Alternative Test for Primes. - Page 2

alpertron · 2006-02-28, 11:57

These numbers are composite: in order to show that, I can give you a prime factor for each of them.

A prime factor of 2^{2^8} + 1 = 604944512477 * 2¹¹ + 1

A prime factor of 2^{2^16} + 1 = 1575 * 2¹⁹ + 1

A prime factor of 2^{2^32} + 1 = 1479 * 2³⁴ + 1

A prime factor of 2^{2^64} + 1 = 17853639 * 2⁶⁷ + 1

There is no known prime factor of 2^{2^128} + 1.

A prime factor of 2^{2^256} + 1 = 36986355 * 2²⁵⁸ + 1

See more factorizations of Fermat numbers on: http://www.prothsearch.net/fermat.html

yucell · 2006-02-28, 19:32

but which ı sent k can not be 128

k can be 2,4,16,256,65536.....

2^2=4
4^2=16
16^2=256
256^2=65536
65536^2=x
x^2=y
y^2=z
.........
it has own rule

cheesehead · 2006-02-28, 21:01

yucell,

alpertron was referring to the form 2^{2[sup]2[sup]j}[/sup][/sup]+1, where your 2^k = alpertron's 2^{2[sup]2[sup]j}[/sup][/sup].

Your list k = 2, 4, 16, 256, 65536, 4294967296, ... is the same as k = 2¹, 2², 2⁴, 2⁸, 2¹⁶, 2³², ... where each term is the square of the preceding term.

So your (2^k)+1 is the same series as (2^(2^(2^j)))+1 for j = 0, 1, 2, 3, 4, 5 ...

It turns out that Pierre de Fermat, one of the greatest mathematicians of all time, studied your series of numbers over 350 years ago. He also conjectured that they were all prime, but he didn't have the ability, given by modern calculators/computers, to determine whether that was true for k > 256 ( j > 3 ).

(Actually, Fermat studied 2^2[sup]i[/sup]+1 for i = 0, 1, 2, 3, 4, ... which includes your series as a subset.)

R.D. Silverman · 2006-02-28, 21:19

Quote:

Originally Posted by cheesehead

Pierre de Fermat, one of the greatest mathematicians of all time,

Not even close.

philmoore · 2006-02-28, 22:13

Quote:

Originally Posted by R.D. Silverman

Not even close.

OK, how about one of the greatest number theoreticians of all time then?

Fermat did develop analytic geometry independently of Descartes, but his flaw was that he lacked the ability to communicate his ideas in a way that invited others to participate in his mathematics. Certainly a brilliant and original thinker, though.

akruppa · 2006-02-28, 22:15

If only he had done more with his discoveries than use them to taunt and tease his contemporaries...

Alex

R.D. Silverman · 2006-02-28, 23:44

[QUOTE=philmoore]OK, how about one of the greatest number theoreticians of all time then?

QUOTE]

Not even close.

ewmayer · 2006-03-01, 00:05

OK, how about "at least as deserving of his celebrity as American skier Bode Miller"?

Peter Nelson · 2006-03-01, 03:34

Quote:

Originally Posted by ewmayer

OK, how about "at least as deserving of his celebrity as American skier Bode Miller"?

At least I know who Fermat was!

I could not name any skiers from any nation nor would I recognise them if I heard their name.

So at least for me, Fermat is streets ahead, and he's mentioned in several books I've read.

philmoore · 2006-03-01, 05:24

Quote:

Originally Posted by R.D. Silverman

Quote:

Originally Posted by philmoore

OK, how about one of the greatest number theoreticians of all time then?

Not even close.

OK, not a Gauss or Dirichlet, but certainly a highly original thinker. I don't think that it is without reason that Fermat has been variously called "the father of number theory" or "the father of modern number theory".

wblipp · 2006-03-01, 05:58

Harold M. Edward's Springer-Verlag book Fermat's Last Theorem, a Genetic Introduction to Algebraic Number Theory begins

"When Pierre de Fermat died in 1655 he was one of the most famous mathematicians in Europe."

