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Number Buffet
de

Die ersten 33 Fermat-Zahlen

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097, 179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137217, 32317006071311007300714876688669951960444102669715484032130345427524655138867890893197201411522913463688717960921898019494119559150490921095088152386448283120630877367300996091750197750389652106796057638384067568276792218642619756161838094338476170470581645852036305042887575891541065808607552399123930385521914333389668342420684974786564569494856176035326322058077805659331026192708460314150258592864177116725943603718461857357598351152301645904403697613233287231227125684710820209725157101726931323469678542580656697935045997268352998638215525166389437335543602135433229604645318478604952148193555853611059596230657, 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2^32768 + 1 (9865 digits), 2^65536 + 1 (19729 digits), 2^131072 + 1 (39457 digits), 2^262144 + 1 (78914 digits), 2^524288 + 1 (157827 digits), 2^1048576 + 1 (315653 digits), 2^2097152 + 1 (631306 digits), 2^4194304 + 1 (1262612 digits), 2^8388608 + 1 (2525223 digits), 2^16777216 + 1 (5050446 digits), 2^33554432 + 1 (10100891 digits), 2^67108864 + 1 (20201782 digits), 2^134217728 + 1 (40403563 digits), 2^268435456 + 1 (80807125 digits), 2^536870912 + 1 (161614249 digits), 2^1073741824 + 1 (323228497 digits), 2^2147483648 + 1 (646456994 digits), 2^4294967296 + 1 (1292913987 digits)

18 of these are shown in power form rather than written out: F(14) already has 4,933 digits and each step after it squares the value. Only F(0) through F(4) — 3, 5, 17, 257 and 65537 — are prime, and no larger Fermat prime has ever been found.

Einstellungen

Schnellvorlagen

F(0) through F(32) — 33 numbers, after which primality is unsettled.

F(0) = 3. Set 5 to begin at Euler's counterexample.

Full decimal form switches to power notation once a value is too long to print.

Group digits as 4,294,967,297.

Aussehen feinjustieren

Wähle zuerst eine Vorlage neben dem Bild — diese Regler passen sie an.

Frame

A border drawn inside the edge of the image.

Erweitert

Ergebnisse

33 Werte

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097, 179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137217, 32317006071311007300714876688669951960444102669715484032130345427524655138867890893197201411522913463688717960921898019494119559150490921095088152386448283120630877367300996091750197750389652106796057638384067568276792218642619756161838094338476170470581645852036305042887575891541065808607552399123930385521914333389668342420684974786564569494856176035326322058077805659331026192708460314150258592864177116725943603718461857357598351152301645904403697613233287231227125684710820209725157101726931323469678542580656697935045997268352998638215525166389437335543602135433229604645318478604952148193555853611059596230657, 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2^32768 + 1 (9865 digits), 2^65536 + 1 (19729 digits), 2^131072 + 1 (39457 digits), 2^262144 + 1 (78914 digits), 2^524288 + 1 (157827 digits), 2^1048576 + 1 (315653 digits), 2^2097152 + 1 (631306 digits), 2^4194304 + 1 (1262612 digits), 2^8388608 + 1 (2525223 digits), 2^16777216 + 1 (5050446 digits), 2^33554432 + 1 (10100891 digits), 2^67108864 + 1 (20201782 digits), 2^134217728 + 1 (40403563 digits), 2^268435456 + 1 (80807125 digits), 2^536870912 + 1 (161614249 digits), 2^1073741824 + 1 (323228497 digits), 2^2147483648 + 1 (646456994 digits), 2^4294967296 + 1 (1292913987 digits)

18 of these are shown in power form rather than written out: F(14) already has 4,933 digits and each step after it squares the value. Only F(0) through F(4) — 3, 5, 17, 257 and 65537 — are prime, and no larger Fermat prime has ever been found.

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Was sind die ersten 33 Fermat-Zahlen?

