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Let $k$ be a sufficiently large positive even integer. (I think $k > 14$ should do.) Can one always find a prime $p$ such that $p$ divides the numerator of the $k$-th Bernoulli number $B_k$ and $k \leq p-3$? A quick inspection of the table of Bernoulli numerators suggests that this is indeed the case. Am I missing something obvious that would prove this or is it one of those notorious "easy-to-state-but-hard-to-prove" questions?

Certainly, the lower bound $|B_k| > 2(k/\pi e)^{k}$ tells us that the numerator of $|B_k|$ is way larger than $k$ but this doesn't discount the possibility that the numerator consists only of (powers of) small primes.

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All pertinent results on the divisibility of Bernoulli numbers I have come across in the literature are negative in nature, including what is possibly the only one pertaining to an index of a specific form: that $B_{(p+1)/2}$ is not divisible by $p$ if $p \equiv 3 \bmod{4}$. The idea behind this statement traces back to Augustin Cauchy, “Mémoires sur la théorie des nombres,” Mémoires de l’Académie Royale des Sciences de l’Institut de France 17 (1840): 249–768, at p. 445, and it is clearly implied in A. Friedmann and J. Tamarkine, “Quelques formules concernant la théorie de la fonction [x] et des nombres de Bernoulli,” Journal für die reine und angewandte Mathematik 135 (1909): 146–56, at p. 156 (which however contains a misprint). More modern-style proofs appear in Leonard Carlitz, “The first factor of the class number of a cyclic field,” Canadian Journal of Mathematics 6 (1954): 23–26, at p. 25, and in Wells Johnson, “$p$-adic Proofs of Congruences for the Bernoulli Numbers,” Journal of Number Theory 7 (1975): 251–265, at p. 257, where it is described as “one of the few general results which seems to be known along these lines.” Stephen V. Ullom, “Upper bounds for $p$-divisibility of sets of Bernoulli numbers,” Journal of Number Theory 12 (1980): 197–200, showed that at most one-half of the $B_{2k}$ with $2k \le p-3$ are divisible by $p$, and Ullom’s result is sharpened in Samuel S. Wagstaff, Jr., “$p$-Divisibility of Certain Sets of Bernoulli Numbers,” Mathematics of Computation 34 (1980): 647–649.

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