Prove Fermat's theorem by expanding via the ... - Cs.ioc.ee

Urmas Repinski
Tallinn University of Technology
Department of Computer Engineering
Prove Fermat's theorem by
expanding via the
multinomial theorem
Concrete Mathematics
Homework 2
2012-10-31

Task definition
Prove Fermat's theorem n p ≡n(mod p) by
expanding (1+1+...+1) p
via the multinomial
theorem.
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Proof
Multinomial Theorem:
(x 1
+x 2
+...+x m
) n = ∑
k 1
+k 2
+...+k m
=n(
n
k 1
, k 2
,... , k m) ∏
1⩽t ⩽m
x t
k t
where
(
n
m) k 1
, k 2
,... , k = n!
k 1
!⋅k 2
!⋅...⋅k m
!
Every sum coefficient is divisible by n except if k j
=n,
1≤j≤m. Therefore
(x 1
+x 2
+...+x n
) p ≡x 1 p +x 2 p +...+x n p (mod p)
Setting all x-s to 1 we get
n p ≡n(mod p)
Proven.
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Multinomial Theorem Informal proof
Multinomial Theorem can be proven by
induction, and
Coefficients are actually number of distinct ways
how to permute a multiset of n elements, and k i
are multiplicities of each of the distinct element.
For example, number of distinct permutations of
letters of the world MISSISSIPPI, which has 1M,
4I-s, 4S-s and 2P-s is:
(
11
1,4, 4, 2) = 11!
1!⋅4!⋅4!⋅2! =34650
That’s why it is practically impossible to solve
permutation tasks using random permutations.
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n p ≡n(mod p)
Thank You
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