Meaning and format of Phi, GF

[roger]

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 = 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

[mfgoode]

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

[geoff]

I think GF(n,m) = n^(2^m)+1, so GF(2,m) = Fm.

[axn]

I think Phi here refers to Cylotomic Polynomials

[mfgoode]

Originally Posted by axn1

I think Phi here refers to Cylotomic Polynomials

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

Originally Posted by axn1

I think Phi here refers to Cylotomic Polynomials

But where does the abbreviation "Phi" come from?

[rogue]

Originally Posted by Unregistered

But where does the abbreviation "Phi" come from?

It is not an abbreviation, it is a Greek letter.

[akruppa]

"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 Φ_k(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

[mfgoode]

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

[mfgoode]

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