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 Euler function, DYK Fact #560
JaneFairfax
Posted: 15:30 Wednesday 01 April 2009


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Did you know? If for each positive integer n we let denote the number of integers less than or equal to n and coprime with n, then is called the Euler function.

We have this interesting little result:

The proof of this given in A Course in Number Theory by H.E. Rose is only a few lines long.

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Athene_noctua
Posted: 16:53 Wednesday 01 April 2009


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Did you know? Leonhard Euler (1707Ė83) was a Swiss mathematician. smile.gif



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JaneFairfax
Posted: 18:03 Wednesday 01 April 2009


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Did you know? If gcd(m,n) = 1, . This result can be proved by abstract algebra.

In particular, if denotes the multiplicative group of congruence classes modulo k of integers coprime with k, then if gcd(m,n) = 1, is isomorphic to . In fact, and are isomorphic as rings, and it can be proved from this that the multiplicative group of units modulo mn is isomorphic to the product of the multiplicative groups of units modulo m and modulo n.

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Athene_noctua
Posted: 19:01 Wednesday 01 April 2009


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Did you know? Euler became blind in the latter part of his life. sad.gif



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JaneFairfax
Posted: 19:37 Wednesday 01 April 2009


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Did you know? If gcd(a,n) = 1, .

Proof: Let be integers such that gcd(ci, n) = 1 and if ij. Then, since gcd(a,n) = 1, are also integers such that gcd(aci, n) = 1 and if ij. Hence Ė and the result follows as .

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algebraic topology
Posted: 14:04 Thursday 02 April 2009


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Did you know?

If p is a prime, φ(p) = p−1 and this becomes Fermatís little theorem: ap−11 (mod p) provided a is not a multiple of p.

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JaneFairfax
Posted: 14:45 Thursday 02 April 2009


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Did you know? In the language of Galois theory, Fermatís little theorem says that if p is prime, then is a splitting field for the polynomial in .

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algebraic topology
Posted: 20:26 Thursday 02 April 2009


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Did you know?

That means

Letting x = 0 gives . If p > 2, we have ; if p = 2, the same equation is true as . This proves Wilsonís theorem.

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JaneFairfax
Posted: 05:50 Friday 03 April 2009


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Did you know? The converse of Wilsonís theorem is also true: if then p is a prime.

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