Die ersten 33 Fermat-Zahlen sind:

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097, 179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137217, 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2^32768 + 1 (9865 digits), 2^65536 + 1 (19729 digits), 2^131072 + 1 (39457 digits), 2^262144 + 1 (78914 digits), 2^524288 + 1 (157827 digits), 2^1048576 + 1 (315653 digits), 2^2097152 + 1 (631306 digits), 2^4194304 + 1 (1262612 digits), 2^8388608 + 1 (2525223 digits), 2^16777216 + 1 (5050446 digits), 2^33554432 + 1 (10100891 digits), 2^67108864 + 1 (20201782 digits), 2^134217728 + 1 (40403563 digits), 2^268435456 + 1 (80807125 digits), 2^536870912 + 1 (161614249 digits), 2^1073741824 + 1 (323228497 digits), 2^2147483648 + 1 (646456994 digits), 2^4294967296 + 1 (1292913987 digits)

Der ausführliche Hintergrundartikel unten ist noch nicht übersetzt und erscheint auf Englisch.

Über Fermat-Zahlen

Pierre de Fermat was a magistrate in Toulouse who did mathematics in letters rather than in books. In correspondence around 1640 with Bernard Frénicle de Bessy, Marin Mersenne and Blaise Pascal he claimed that every number of the form 2^(2^n) + 1 is prime, and was unusually candid about not being able to prove it — he told Pascal he was convinced of the result but could not establish it. The first five values, 3, 5, 17, 257 and 65537, are indeed prime, which is an uncomfortably persuasive amount of evidence.

Leonhard Euler ended it in 1732, aged twenty-five, by publishing that 4,294,967,297 = 641 × 6,700,417. He did not stumble on 641: he had proved that any factor of F(n) must have a restricted form, which cut the candidates for F(5) to a short list he could test by hand. François Édouard Anatole Lucas sharpened that constraint in 1878, and Théophile Pépin gave a clean primality criterion for Fermat numbers in 1877.

The sequence nonetheless produced one of the great results in classical geometry. On 30 March 1796, at nineteen, Carl Friedrich Gauss found that a regular 17-gon can be constructed with compass and straightedge — the first advance on the Greek constructions in two thousand years, and by his own account the thing that decided him on mathematics over philology. In the Disquisitiones Arithmeticae of 1801 he tied constructibility to the Fermat primes; Pierre Wantzel proved the converse in 1837.

Progress since has been a factoring story. Fortuné Landry split F(6) in 1880 at the age of eighty-two, a factorisation Thomas Clausen appears to have found privately in 1854 without publishing. Michael Morrison and John Brillhart cracked F(7) in 1970 with the continued fraction method, and Richard Brent and John Pollard took F(8) in 1980. Not one new Fermat prime has turned up in nearly three centuries.

Wichtige Eigenschaften

  • F(n) = 2^(2^n) + 1, giving 3, 5, 17, 257, 65537, 4294967297, …
  • Only F(0) through F(4) are known to be prime, and every F(n) from n = 5 to n = 32 has been proved composite.
  • F(5) = 4,294,967,297 = 641 × 6,700,417, the factorisation Euler published in 1732.
  • F(n) = F(0)·F(1)·…·F(n-1) + 2 for every n ≥ 1, and also F(n) = (F(n-1) - 1)² + 1.
  • Any two distinct Fermat numbers are coprime, which gives Goldbach his proof that there are infinitely many primes.
  • Every prime factor of F(n) for n ≥ 2 has the form k·2^(n+2) + 1 — Euler used the weaker version of this to find 641.
  • Pépin's test: for n ≥ 1, F(n) is prime if and only if 3^((F(n)-1)/2) ≡ -1 (mod F(n)).
  • A regular polygon with N sides is constructible with compass and straightedge exactly when N is a power of two times a product of distinct Fermat primes, so 17 and 257 sides are constructible and 7 and 9 are not.
  • F(n) has floor(2^n · log10 2) + 1 decimal digits: F(5) has 10 and F(12) has 1,234.

Andere Längen

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