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Conjectured least prime p for which binomial(n*q,q) (mod q^3) = n for all primes q >= p.
1

%I #8 Nov 02 2024 03:21:41

%S 2,5,3,2,5,3,3,5,3,3,5,3,3,5,3,3,5,3,3,5,3,3,5,3,3,5,5,5,5,5,5,5,5,5,

%T 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,

%U 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5

%N Conjectured least prime p for which binomial(n*q,q) (mod q^3) = n for all primes q >= p.

%C The n = 2 case is mentioned in Eric Weisstein's website.

%H T. D. Noe, <a href="/A222760/b222760.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/CentralBinomialCoefficient.html">MathWorld: Central Binomial Coefficient</a>

%t lim = 100; Table[r = Table[Mod[Binomial[n*p, p], p^3] == n, {p, Prime[Range[lim]]}]; i = lim; While[i > 0 && r[[i]], i--]; Prime[i + 1], {n, 87}]

%Y Cf. A096328 (prime(prime(n)^3)), A222759.

%K nonn

%O 1,1

%A _T. D. Noe_, Mar 13 2013