Mirimanoff's congruence

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In number theory, a branch of mathematics, a Mirimanoff's congruence is one of a collection of expressions in modular arithmetic which, if they hold, entail the truth of Fermat's Last Theorem. Since the theorem has now been proven, these are now of mainly historical significance, though the Mirimanoff polynomials are interesting in their own right. The theorem is due to Dimitri Mirimanoff.

[edit] Definition

The nth Mirimanoff polynomial for the prime p is

φ_n(t) = 1^{n − 1}t + 2^{n − 1}t² + ... + (p − 1)^{n − 1}t^{p − 1}.

In terms of these polynomials, if t is one of the six values {-X/Y, -Y/X, -X/Z, -Z/X, -Y/Z, -Z/Y} where X^p+Y^p+Z^p=0 is a solution to Fermat's Last Theorem, then

• φ_p-1(t) ≡ 0 (mod p)
• φ_p-2(t)φ₂(t) ≡ 0 (mod p)
• φ_p-3(t)φ₃(t) ≡ 0 (mod p)

...

• φ_(p+1)/2(t)φ_(p-1)/2(t) ≡ 0 (mod p)

[edit] Other congruences

Mirimanoff also proved the following:

• If an odd prime p does not divide one of the numerators of the Bernoulli numbers B_p-3, B_p-5, B_p-7 or B_p-9, then the first case of Fermat's Last Theorem, where p does not divide X, Y or Z in the equation X^p+Y^p+Z^p=0, holds.

• If the first case of Fermat's Last Theorem fails for the prime p, then 3^p-1 ≡ 1 (mod p²). A prime number with this property is sometimes called a Mirimanoff prime, in analogy to a Wieferich prime which is a prime such that 2^p-1 ≡ 1 (mod p²).

[edit] References

• Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, 1979