Divisor function
Integer sequences
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