20050112, 15:33  #1 
Nov 2004
2^{4} Posts 
Fermat's Theorem
We all know fermat's theorem that states: if p is a prime and (p,a)=1 then p divides a^(p1)1. I noticed that p divides also: a^k(p1)1. Is this a well known characteristic?
Second question: what about the infinite sums c, and in particularly when c=1 ? like 11+11+... 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 
20050112, 16:34  #2  
Jun 2003
The Texas Hill Country
2101_{8} Posts 
Quote:
a^(M*N)1 = (a^M1)*(a^(M*(N1)) + a^(M*(N2)) + ... + a^M + 1) 

20050324, 16:41  #3  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Fermat's theorem
Quote:
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 

20050325, 02:38  #4  
∂^{2}ω=0
Sep 2002
República de California
3×3,877 Posts 
Quote:


20050407, 17:28  #5 
Bronze Medalist
Jan 2004
Mumbai,India
2052_{10} Posts 
Fermat's theorem
Crook
[Second question: what about the infinite sums c, and in particularly when c=1 ? like 11+11+./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 = 11+11 ... = 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 
20050505, 17:18  #6 
Bronze Medalist
Jan 2004
Mumbai,India
804_{16} Posts 
Fermat's theorem
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 = (aa) +(aa) ......... = 0 On the other hand we group the terms in a 2nd. way We can write S =a (aa) (aa) (aa)......... a000 =a Again by still another grouping S =a (aa+aa +aa............. 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 Last fiddled with by mfgoode on 20050505 at 17:26 Reason: typo error 
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