Fermat's Theorem

[Crook]

We all know fermat's theorem that states: if p is a prime and (p,a)=1 then p divides a^(p-1)-1. I noticed that p divides also: a^k(p-1)-1. Is this a well known characteristic?

Second question: what about the infinite sums -c, and in particularly when c=1 ? like 1-1+1-1+...

Last question: does anybody know if new progresses were made in defining the necessary conditions for a function to be defined with a Fourier series?

Greetings

[Wacky]

Originally Posted by Crook

We all know fermat's theorem that states: if p is a prime and (p,a)=1 then p divides a^(p-1)-1. I noticed that p divides also: a^k(p-1)-1. Is this a well known characteristic?

Certainly.

a^(M*N)-1 = (a^M-1)*(a^(M*(N-1)) + a^(M*(N-2)) + ... + a^M + 1)

[mfgoode]

Originally Posted by Crook

We all know fermat's theorem that states: if p is a prime and (p,a)=1 then p divides a^(p-1)-1. I noticed that p divides also: a^k(p-1)-1. Is this a well known characteristic?

Second question: what about the infinite sums -c, and in particularly when c=1 ? like 1-1+1-1+...

Last question: does anybody know if new progresses were made in defining the necessary conditions for a function to be defined with a Fourier series?

Greetings

2nd. Question: take the first term separately and group the next in twos
We get the sum as (1)
Any other grouping will give (0)
Mally

[ewmayer]

Originally Posted by Crook

Last question: does anybody know if new progresses were made in defining the necessary conditions for a function to be defined with a Fourier series?

Not sure I understand you correctly - the classic Fourier series is an example of approximation in a Hilbert space (L^2), and as such, any function defined in L^2 (i.e. with finite L^2 norm on the interval of definition) can be approximated to any desired degree of accuracy via Fourier series, though the convergence properties of the series depend on the smoothness of the function - specifically, for a function that is C^k (continuous derivatives up to kth order) the coefficients (typically written as a_n and b_n, where one set is the coefficients of the cosine terms and the other set is the coefficients of the sine terms) of the series approximation decay proportionally to 1/n^(k+2). Also note (and this is very important and frequently misunderstood) that when we say the FS approximation "converges to the function" we mean CONVERGENCE IN THE SENSE OF THE L^2 NORM, i.e. as n->oo, the norm of the difference between the function and the FS approximation vanishes. That need not not imply pointwise convergence: a classic example is the FS approximation to a step function (which is discontinuous, i.e. only C^0 smooth), where the FS exhibits oscillations around the discontinuity ("Gibbs' phenomenon") whose amplitude does not go to zero as the number of terms in the FS approximation goes to infinity - rather the width of the wiggles gets increasingly smaller, such that their L^2 integral vanishes as n->oo.

[mfgoode]

Crook
[Second question: what about the infinite sums -c, and in particularly when c=1 ? like 1-1+1-1+./UNQUOTE]..

It will be interesting to see a different and surprising answer to the ones Ive given. The answer was given by Guido Grandi a priest and professor of Pisa known for his study of rosaces (r=sin(n*theta) and other curves which resemble flowers. In the 18th century, also known for its mysticism, Guido considered the formula
1/2 = 1-1+1-1 ...
= 0+0+0..
as the symbol for Creation from Nothing.
He obtained the result 1/2 by considering the case of a father who bequeaths a gem to his two sons who each may keep the bauble for one year in alternation. It then belongs to each son for one half!
Source: 'A concise history of mathematics' by Dirk J. Struick
Mally

[mfgoode]

Thank you Crook for investigating the mystery of the infinite series.
As a result I have been able to go further in this investigation. However I must restrict this post by giving just one more rendition.

In the 19th century Bernard Bolzano was the first to treat this problem on a sound and logical manner. Since Zeno's paradoxes had put mathem'cians in a flummux there was a lot of speculation as to how to relate to infinity. Then Bolzano came along and treated the problem on a war footing.

Consider the series S = a -a + a -a +a -a +..............
If we group the terms thus we get
S = (a-a) +(a-a) .........
= 0

On the other hand we group the terms in a 2nd. way
We can write S =a -(a-a) -(a-a) -(a-a).........
a-0-0-0
=a

Again by still another grouping
S =a -(a-a+a-a +a-a.............
S =a- S
Hence 2S=a or S=a/2 (so the learned proffessor/priest of Pisa Guido Grandi mentioned in an earlier post was not so wrong after all)

Today with maths on a firmer footing we can label it as a class of oscillating series between the values of 0 and a

Even more startling are the results obtained from the series in the special case when a = 1
I will reserve this for another post.

For further reading;
'Riddles in maths' by Eugene Northrop 1960
'The Paradoxes of the Infinite' by Bernard Bolzano1851 .
Mally