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Fermat numbers
The nth Fermat quantity is outlined by
Pierre Fermat conjectured that the F(n) have been prime for all n, and they’re for n = 0, 1, 2, 3, and 4, however Leonard Euler factored F(5), exhibiting that it isn’t prime.
Tangent numbers
The nth tangent quantity is outlined by the Taylor collection for tangent:
One other approach to put it’s that the exponential producing operate for T(n) is tan(z).
Fermat primes and tangent numbers
Right here’s a exceptional connection between Fermat numbers and tangent numbers, found by Richard McIntosh as an undergraduate [1]:
F(n) is prime if and provided that F(n) doesn’t divide T(F(n) − 2).
That’s, the nth Fermat quantity is prime if and provided that it doesn’t divide the (F(n) − 2)th tangent quantity.
We might duplicate Euler’s evaluation that F(5) shouldn’t be prime by exhibiting that 4294967297 doesn’t divide the 4294967295th tangent quantity. That doesn’t sound very sensible, however it’s attention-grabbing.
Replace: To see simply how impractical the outcome on this put up could be for testing whether or not a Fermat quantity is prime, I discovered an asymptotic estimate of tangent numbers on OEIS, and estimated that the 4294967295th tangent quantity has about 80 billion digits.
[1] Richard McIntosh. A Obligatory and Ample Situation for the Primality of Fermat Numbers. The American Mathematical Month-to-month, Vol. 90, No. 2 (Feb., 1983), pp. 98–99
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