The second paragraph begins

"There are two surprising facts about Fermat's fame as a mathematician. The first is that he was not a mathematician at all, but a jurist. Throughout his mature life he held rather important judicial positions in Toulouse, and his mathematical work was done as an avocation. The second is that he never published a single^* mathematical work."

So how about "Fermat was one of the most famous mathematician of the 17th century" or perhaps "Fermat was one the greatest mathematical hobbiests of all time?"

2006-02-28, 21:01	#14
cheesehead "Richard B. Woods" Aug 2002 Wisconsin USA 1111000001100₂ Posts	yucell, alpertron was referring to the form 2^{2[sup]2[sup]j}[/sup][/sup]+1, where your 2^k = alpertron's 2^{2[sup]2[sup]j}[/sup][/sup]. Your list k = 2, 4, 16, 256, 65536, 4294967296, ... is the same as k = 2¹, 2², 2⁴, 2⁸, 2¹⁶, 2³², ... where each term is the square of the preceding term. So your (2^k)+1 is the same series as (2^(2^(2^j)))+1 for j = 0, 1, 2, 3, 4, 5 ... It turns out that Pierre de Fermat, one of the greatest mathematicians of all time, studied your series of numbers over 350 years ago. He also conjectured that they were all prime, but he didn't have the ability, given by modern calculators/computers, to determine whether that was true for k > 256 ( j > 3 ). (Actually, Fermat studied 2^2[sup]i[/sup]+1 for i = 0, 1, 2, 3, 4, ... which includes your series as a subset.) Last fiddled with by cheesehead on 2006-02-28 at 21:16

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2006-02-28, 11:57	#12
alpertron Aug 2002 Buenos Aires, Argentina 2·683 Posts	These numbers are composite: in order to show that, I can give you a prime factor for each of them. A prime factor of 2^{2^8} + 1 = 604944512477 * 2¹¹ + 1 A prime factor of 2^{2^16} + 1 = 1575 * 2¹⁹ + 1 A prime factor of 2^{2^32} + 1 = 1479 * 2³⁴ + 1 A prime factor of 2^{2^64} + 1 = 17853639 * 2⁶⁷ + 1 There is no known prime factor of 2^{2^128} + 1. A prime factor of 2^{2^256} + 1 = 36986355 * 2²⁵⁸ + 1 See more factorizations of Fermat numbers on: http://www.prothsearch.net/fermat.html

2006-02-28, 19:32	#13
yucell Feb 2006 İSTANBUL 5 Posts	but which ı sent k can not be 128 k can be 2,4,16,256,65536..... 2^2=4 4^2=16 16^2=256 256^2=65536 65536^2=x x^2=y y^2=z ......... it has own rule

2006-02-28, 22:15	#17
akruppa "Nancy" Aug 2002 Alexandria 2467₁₀ Posts	If only he had done more with his discoveries than use them to taunt and tease his contemporaries... Alex

2006-02-28, 23:44	#18
R.D. Silverman Nov 2003 2²×5×373 Posts	[QUOTE=philmoore]OK, how about one of the greatest number theoreticians of all time then? QUOTE] Not even close.

2006-03-01, 00:05	#19
ewmayer ∂²ω=0 Sep 2002 República de California 103×113 Posts	OK, how about "at least as deserving of his celebrity as American skier Bode Miller"?

2006-03-01, 05:58	#22
wblipp "William" May 2003 New Haven 2×7×13² Posts	Harold M. Edward's Springer-Verlag book Fermat's Last Theorem, a Genetic Introduction to Algebraic Number Theory begins "When Pierre de Fermat died in 1655 he was one of the most famous mathematicians in Europe." The second paragraph begins "There are two surprising facts about Fermat's fame as a mathematician. The first is that he was not a mathematician at all, but a jurist. Throughout his mature life he held rather important judicial positions in Toulouse, and his mathematical work was done as an avocation. The second is that he never published a single^* mathematical work." So how about "Fermat was one of the most famous mathematician of the 17th century" or perhaps "Fermat was one the greatest mathematical hobbiests of all time?"