20070112, 22:48  #1 
Oct 2006
2^{2}×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(a^{n}, 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 = 2^{2[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 
20070114, 16:36  #2  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Quote:
It can be expressed as (1 + sq.rt. 5) /2 = 1.618033989..... Mally 

20070115, 03:53  #3 
Mar 2003
New Zealand
2205_{8} Posts 
I think GF(n,m) = n^(2^m)+1, so GF(2,m) = Fm.

20070115, 06:06  #4 
Jun 2003
2×3^{2}×293 Posts 
I think Phi here refers to Cylotomic Polynomials

20070115, 17:12  #5  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Quote:
Quote:


20070117, 23:36  #6  
2^{2}·3^{2}·7^{2} Posts 
Quote:


20070118, 12:12  #7 
"Mark"
Apr 2003
Between here and the
5×1,307 Posts 

20070118, 12:29  #8 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
"cyclotomic" means "circle cutting", because the complex roots of x^n1 cut the unit circle into regular sections, and the divisors of x^n1 are the cyclotomic polynomials Φ_{k}(x) with kn. 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^41 cuts the unit circle vertically in the complex plane, so that would look just like the Φ)
Alex Last fiddled with by akruppa on 20070119 at 12:44 Reason: the *primitive* roots don't cut into *regular* sections 
20070120, 15:46  #9 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
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 
20070121, 06:33  #10  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Phidias
Quote:
For more on this Greek Character refer to http://en.wikipedia.org/wiki/Phidias Mally 

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