Proofs of Fermat's Little Theorem - Proof Using The Multinomial Expansion

Proof Using The Multinomial Expansion

The proof is a very simple application of the Multinomial formula which is brought here for the sake of simplicity.

$(x_1 + x_2 + \cdots + x_m)^n = \sum_{k_1,k_2,\ldots,k_m} {n \choose k_1, k_2, \ldots, k_m} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m}.$

The summation is taken over all sequences of nonnegative integer indices k₁ through k_m such the sum of all k_i is n.

Thus if we express a as a sum of 1s (ones), we obtain

$a^p = \sum_{k_1,k_2,\ldots,k_a} {p \choose k_1, k_2, \ldots, k_a}$

Clearly, if p is prime, and if k_j not equal to p for any j, we have

and

if k_j equal to p for some j

Since there are exactly a elements such that the theorem follows.

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