Fermat last theorem : proof of a criterion by Cauchy

Question

In 13 Lectures on Fermat's Last Theorem, Ribenboim states the following theorem (on page 7) attributed to Cauchy:

If the first case of Fermat's theorem fails for the exponent $p$, then the sum: $$ 1^{p-4} + 2^{p-4} + \cdots +\left( \frac{p-1}{2} \right) ^{p-4} $$ is a multiple of $ p $.

Recall The first case holds for the exponent $p$ when there do not exist integers $x, y, z$ such that $p \mid xyz$ and $x^p + y^p = z^p$.

Ribenboim also states that Genocchi proved this theorem (p. 6).

I went through various articles by Cauchy, where the theorem is not clearly proved. Instead, after recalling various properties of cyclotomic integers, Cauchy states that one could prove the criterion.

I also looked for the articles by Genocchi but I could only find a proof that relies on the theorem by Cauchy (summary in French here) to simplify the criterion with a divisibility condition of Bernoulli numbers.

I then went through Fermat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers by Dickson, who seems to be less optimistic about Cauchy's proof, according to him Cauchy only stated the criterion (p. 180).

Are there any reference that derive Cauchy's criterion in details?

Edit

I do not doubt that the criterion is correct (and there is no hope in finding counter examples) but I am particularly interested in proofs of the criterion that would not rely on Kummer theory of ideals.

Either a full proof by Cauchy (but after hours of research, I am losing hope to find any) or a derivation by the same means (I hoped to find one by Genocchi's work but the references provided by Ribenboim did not lead to such proof).

Answer 1

Cauchy's criterion is a special case$^\ast$ of a more general criterion proven by Kummer in 1857 [1], and by Fueter in 1922 [2]. A description of Kummer's derivation and how it implies the Cauchy criterion is in [3] (pages 121-126).

• [1] E.E. Kummer, Einige Sätze über die aus den Wurzeln der Gleichung der Gleichung $\alpha^\lambda=1$ gebildeten complexen Zahlen, für den Fall dass die Klassenanzahl durch $\lambda$ theilbar ist, nebst Anwendung derselben auf einen weiteren Beweis des letzten Fermatschen Lehrsatzes (1857). _{On page 64 Kummer notes Cauchy's criterion as a special case.}
• [2] R. Fueter, Kummers Kriterium zum letzten Theorem von Fermat (1922).
• [3] P. Ribenboim, Kummer exits (1979).

---

$^\ast$ _{Cauchy (1847) and Genocchi (1852) both state (without proof) that, if the first case of Fermat's theorem fails for the exponent $p$, then the Bernoulli number $B_{p-3}$ is a multiple of $p$. Kummer (1857) proves that both $B_{p-3}$ and $B_{p-5}$ must be multiples of $p$. That Kummer gave the first proof is noted by Vandiver: Fermat's Last Theorem: Its History and the Nature of the Known Results Concerning It -- page 562. He speaks of "Kummer's criterion" rather than "Cauchy's criterion".}