mersenneforum.org Meaning and format of Phi, GF
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 2007-01-12, 22:48 #1 roger     Oct 2006 22×5×13 Posts Meaning and format of Phi, GF I've been using Proth's program, and unfortunately do not understand the format of both GF(n, m) and Phi(an, m). I know that F29 or whatever number in place of 29 is the nth Fermat, but not how GF(n, m) corresponds to an expression, eg F29 = 22[sup]29[/sup]+1. For Phi, I know that it is an irregular decimal number, like pi, but not how it can be expressed either. I tried looking for definitions of the two, but didn't get too much of the basics. Thanks for your help, and sorry for my ignorance, Roger
2007-01-14, 16:36   #2
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

22×33×19 Posts

Quote:
 Originally Posted by roger For Phi, I know that it is an irregular decimal number, like pi, but not how it can be expressed either. I tried looking for definitions of the two, but didn't get too much of the basics. Roger
The natural golden mean constant is known as Phi.

It can be expressed as (1 + sq.rt. 5) /2 = 1.618033989.....

Mally

 2007-01-15, 03:53 #3 geoff     Mar 2003 New Zealand 48516 Posts I think GF(n,m) = n^(2^m)+1, so GF(2,m) = Fm.
 2007-01-15, 06:06 #4 axn     Jun 2003 153B16 Posts I think Phi here refers to Cylotomic Polynomials
2007-01-15, 17:12   #5
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

1000000001002 Posts

Quote:
 Originally Posted by axn1 I think Phi here refers to Cylotomic Polynomials

Quote:
 Originally Posted by Roger For Phi, I know that it is an irregular decimal number, like pi, but not how it can be expressed either.
Mally

2007-01-17, 23:36   #6
Unregistered

24·33·19 Posts

Quote:
 Originally Posted by axn1 I think Phi here refers to Cylotomic Polynomials
But where does the abbreviation "Phi" come from?

2007-01-18, 12:12   #7
rogue

"Mark"
Apr 2003
Between here and the

2×72×71 Posts

Quote:
 Originally Posted by Unregistered But where does the abbreviation "Phi" come from?
It is not an abbreviation, it is a Greek letter.

 2007-01-18, 12:29 #8 akruppa     "Nancy" Aug 2002 Alexandria 2,467 Posts "cyclotomic" means "circle cutting", because the complex roots of x^n-1 cut the unit circle into regular sections, and the divisors of x^n-1 are the cyclotomic polynomials Φk(x) with k|n. The Greek letter Φ (Phi) looks like a circle with a line cutting it into two pieces, so I suppose that's why this letter was chosen for cyclotomic polynomials. (Specifically, the line going through the two primitive roots i, -i of x^4-1 cuts the unit circle vertically in the complex plane, so that would look just like the Φ) Alex Last fiddled with by akruppa on 2007-01-19 at 12:44 Reason: the *primitive* roots don't cut into *regular* sections
2007-01-20, 15:46   #9
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

1000000001002 Posts

Quote:
 Originally Posted by rogue It is not an abbreviation, it is a Greek letter.

Yes its a letter of the Greek alphabet. It was suggested in the early days of the last century that the Greek letter phi- the initial letter of Phidias's name should be adopted to designate the golden ratio. The ubiquity of phi in mathematics aroused the interest of many math'cians in the Middle ages and during the Renaissance. So first and foremost it denotes the golden ratio = (sq.rt.5 +1)/2 =1.618033989.....though it is also used in other calculations as the one cited above/below.

Mally

2007-01-21, 06:33   #10
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

22·33·19 Posts
Phidias

Quote:
 Originally Posted by mfgoode Yes its a letter of the Greek alphabet. It was suggested in the early days of the last century that the Greek letter phi- the initial letter of Phidias's name should be adopted to designate the golden ratio. Mally

For more on this Greek Character refer to

http://en.wikipedia.org/wiki/Phidias

Mally

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