@prefix psr: . @prefix skos: . @prefix dc: . @prefix xsd: . psr: a skos:ConceptScheme . psr:-D4PWF2PN-5 skos:prefLabel "Descartes number"@en, "nombre de Descartes"@fr ; skos:inScheme psr: ; skos:broader psr:-CVDPQB0Q-M, psr:-FM1M1PDT-5 ; dc:created "2023-07-26"^^xsd:date ; skos:definition """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 \cdot7 2 \cdot11 2 \cdot13 2 \cdot22021 = (3\cdot1001) 2 \cdot (22\cdot1001 - 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,

{\\\\displaystyle {\\egin{aligned}\\\\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\\\\cdot 19)\\\\cdot (7\\\\cdot 19)\\\\cdot (3\\\\cdot 61)\\\\cdot (22\\\\cdot 1001)\\\\\\\\&=3^{2}\\\\cdot 7\\\\cdot 13\\\\cdot 19^{2}\\\\cdot 61\\\\cdot (22\\\\cdot 7\\\\cdot 11\\\\cdot 13)\\\\\\\\&=2\\\\cdot (3^{2}\\\\cdot 7^{2}\\\\cdot 11^{2}\\\\cdot 13^{2})\\\\cdot (19^{2}\\\\cdot 61)\\\\\\\\&=2\\\\cdot (3^{2}\\\\cdot 7^{2}\\\\cdot 11^{2}\\\\cdot 13^{2})\\\\cdot 22021=2D,\\\\end{aligned}}}

where we ignore the fact that 22021 is composite (

22021 = 19 2 \cdot 61

). A Descartes number is defined as an odd number

n = m \cdot p

where

m

and

p

are coprime and

2 n = σ(m) \cdot (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,

σ(m) = 2 m - 1

and

2 m - 1

is taken as a 'spoof' prime, then

n = m \cdot (2 m - 1)

is a Descartes number, since

σ(n) = σ(m \cdot (2 m - 1)) = σ(m) \cdot 2 m = (2 m - 1) \cdot 2 m = 2 n

. If

2 m - 1

were prime,

n

would be an odd perfect number.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Descartes_number)"""@en, """En théorie des nombres, un nombre de Descartes est un nombre impair qui serait un nombre parfait impair, si l'un de ses diviseurs composés était considéré comme premier. Ces nombres portent le nom de René Descartes qui a observé que le nombre

D = 3 272 \cdot11 2 \cdot13 2 \cdot22 021 = (3\cdot1 001) 2 \cdot (22\cdot1001 - 1) = 198 585 576 189

serait un nombre parfait impair si son diviseur

22 021

était premier ; en effet, la somme de ses diviseurs serait égale à son double :

{\\\\displaystyle {\\egin{aligned}\\\\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\\\\cdot 19)\\\\cdot (7\\\\cdot 19)\\\\cdot (3\\\\cdot 61)\\\\cdot (22\\\\cdot 1001)\\\\\\\\&=3^{2}\\\\cdot 7\\\\cdot 13\\\\cdot 19^{2}\\\\cdot 61\\\\cdot (22\\\\cdot 7\\\\cdot 11\\\\cdot 13)\\\\\\\\&=2\\\\cdot (3^{2}\\\\cdot 7^{2}\\\\cdot 11^{2}\\\\cdot 13^{2})\\\\cdot (19^{2}\\\\cdot 61)\\\\\\\\&=2\\\\cdot (3^{2}\\\\cdot 7^{2}\\\\cdot 11^{2}\\\\cdot 13^{2})\\\\cdot 22021=2D,\\\\end{aligned}}}

Mais 22 021 est composé (

22 021 = 19 2 \cdot 61

). Un nombre de Descartes est donc défini comme étant un nombre impair

n = m \cdot p

où

m

p

sont premiers entre eux,

p

non premier et

2 n = σ(m) \cdot (p + 1)

; en effet, si

p

était premier, on aurait

{\\\\displaystyle \\\\sigma (n)=2n}

.
(Wikipedia, L'Encylopédie Libre, https://fr.wikipedia.org/wiki/Nombre_de_Descartes)"""@fr ; dc:modified "2024-10-18"^^xsd:date ; a skos:Concept ; skos:exactMatch , . psr:-CVDPQB0Q-M skos:prefLabel "natural numbers"@en, "entier naturel"@fr ; a skos:Concept ; skos:narrower psr:-D4PWF2PN-5 . psr:-FM1M1PDT-5 skos:prefLabel "suite d'entiers"@fr, "integer sequence"@en ; a skos:Concept ; skos:narrower psr:-D4PWF2PN-5 .