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January 9th, 2015, 06:11 AM  #1 
Newbie Joined: Jan 2015 From: Athens Posts: 4 Thanks: 0  (a^p)(b^p)(c^p) congruent to 0 (mod p) (with mild restrictive conditions)
Hello there, I recently found out about a very interesting relationship in number theory: Given a = b + c where a, b and c are integers, it is true that: (a^p)(b^p)(c^p) is congruent to 0, modulo p, for any prime p. *Note that this relationship is a general version of Fermat's little theorem as: (a^p)((a1)^p)1 is congruent to 0 modulo p So (a^p)((a2)^p)(1^p)1 is congruent to 0 modulo p (a^p)((a2)^p)2 is congruent to 0 modulo p ... ... And so on until (a^p)a is congruent to 0 modulo p Thereby probing Fermat's little theorem I have uploaded the proof for this, along with two other resulting lemmas on the following blog: Things Mathematical I would appreciate it if you: a) Read the proof and give me some feedback b) Let me know if you have knowledge of this relationship or a generalization of it. 
January 9th, 2015, 07:29 AM  #2 
Senior Member Joined: May 2013 Posts: 118 Thanks: 10 
Dear Stratos At a first glance,it seems to me that all of the three theorems are a direct consequence of Fermat's little theorem 
January 9th, 2015, 10:11 AM  #3 
Newbie Joined: Jan 2015 From: Athens Posts: 4 Thanks: 0  Thanks for the observation
As i think you can see through the proof, the derivation is completely unrelated to Fermat's Little theorem. However after your observation I realized that using Fermat's little theorem as a starting point: (a^p) = a (mod p) So assuming a = b + c (where b and c are both integers) We may say (a^p) = b + c (mod p) So it follows that (a^p) = b + c = (b^p) +(c^p) (mod p) And from that: (a^p)  (b^p)  (c^p) = 0 (mod p) That is an interesting observation, and thank you for pointing that out. It is interesting however, that the derivation that I have given does not involve Fermat's Little Theorem as a given, as it uses binomial expansion and a simple divisibility theorem. 

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apbpcp, conditions, congruence, congruent, fermat's little theorem, mild, mod, number theory, powers, prime, restrictive 
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