Is there a nice cozy formula that will always spit
out primes? Try this one:
f(n) = n^{2} + n + 41.
Euler discovered that this formula has a long string of prime
values: it is prime for all n between 0 and 39 inclusive.
However, it is not prime for all
integers. In fact, it can be shown that no
nonconstant polynomial with integral coefficients will
always spit out primes at the natural numbers.
There are formulas which always spit out primes when you
plug in a natural number... here's one (Mills, 1947):
greatest integer less than (X raised to 3^{n}),
where X is approximately 1.3064... Surprised?
See the remark below!
The Math Behind the Fact:
It is worth pointing out that while the formula above
looks nice, it is useless... it grows too quickly,
and to determine X is tantamount to knowing the primes
in its range!
How to Cite this Page:
Su, Francis E., et al. "Formula for Primes?."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
