A new criterion for the first case of Fermat’s last theorem. (English) Zbl 0817.11022

For an odd prime \(l\), consider the congruence system \((*)\) \(B_{2j} \varphi_{l- 2j} (t)\equiv 0\pmod l\) \((j=1,\dots, (l-3) /2)\), where \(B_{2j}\) are Bernoulli numbers and \(\varphi_ i (t)\) are Mirimanoff polynomials. A classical result by Kummer shows that if the Fermat equation \(x^ l+ y^ l+ z^ l =0\) has a solution in Case I (i.e. \(l\nmid xyz\)), then \(-x/y\bmod l\) is a solution of \((*)\). The authors prove that if \(\tau\) and \(1-\tau\) are solutions \(\not\equiv 0, \pm1 \pmod l\) of \((*)\) and their orders \(\text{mod } l\) are \(>16\), then the sums \(\sum 1/j\) with \(kl/N< j<(k+1) l/N\) vanish \(\text{mod }l\) for \(k=0, \dots, N-1\) whenever \(1\leq N\leq 46\).
An alternative formulation, in fact the authors’ main theorem, states that the vanishing \(\text{mod } l\) of all these sums would follow from the solvability of Fermat’s equation in Case I. The proof is based on a previous result by the second author [Comment. Math. Univ. St. Pauli 41, No. 1, 35-54 (1992; Zbl 0753.11016)] and requires a lot of sophisticated computation. Earlier, P. Cikánek [Math. Comput. 62, 923-930 (1994; Zbl 0805.11027)] had shown by a more straightforward computation that the same results are true up to \(N\leq 94\) if \(l\) is large enough.


11D41 Higher degree equations; Fermat’s equation
11Y40 Algebraic number theory computations
11B68 Bernoulli and Euler numbers and polynomials
11Y16 Number-theoretic algorithms; complexity
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