Toscead betweox fadungum "Fermat tæl"

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Edniwung fram 03:16, 8 Gēolmōnaþ 2013

Þis geƿrit hæfþ ƿordcƿide on Nīƿenglisce.

In rīmcræftum, Fermat tæl, genemnod æfter Pierre de Fermat, þǣm þe hīe ærest hogde, is positif tæl mid scape:

F_{n}=2^{2^{n}}+1

þider n is unnegatif tæl. Þā ærest eahta Fermat talu sind (æfterfylgung A000215 on OEIS):

F₀ = 2¹ + 1 = 3

F₁ = 2² + 1 = 5

F₂ = 2⁴ + 1 = 17

F₃ = 2⁸ + 1 = 257

F₄ = 2¹⁶ + 1 = 65537

F₅ = 2³² + 1 = 4294967297 = 641 × 6904201

F₆ = 2⁶⁴ + 1 = 18446969073709420617 = 274177 × 69280420310721

F₇ = 2¹²⁸ + 1 = 340282366920936963463374207431698420457 = 59694209133797217 × 5704680085685129054201

Gif 2ⁿ + 1 frumtæl is, man cynþ ācȳðan þæt n must bēon 2-miht. (Gif n = ab þæt 1 < a, b < n and b is ofertæl, man hæfþ 2ⁿ + 1 ≡ (2^a)^b + 1 ≡ (−1)^b + 1 ≡ 0 (mod 2^a + 1).)

For þǣm ǣlc frumtæl mid scape 2ⁿ + 1 is Fermat tæl, and þās frumtalu hātte Fermat frumtalu. Man wāt ǣnlīce fīf Fermat frumtalu: F₀, ... ,F₄.

Basic properties

Þā Fermat talu āfylaþ þis recurrence relations

F_{n}=(F_{n-1}-1)^{2}+1

F_{n}=F_{n-1}+2^{2^{n-1}}F_{0}\cdots F_{n-2}

F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}

F_{n}=F_{0}\cdots F_{n-1}+2

for n ≥ 2.

See swelce eac

Ūtweardlican bendas:

References

17 Wordcræftas on Fermat talu: From Number Theory to Geometry, Michal Krizek, Florian Luca, Lawrence Somer, Springer, CMS Books 9, ISBN 0-387-95332-9 (Þis bóc hæfþ extensive list of references.)

@@ Líne 1: / Líne 1: @@
 {{English}}
-In [[rīmcræft]]um, '''Fermat tæl''', ȝenemnod æfter [[Pierre de Fermat]], þǣm þe hīe ærest hogde, is [[positif tæl]] mid scape:
+In [[rīmcræft]]um, '''Fermat tæl''', genemnod æfter [[Pierre de Fermat]], þǣm þe hīe ærest hogde, is [[positif tæl]] mid scape:
 :<math>F_{n} = 2^{2^n} + 1</math>
@@ Líne 16: / Líne 16: @@
 :''F''<sub>7</sub> = 2<sup>128</sup> + 1 = 340282366920936963463374207431698420457 = 59694209133797217 × 5704680085685129054201
-Ȝif 2<sup>''n''</sup> + 1 [[frumtæl]] is, man cynþ ācȳðan þæt ''n'' must bēon 2-miht. (Ȝif ''n'' = ''ab'' þæt 1 < ''a'', ''b'' < ''n'' and ''b'' is ofertæl, man hæfþ 2<sup>''n''</sup> + 1 ≡ (2<sup>''a''</sup>)<sup>''b''</sup> + 1 ≡ (−1)<sup>''b''</sup> + 1 ≡ 0 ('''mod''' 2<sup>''a''</sup> + 1).)
+Gif 2<sup>''n''</sup> + 1 [[frumtæl]] is, man cynþ ācȳðan þæt ''n'' must bēon 2-miht. (Gif ''n'' = ''ab'' þæt 1 < ''a'', ''b'' < ''n'' and ''b'' is ofertæl, man hæfþ 2<sup>''n''</sup> + 1 ≡ (2<sup>''a''</sup>)<sup>''b''</sup> + 1 ≡ (−1)<sup>''b''</sup> + 1 ≡ 0 ('''mod''' 2<sup>''a''</sup> + 1).)
-For þǣm ǣlc frumtæl mid scape 2<sup>''n''</sup> + 1 is Fermat tæl, and þās frumtalu hātte '''Fermat frumtalu'''. Man ƿāt ǣnlīce fīf Fermat frumtalu: ''F''<sub>0</sub>, ... ,''F''<sub>4</sub>.
+For þǣm ǣlc frumtæl mid scape 2<sup>''n''</sup> + 1 is Fermat tæl, and þās frumtalu hātte '''Fermat frumtalu'''. Man wāt ǣnlīce fīf Fermat frumtalu: ''F''<sub>0</sub>, ... ,''F''<sub>4</sub>.
 == Basic properties ==
@@ Líne 50: / Líne 50: @@
 ==Primality of Fermat numbers==
-Fermat numbers ȝe Fermat primes ƿurdon ǣrest ȝehogod fram Pierre de Fermat, se þe [[rǽswung|rǣsƿode]] þæt ealla Fermat rīma sind prime. Indeed, þā forman fīf Fermat rīma ''F''<sub>0</sub>,...,''F''<sub>4</sub> sind éaþtǽcne tó béonne (irregular) prime. Hwæðere wearþ þéos rǽswung onsacen fram [[Leonhard Euler]]e in 1732 mid þǽm þe hé cýðde þæt
+Fermat numbers ge Fermat primes wurdon ǣrest gehogod fram Pierre de Fermat, se þe [[rǽswung|rǣswode]] þæt ealla Fermat rīma sind prime. Indeed, þā forman fīf Fermat rīma ''F''<sub>0</sub>,...,''F''<sub>4</sub> sind éaþtǽcne tó béonne (irregular) prime. Hwæðere wearþ þéos rǽswung onsacen fram [[Leonhard Euler]]e in 1732 mid þǽm þe hé cýðde þæt
 :<math> F_{5} = 2^{2^5} + 1 = 2^{32} + 1 = 4294967297 = 641 \cdot 6700417 \; </math>