Fermat Numbers - Revision history

Admin: /* Properties */

2019-03-16T06:55:51Z

Properties

← Older revision		Revision as of 06:55, 16 March 2019
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	==Properties==		==Properties==

			Fermat Numbers obey the recursion:

			<math>
			F_{n+1} = (F_n - 1)^2 + 1
			</math>

			This leads to the product:

			<math>
			F_{n+1} - 2 = F_0 F_1 \cdots F_n
			</math>

			So <math>F_{n+1} - 2</math> is divisible by all lower Fermat numbers.

			<math>n = 0, 1, 2, 3, 4</math> are each prime

			<math>F_5</math> was factored by Euler in 1732

			<math>F_6</math> was factored by Dandry and Lasseur in 1880

			<math>F_7</math> was factored by Morrison and Brillhart in 1970

			<math>F_8</math> was factored by Brent and Pollard in 1980

			<Math>F_9</math> was factored by Lenstra et al in 1990

			<math>F_{10}</math> was factored by Brent in 1995

			<math>F_{11}</math> was factored by Brent in 1989

			<math>F_{14}</math> is known to be composite but not factored

			<math>F_{20}</math> is known to be composite but not factored

			<math>F_{22}</math> is known to be composite but not factored

			<math>F_{24}</math> is known to be composite but not factored

			All Fermat numbers from 5 - 24 are known to be composite nad are factored or partially factored

			==Open Problems==

			1. Are there infinitely many prime Fermat numbers?

			2. Are there infinitely many composite Fermat numbers?

			3. Is every Fermat number square-free?

	==First 20 Fermat Numbers==		==First 20 Fermat Numbers==

Admin at 06:50, 16 March 2019

2019-03-16T06:50:27Z

← Older revision		Revision as of 06:50, 16 March 2019
Line 2:		Line 2:

	Fermat numbers are numbers of the form <math>2^{2^n} + 1</math>		Fermat numbers are numbers of the form <math>2^{2^n} + 1</math>

			==Properties==

			==First 20 Fermat Numbers==

	Here are the first 20 Fermat numbers:		Here are the first 20 Fermat numbers:

Admin at 06:50, 16 March 2019

2019-03-16T06:50:00Z

← Older revision		Revision as of 06:50, 16 March 2019
Line 2:		Line 2:

	Fermat numbers are numbers of the form <math>2^{2^n} + 1</math>		Fermat numbers are numbers of the form <math>2^{2^n} + 1</math>

			Here are the first 20 Fermat numbers:

	<pre style="white-space: pre-wrap; background-color: #000; color: #fff;">		<pre style="white-space: pre-wrap; background-color: #000; color: #fff;">

Admin at 11:55, 16 July 2017

2017-07-16T11:55:27Z

Show changes

Admin: Created page with "{{ConstantsFlag}} Fermat numbers are numbers of the form

2^{2^n} + 1

 3   5  17  257  6..."

2017-07-16T05:41:48Z

Created page with "{{ConstantsFlag}} Fermat numbers are numbers of the form <math>2^{2^n} + 1</math> <pre style="white-space: pre-wrap; background-color: #000; color: #fff;"> 3 5 17 257 6..."

New page

{{ConstantsFlag}}

Fermat numbers are numbers of the form <math>2^{2^n} + 1</math>

<pre style="white-space: pre-wrap; background-color: #000; color: #fff;">
3

5

17

257

65537

4294967297

18446744073709551616

340282366920938463463374607431768211457

115792089237316195423570985008687907853269984665640564039457584007913129639937

13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097
</pre>