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It is conjectured that if {2n-1 \choose n-1} \equiv 1 \pmod{n^k}, when k=3, then n is prime. The conjecture can be understood by considering k = 1 and 2 as well as 3. When k = 1, Babbage's theorem implies that it holds for n = p2 for p an odd prime, while Wolstenholme's theorem implies that it holds for n = p3 for p > 3. When k = 2, it holds for n = p2 if p is a Wolstenholme prime. Only a few other composite values of n are known when k = 1 (sequence A228562 in OEIS), and none are known when k = 2, much less k = 3. Thus the conjecture is considered likely because Wolstenholme's congruence seems over-constrained and artificial for composite numbers. Moreover, if the congruence does hold for any particular n other than a prime or prime power, and any particular k, it does not imply that |
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